Topological centres for group algebras,actions, and quantum groups
Matthias Neufang
Carleton University (Ottawa)
Topological centre basics Topological centre problems Topological centres as a tool
1 Topological centre basics
2 Topological centre problems
3 Topological centres as a tool
Topological centre basics Topological centre problems Topological centres as a tool
1 Topological centre basics
2 Topological centre problems
3 Topological centres as a tool
Topological centre basics Topological centre problems Topological centres as a tool
Arens products: Algebraic description
A Banach algebra; as Banach space: A ↪→ A∗∗
∃ 2 canonical extensions of product to A∗∗ (Arens ’51)
X ,Y ∈ A∗∗, f ∈ A∗, a, b ∈ A
〈X2Y , f 〉 = 〈X ,Y 2f 〉〈Y 2f , a〉 = 〈Y , f 2a〉〈f 2a, b〉 = 〈f , a · b〉
. . . and the other way around:
〈X3Y , f 〉 = 〈Y , f 3X 〉〈f 3X , a〉 = 〈X , a3f 〉〈a3f , b〉 = 〈f , b · a〉
Topological centre basics Topological centre problems Topological centres as a tool
Arens products: Algebraic description
A Banach algebra; as Banach space: A ↪→ A∗∗
∃ 2 canonical extensions of product to A∗∗ (Arens ’51)
X ,Y ∈ A∗∗, f ∈ A∗, a, b ∈ A
〈X2Y , f 〉 = 〈X ,Y 2f 〉〈Y 2f , a〉 = 〈Y , f 2a〉〈f 2a, b〉 = 〈f , a · b〉
. . . and the other way around:
〈X3Y , f 〉 = 〈Y , f 3X 〉〈f 3X , a〉 = 〈X , a3f 〉〈a3f , b〉 = 〈f , b · a〉
Topological centre basics Topological centre problems Topological centres as a tool
Arens products: Algebraic description
A Banach algebra; as Banach space: A ↪→ A∗∗
∃ 2 canonical extensions of product to A∗∗ (Arens ’51)
X ,Y ∈ A∗∗, f ∈ A∗, a, b ∈ A
〈X2Y , f 〉 = 〈X ,Y 2f 〉〈Y 2f , a〉 = 〈Y , f 2a〉〈f 2a, b〉 = 〈f , a · b〉
. . . and the other way around:
〈X3Y , f 〉 = 〈Y , f 3X 〉〈f 3X , a〉 = 〈X , a3f 〉〈a3f , b〉 = 〈f , b · a〉
Topological centre basics Topological centre problems Topological centres as a tool
Arens products: Algebraic description
A Banach algebra; as Banach space: A ↪→ A∗∗
∃ 2 canonical extensions of product to A∗∗ (Arens ’51)
X ,Y ∈ A∗∗, f ∈ A∗, a, b ∈ A
〈X2Y , f 〉 = 〈X ,Y 2f 〉
〈Y 2f , a〉 = 〈Y , f 2a〉〈f 2a, b〉 = 〈f , a · b〉
. . . and the other way around:
〈X3Y , f 〉 = 〈Y , f 3X 〉〈f 3X , a〉 = 〈X , a3f 〉〈a3f , b〉 = 〈f , b · a〉
Topological centre basics Topological centre problems Topological centres as a tool
Arens products: Algebraic description
A Banach algebra; as Banach space: A ↪→ A∗∗
∃ 2 canonical extensions of product to A∗∗ (Arens ’51)
X ,Y ∈ A∗∗, f ∈ A∗, a, b ∈ A
〈X2Y , f 〉 = 〈X ,Y 2f 〉〈Y 2f , a〉 = 〈Y , f 2a〉
〈f 2a, b〉 = 〈f , a · b〉
. . . and the other way around:
〈X3Y , f 〉 = 〈Y , f 3X 〉〈f 3X , a〉 = 〈X , a3f 〉〈a3f , b〉 = 〈f , b · a〉
Topological centre basics Topological centre problems Topological centres as a tool
Arens products: Algebraic description
A Banach algebra; as Banach space: A ↪→ A∗∗
∃ 2 canonical extensions of product to A∗∗ (Arens ’51)
X ,Y ∈ A∗∗, f ∈ A∗, a, b ∈ A
〈X2Y , f 〉 = 〈X ,Y 2f 〉〈Y 2f , a〉 = 〈Y , f 2a〉〈f 2a, b〉 = 〈f , a · b〉
. . . and the other way around:
〈X3Y , f 〉 = 〈Y , f 3X 〉〈f 3X , a〉 = 〈X , a3f 〉〈a3f , b〉 = 〈f , b · a〉
Topological centre basics Topological centre problems Topological centres as a tool
Arens products: Algebraic description
A Banach algebra; as Banach space: A ↪→ A∗∗
∃ 2 canonical extensions of product to A∗∗ (Arens ’51)
X ,Y ∈ A∗∗, f ∈ A∗, a, b ∈ A
〈X2Y , f 〉 = 〈X ,Y 2f 〉〈Y 2f , a〉 = 〈Y , f 2a〉〈f 2a, b〉 = 〈f , a · b〉
. . . and the other way around:
〈X3Y , f 〉 = 〈Y , f 3X 〉〈f 3X , a〉 = 〈X , a3f 〉〈a3f , b〉 = 〈f , b · a〉
Topological centre basics Topological centre problems Topological centres as a tool
Arens products: Topological description
A 3 xi −→ X ∈ A∗∗ (w∗)
A 3 yj −→ Y ∈ A∗∗ (w∗)
X2Y = limi limj xi · yj
X3Y = limj limi xi · yj
2 = 3 ⇔: A Arens regular (e.g., operator algebras)
But for algebras closest to the heart of harmonic analysts:
X2Y 6= X3Y
; How to measure the degree of non-regularity?
Topological centre basics Topological centre problems Topological centres as a tool
Arens products: Topological description
A 3 xi −→ X ∈ A∗∗ (w∗)
A 3 yj −→ Y ∈ A∗∗ (w∗)
X2Y = limi limj xi · yj
X3Y = limj limi xi · yj
2 = 3 ⇔: A Arens regular (e.g., operator algebras)
But for algebras closest to the heart of harmonic analysts:
X2Y 6= X3Y
; How to measure the degree of non-regularity?
Topological centre basics Topological centre problems Topological centres as a tool
Arens products: Topological description
A 3 xi −→ X ∈ A∗∗ (w∗)
A 3 yj −→ Y ∈ A∗∗ (w∗)
X2Y = limi limj xi · yj
X3Y = limj limi xi · yj
2 = 3 ⇔: A Arens regular (e.g., operator algebras)
But for algebras closest to the heart of harmonic analysts:
X2Y 6= X3Y
; How to measure the degree of non-regularity?
Topological centre basics Topological centre problems Topological centres as a tool
Arens products: Topological description
A 3 xi −→ X ∈ A∗∗ (w∗)
A 3 yj −→ Y ∈ A∗∗ (w∗)
X2Y = limi limj xi · yj
X3Y = limj limi xi · yj
2 = 3 ⇔: A Arens regular (e.g., operator algebras)
But for algebras closest to the heart of harmonic analysts:
X2Y 6= X3Y
; How to measure the degree of non-regularity?
Topological centre basics Topological centre problems Topological centres as a tool
Arens products: Topological description
A 3 xi −→ X ∈ A∗∗ (w∗)
A 3 yj −→ Y ∈ A∗∗ (w∗)
X2Y = limi limj xi · yj
X3Y = limj limi xi · yj
2 = 3 ⇔: A Arens regular (e.g., operator algebras)
But for algebras closest to the heart of harmonic analysts:
X2Y 6= X3Y
; How to measure the degree of non-regularity?
Topological centre basics Topological centre problems Topological centres as a tool
Arens products: Topological description
A 3 xi −→ X ∈ A∗∗ (w∗)
A 3 yj −→ Y ∈ A∗∗ (w∗)
X2Y = limi limj xi · yj
X3Y = limj limi xi · yj
2 = 3 ⇔: A Arens regular (e.g., operator algebras)
But for algebras closest to the heart of harmonic analysts:
X2Y 6= X3Y
; How to measure the degree of non-regularity?
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres
Z`(A∗∗) := { X | X2Y = X3Y ∀ Y }= { X | Y 7→ X2Y w∗-cont. }
Zr (A∗∗) := { X | Y 2X = Y 3X ∀ Y }= { X | Y 7→ Y 3X w∗-cont. }
A Arens regular :⇔ Z` = Zr = A∗∗
Definition (Dales–Lau ’05)
A Left Strongly Arens Irregular (LSAI) :⇔ Z` = AA Right Strongly Arens Irregular (RSAI) :⇔ Zr = AA Strongly Arens Irregular (SAI) :⇔ Z` = Zr = A
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres
Z`(A∗∗) := { X | X2Y = X3Y ∀ Y }
= { X | Y 7→ X2Y w∗-cont. }
Zr (A∗∗) := { X | Y 2X = Y 3X ∀ Y }= { X | Y 7→ Y 3X w∗-cont. }
A Arens regular :⇔ Z` = Zr = A∗∗
Definition (Dales–Lau ’05)
A Left Strongly Arens Irregular (LSAI) :⇔ Z` = AA Right Strongly Arens Irregular (RSAI) :⇔ Zr = AA Strongly Arens Irregular (SAI) :⇔ Z` = Zr = A
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres
Z`(A∗∗) := { X | X2Y = X3Y ∀ Y }= { X | Y 7→ X2Y w∗-cont. }
Zr (A∗∗) := { X | Y 2X = Y 3X ∀ Y }= { X | Y 7→ Y 3X w∗-cont. }
A Arens regular :⇔ Z` = Zr = A∗∗
Definition (Dales–Lau ’05)
A Left Strongly Arens Irregular (LSAI) :⇔ Z` = AA Right Strongly Arens Irregular (RSAI) :⇔ Zr = AA Strongly Arens Irregular (SAI) :⇔ Z` = Zr = A
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres
Z`(A∗∗) := { X | X2Y = X3Y ∀ Y }= { X | Y 7→ X2Y w∗-cont. }
Zr (A∗∗) := { X | Y 2X = Y 3X ∀ Y }= { X | Y 7→ Y 3X w∗-cont. }
A Arens regular :⇔ Z` = Zr = A∗∗
Definition (Dales–Lau ’05)
A Left Strongly Arens Irregular (LSAI) :⇔ Z` = AA Right Strongly Arens Irregular (RSAI) :⇔ Zr = AA Strongly Arens Irregular (SAI) :⇔ Z` = Zr = A
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres
Z`(A∗∗) := { X | X2Y = X3Y ∀ Y }= { X | Y 7→ X2Y w∗-cont. }
Zr (A∗∗) := { X | Y 2X = Y 3X ∀ Y }= { X | Y 7→ Y 3X w∗-cont. }
A Arens regular :⇔ Z` = Zr = A∗∗
Definition (Dales–Lau ’05)
A Left Strongly Arens Irregular (LSAI) :⇔ Z` = AA Right Strongly Arens Irregular (RSAI) :⇔ Zr = AA Strongly Arens Irregular (SAI) :⇔ Z` = Zr = A
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres
Z`(A∗∗) := { X | X2Y = X3Y ∀ Y }= { X | Y 7→ X2Y w∗-cont. }
Zr (A∗∗) := { X | Y 2X = Y 3X ∀ Y }= { X | Y 7→ Y 3X w∗-cont. }
A Arens regular :⇔ Z` = Zr = A∗∗
Definition (Dales–Lau ’05)
A Left Strongly Arens Irregular (LSAI) :⇔ Z` = A
A Right Strongly Arens Irregular (RSAI) :⇔ Zr = AA Strongly Arens Irregular (SAI) :⇔ Z` = Zr = A
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres
Z`(A∗∗) := { X | X2Y = X3Y ∀ Y }= { X | Y 7→ X2Y w∗-cont. }
Zr (A∗∗) := { X | Y 2X = Y 3X ∀ Y }= { X | Y 7→ Y 3X w∗-cont. }
A Arens regular :⇔ Z` = Zr = A∗∗
Definition (Dales–Lau ’05)
A Left Strongly Arens Irregular (LSAI) :⇔ Z` = AA Right Strongly Arens Irregular (RSAI) :⇔ Zr = A
A Strongly Arens Irregular (SAI) :⇔ Z` = Zr = A
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres
Z`(A∗∗) := { X | X2Y = X3Y ∀ Y }= { X | Y 7→ X2Y w∗-cont. }
Zr (A∗∗) := { X | Y 2X = Y 3X ∀ Y }= { X | Y 7→ Y 3X w∗-cont. }
A Arens regular :⇔ Z` = Zr = A∗∗
Definition (Dales–Lau ’05)
A Left Strongly Arens Irregular (LSAI) :⇔ Z` = AA Right Strongly Arens Irregular (RSAI) :⇔ Zr = AA Strongly Arens Irregular (SAI) :⇔ Z` = Zr = A
Topological centre basics Topological centre problems Topological centres as a tool
Asymmetries
Obviously: Z` = A∗∗ ⇔ Zr = A∗∗However: Z` = A 6⇒ Zr = A
Proposition (Dales–Lau ’05; N. ’05)
LSAI 6⇒ RSAI
Example convolution algebra T (G) = (T (L2(G)), ∗)
ρ ∗ τ :=
∫G
LxρLx−1π(τ)(x) dx
G non-compact, second countable ⇒ T (G) LSAI but not RSAI
Topological centre basics Topological centre problems Topological centres as a tool
Asymmetries
Obviously: Z` = A∗∗ ⇔ Zr = A∗∗
However: Z` = A 6⇒ Zr = A
Proposition (Dales–Lau ’05; N. ’05)
LSAI 6⇒ RSAI
Example convolution algebra T (G) = (T (L2(G)), ∗)
ρ ∗ τ :=
∫G
LxρLx−1π(τ)(x) dx
G non-compact, second countable ⇒ T (G) LSAI but not RSAI
Topological centre basics Topological centre problems Topological centres as a tool
Asymmetries
Obviously: Z` = A∗∗ ⇔ Zr = A∗∗However: Z` = A 6⇒ Zr = A
Proposition (Dales–Lau ’05; N. ’05)
LSAI 6⇒ RSAI
Example convolution algebra T (G) = (T (L2(G)), ∗)
ρ ∗ τ :=
∫G
LxρLx−1π(τ)(x) dx
G non-compact, second countable ⇒ T (G) LSAI but not RSAI
Topological centre basics Topological centre problems Topological centres as a tool
Asymmetries
Obviously: Z` = A∗∗ ⇔ Zr = A∗∗However: Z` = A 6⇒ Zr = A
Proposition (Dales–Lau ’05; N. ’05)
LSAI 6⇒ RSAI
Example convolution algebra T (G) = (T (L2(G)), ∗)
ρ ∗ τ :=
∫G
LxρLx−1π(τ)(x) dx
G non-compact, second countable ⇒ T (G) LSAI but not RSAI
Topological centre basics Topological centre problems Topological centres as a tool
Arens products & Kadison–Singer Conjecture
Kadison–Singer Conjecture (’59)
Let m ∈ `∞(N)∗ be a pure state, i.e., m ∈ βN .Does m extend uniquely to a state on B(`2(N)) ?
Equivalent conjecture about Bessel sequences (Weaver ’04)
∃ M ≥ 2 and ε > 0 such that:given x1, . . . , xn ∈ Ck (n ≥ 2) with `2-norm ≤ 1 and∑
i
|〈xi , y〉|2 ≤ M ∀ unit vector y ∈ Ck
⇒ ∃ partition A1, . . . ,A` (` ≥ 2) of {1, . . . , n} with∑i∈Aj
|〈xi , y〉|2 ≤ M − ε ∀ unit vectors y ∈ Ck , ∀ j
Topological centre basics Topological centre problems Topological centres as a tool
Arens products & Kadison–Singer Conjecture
Kadison–Singer Conjecture (’59)
Let m ∈ `∞(N)∗ be a pure state, i.e., m ∈ βN .
Does m extend uniquely to a state on B(`2(N)) ?
Equivalent conjecture about Bessel sequences (Weaver ’04)
∃ M ≥ 2 and ε > 0 such that:given x1, . . . , xn ∈ Ck (n ≥ 2) with `2-norm ≤ 1 and∑
i
|〈xi , y〉|2 ≤ M ∀ unit vector y ∈ Ck
⇒ ∃ partition A1, . . . ,A` (` ≥ 2) of {1, . . . , n} with∑i∈Aj
|〈xi , y〉|2 ≤ M − ε ∀ unit vectors y ∈ Ck , ∀ j
Topological centre basics Topological centre problems Topological centres as a tool
Arens products & Kadison–Singer Conjecture
Kadison–Singer Conjecture (’59)
Let m ∈ `∞(N)∗ be a pure state, i.e., m ∈ βN .Does m extend uniquely to a state on B(`2(N)) ?
Equivalent conjecture about Bessel sequences (Weaver ’04)
∃ M ≥ 2 and ε > 0 such that:given x1, . . . , xn ∈ Ck (n ≥ 2) with `2-norm ≤ 1 and∑
i
|〈xi , y〉|2 ≤ M ∀ unit vector y ∈ Ck
⇒ ∃ partition A1, . . . ,A` (` ≥ 2) of {1, . . . , n} with∑i∈Aj
|〈xi , y〉|2 ≤ M − ε ∀ unit vectors y ∈ Ck , ∀ j
Topological centre basics Topological centre problems Topological centres as a tool
Arens products & Kadison–Singer Conjecture
Kadison–Singer Conjecture (’59)
Let m ∈ `∞(N)∗ be a pure state, i.e., m ∈ βN .Does m extend uniquely to a state on B(`2(N)) ?
Equivalent conjecture about Bessel sequences (Weaver ’04)
∃ M ≥ 2 and ε > 0 such that:given x1, . . . , xn ∈ Ck (n ≥ 2) with `2-norm ≤ 1 and∑
i
|〈xi , y〉|2 ≤ M ∀ unit vector y ∈ Ck
⇒ ∃ partition A1, . . . ,A` (` ≥ 2) of {1, . . . , n} with∑i∈Aj
|〈xi , y〉|2 ≤ M − ε ∀ unit vectors y ∈ Ck , ∀ j
Topological centre basics Topological centre problems Topological centres as a tool
Arens products & Kadison–Singer Conjecture
Kadison–Singer Conjecture (’59)
Let m ∈ `∞(N)∗ be a pure state, i.e., m ∈ βN .Does m extend uniquely to a state on B(`2(N)) ?
Equivalent conjecture about Bessel sequences (Weaver ’04)
∃ M ≥ 2 and ε > 0 such that:
given x1, . . . , xn ∈ Ck (n ≥ 2) with `2-norm ≤ 1 and∑i
|〈xi , y〉|2 ≤ M ∀ unit vector y ∈ Ck
⇒ ∃ partition A1, . . . ,A` (` ≥ 2) of {1, . . . , n} with∑i∈Aj
|〈xi , y〉|2 ≤ M − ε ∀ unit vectors y ∈ Ck , ∀ j
Topological centre basics Topological centre problems Topological centres as a tool
Arens products & Kadison–Singer Conjecture
Kadison–Singer Conjecture (’59)
Let m ∈ `∞(N)∗ be a pure state, i.e., m ∈ βN .Does m extend uniquely to a state on B(`2(N)) ?
Equivalent conjecture about Bessel sequences (Weaver ’04)
∃ M ≥ 2 and ε > 0 such that:given x1, . . . , xn ∈ Ck (n ≥ 2) with `2-norm ≤ 1
and∑i
|〈xi , y〉|2 ≤ M ∀ unit vector y ∈ Ck
⇒ ∃ partition A1, . . . ,A` (` ≥ 2) of {1, . . . , n} with∑i∈Aj
|〈xi , y〉|2 ≤ M − ε ∀ unit vectors y ∈ Ck , ∀ j
Topological centre basics Topological centre problems Topological centres as a tool
Arens products & Kadison–Singer Conjecture
Kadison–Singer Conjecture (’59)
Let m ∈ `∞(N)∗ be a pure state, i.e., m ∈ βN .Does m extend uniquely to a state on B(`2(N)) ?
Equivalent conjecture about Bessel sequences (Weaver ’04)
∃ M ≥ 2 and ε > 0 such that:given x1, . . . , xn ∈ Ck (n ≥ 2) with `2-norm ≤ 1 and∑
i
|〈xi , y〉|2 ≤ M ∀ unit vector y ∈ Ck
⇒ ∃ partition A1, . . . ,A` (` ≥ 2) of {1, . . . , n} with∑i∈Aj
|〈xi , y〉|2 ≤ M − ε ∀ unit vectors y ∈ Ck , ∀ j
Topological centre basics Topological centre problems Topological centres as a tool
Arens products & Kadison–Singer Conjecture
Kadison–Singer Conjecture (’59)
Let m ∈ `∞(N)∗ be a pure state, i.e., m ∈ βN .Does m extend uniquely to a state on B(`2(N)) ?
Equivalent conjecture about Bessel sequences (Weaver ’04)
∃ M ≥ 2 and ε > 0 such that:given x1, . . . , xn ∈ Ck (n ≥ 2) with `2-norm ≤ 1 and∑
i
|〈xi , y〉|2 ≤ M ∀ unit vector y ∈ Ck
⇒ ∃ partition A1, . . . ,A` (` ≥ 2) of {1, . . . , n}
with∑i∈Aj
|〈xi , y〉|2 ≤ M − ε ∀ unit vectors y ∈ Ck , ∀ j
Topological centre basics Topological centre problems Topological centres as a tool
Arens products & Kadison–Singer Conjecture
Kadison–Singer Conjecture (’59)
Let m ∈ `∞(N)∗ be a pure state, i.e., m ∈ βN .Does m extend uniquely to a state on B(`2(N)) ?
Equivalent conjecture about Bessel sequences (Weaver ’04)
∃ M ≥ 2 and ε > 0 such that:given x1, . . . , xn ∈ Ck (n ≥ 2) with `2-norm ≤ 1 and∑
i
|〈xi , y〉|2 ≤ M ∀ unit vector y ∈ Ck
⇒ ∃ partition A1, . . . ,A` (` ≥ 2) of {1, . . . , n} with∑i∈Aj
|〈xi , y〉|2 ≤ M − ε ∀ unit vectors y ∈ Ck , ∀ j
Topological centre basics Topological centre problems Topological centres as a tool
Kadison–Singer Conjecture & T (G)
Proposition (N. ’08)
Kadison–Singer Conjecture for discrete G⇒ the map
βG 3 m 7→ m ∈ T (`2(G))∗∗
is multiplicative w.r.t. convolution
This is true for G = Z as well as G = F2 . . .
Transfer of topological dynamics!
Topological centre basics Topological centre problems Topological centres as a tool
Kadison–Singer Conjecture & T (G)
Proposition (N. ’08)
Kadison–Singer Conjecture for discrete G
⇒ the mapβG 3 m 7→ m ∈ T (`2(G))∗∗
is multiplicative w.r.t. convolution
This is true for G = Z as well as G = F2 . . .
Transfer of topological dynamics!
Topological centre basics Topological centre problems Topological centres as a tool
Kadison–Singer Conjecture & T (G)
Proposition (N. ’08)
Kadison–Singer Conjecture for discrete G⇒ the map
βG 3 m 7→ m ∈ T (`2(G))∗∗
is multiplicative w.r.t. convolution
This is true for G = Z as well as G = F2 . . .
Transfer of topological dynamics!
Topological centre basics Topological centre problems Topological centres as a tool
Kadison–Singer Conjecture & T (G)
Proposition (N. ’08)
Kadison–Singer Conjecture for discrete G⇒ the map
βG 3 m 7→ m ∈ T (`2(G))∗∗
is multiplicative w.r.t. convolution
This is true for G = Z as well as G = F2 . . .
Transfer of topological dynamics!
Topological centre basics Topological centre problems Topological centres as a tool
Kadison–Singer Conjecture & T (G)
Proposition (N. ’08)
Kadison–Singer Conjecture for discrete G⇒ the map
βG 3 m 7→ m ∈ T (`2(G))∗∗
is multiplicative w.r.t. convolution
This is true for G = Z as well as G = F2 . . .
Transfer of topological dynamics!
Topological centre basics Topological centre problems Topological centres as a tool
Kadison–Singer Conjecture & T (G)
Proposition (N. ’08)
Kadison–Singer Conjecture for discrete G⇒ the map
βG 3 m 7→ m ∈ T (`2(G))∗∗
is multiplicative w.r.t. convolution
This is true for G = Z as well as G = F2 . . .
Transfer of topological dynamics!
Topological centre basics Topological centre problems Topological centres as a tool
1 Topological centre basics
2 Topological centre problems
3 Topological centres as a tool
Topological centre basics Topological centre problems Topological centres as a tool
The Ghahramani–Lau Conjecture
Theorem (Lau–Losert ’88)
L1(G) is SAI for any locally compact group G.
Conjecture (Lau ’94 & Ghahramani–Lau ’95)
M(G) is SAI for any locally compact group G.
Theorem (N. ’05)
The conjecture holds for all non-compact groups G s.t.G has non-measurable cardinality OR k(G) ≥ 2b(G)
One cannot prove in ZFC the existence of measurable cardinals(Ulam ’30)!
Theorem (Losert ’09)
The second condition can be weakened to k(G) ≥ b(G) .
Topological centre basics Topological centre problems Topological centres as a tool
The Ghahramani–Lau Conjecture
Theorem (Lau–Losert ’88)
L1(G) is SAI for any locally compact group G.
Conjecture (Lau ’94 & Ghahramani–Lau ’95)
M(G) is SAI for any locally compact group G.
Theorem (N. ’05)
The conjecture holds for all non-compact groups G s.t.G has non-measurable cardinality OR k(G) ≥ 2b(G)
One cannot prove in ZFC the existence of measurable cardinals(Ulam ’30)!
Theorem (Losert ’09)
The second condition can be weakened to k(G) ≥ b(G) .
Topological centre basics Topological centre problems Topological centres as a tool
The Ghahramani–Lau Conjecture
Theorem (Lau–Losert ’88)
L1(G) is SAI for any locally compact group G.
Conjecture (Lau ’94 & Ghahramani–Lau ’95)
M(G) is SAI for any locally compact group G.
Theorem (N. ’05)
The conjecture holds for all non-compact groups G s.t.G has non-measurable cardinality OR k(G) ≥ 2b(G)
One cannot prove in ZFC the existence of measurable cardinals(Ulam ’30)!
Theorem (Losert ’09)
The second condition can be weakened to k(G) ≥ b(G) .
Topological centre basics Topological centre problems Topological centres as a tool
The Ghahramani–Lau Conjecture
Theorem (Lau–Losert ’88)
L1(G) is SAI for any locally compact group G.
Conjecture (Lau ’94 & Ghahramani–Lau ’95)
M(G) is SAI for any locally compact group G.
Theorem (N. ’05)
The conjecture holds for all non-compact groups G s.t.G has non-measurable cardinality OR k(G) ≥ 2b(G)
One cannot prove in ZFC the existence of measurable cardinals(Ulam ’30)!
Theorem (Losert ’09)
The second condition can be weakened to k(G) ≥ b(G) .
Topological centre basics Topological centre problems Topological centres as a tool
The Ghahramani–Lau Conjecture
Theorem (Lau–Losert ’88)
L1(G) is SAI for any locally compact group G.
Conjecture (Lau ’94 & Ghahramani–Lau ’95)
M(G) is SAI for any locally compact group G.
Theorem (N. ’05)
The conjecture holds for all non-compact groups G s.t.G has non-measurable cardinality OR k(G) ≥ 2b(G)
One cannot prove in ZFC the existence of measurable cardinals(Ulam ’30)!
Theorem (Losert ’09)
The second condition can be weakened to k(G) ≥ b(G) .
Topological centre basics Topological centre problems Topological centres as a tool
Key technique: Factorization
Definition (N. ’05)
A Banach algebra, κ ≥ ℵ0.
1 A has factorization property of level κ (Fκ) if
∀ (hi )i∈I ⊆ Ball(A∗), |I | ≤ κ∃ (Xi )i∈I ⊆ Ball(A∗∗) ∃ h ∈ A∗
hi = Xi2h (i ∈ I )
2 A has Mazur’s property of level κ (Mκ) ifany X ∈ A∗∗ which is w∗-κ-continuous on A∗, lies in A.
Theorem (N. ’05)
A has Fκ and Mκ for some κ ≥ ℵ0 ⇒ A is SAI
Theorem (N. ’05; Hu–N. ’04)
M(G) has Fk(G) (for non-compact G) and M|G|.
Topological centre basics Topological centre problems Topological centres as a tool
Key technique: Factorization
Definition (N. ’05)
A Banach algebra, κ ≥ ℵ0.
1 A has factorization property of level κ (Fκ) if
∀ (hi )i∈I ⊆ Ball(A∗), |I | ≤ κ∃ (Xi )i∈I ⊆ Ball(A∗∗) ∃ h ∈ A∗
hi = Xi2h (i ∈ I )
2 A has Mazur’s property of level κ (Mκ) ifany X ∈ A∗∗ which is w∗-κ-continuous on A∗, lies in A.
Theorem (N. ’05)
A has Fκ and Mκ for some κ ≥ ℵ0 ⇒ A is SAI
Theorem (N. ’05; Hu–N. ’04)
M(G) has Fk(G) (for non-compact G) and M|G|.
Topological centre basics Topological centre problems Topological centres as a tool
Key technique: Factorization
Definition (N. ’05)
A Banach algebra, κ ≥ ℵ0.
1 A has factorization property of level κ (Fκ) if
∀ (hi )i∈I ⊆ Ball(A∗), |I | ≤ κ
∃ (Xi )i∈I ⊆ Ball(A∗∗) ∃ h ∈ A∗
hi = Xi2h (i ∈ I )
2 A has Mazur’s property of level κ (Mκ) ifany X ∈ A∗∗ which is w∗-κ-continuous on A∗, lies in A.
Theorem (N. ’05)
A has Fκ and Mκ for some κ ≥ ℵ0 ⇒ A is SAI
Theorem (N. ’05; Hu–N. ’04)
M(G) has Fk(G) (for non-compact G) and M|G|.
Topological centre basics Topological centre problems Topological centres as a tool
Key technique: Factorization
Definition (N. ’05)
A Banach algebra, κ ≥ ℵ0.
1 A has factorization property of level κ (Fκ) if
∀ (hi )i∈I ⊆ Ball(A∗), |I | ≤ κ∃ (Xi )i∈I ⊆ Ball(A∗∗) ∃ h ∈ A∗
hi = Xi2h (i ∈ I )
2 A has Mazur’s property of level κ (Mκ) ifany X ∈ A∗∗ which is w∗-κ-continuous on A∗, lies in A.
Theorem (N. ’05)
A has Fκ and Mκ for some κ ≥ ℵ0 ⇒ A is SAI
Theorem (N. ’05; Hu–N. ’04)
M(G) has Fk(G) (for non-compact G) and M|G|.
Topological centre basics Topological centre problems Topological centres as a tool
Key technique: Factorization
Definition (N. ’05)
A Banach algebra, κ ≥ ℵ0.
1 A has factorization property of level κ (Fκ) if
∀ (hi )i∈I ⊆ Ball(A∗), |I | ≤ κ∃ (Xi )i∈I ⊆ Ball(A∗∗) ∃ h ∈ A∗
hi = Xi2h (i ∈ I )
2 A has Mazur’s property of level κ (Mκ) ifany X ∈ A∗∗ which is w∗-κ-continuous on A∗, lies in A.
Theorem (N. ’05)
A has Fκ and Mκ for some κ ≥ ℵ0 ⇒ A is SAI
Theorem (N. ’05; Hu–N. ’04)
M(G) has Fk(G) (for non-compact G) and M|G|.
Topological centre basics Topological centre problems Topological centres as a tool
Key technique: Factorization
Definition (N. ’05)
A Banach algebra, κ ≥ ℵ0.
1 A has factorization property of level κ (Fκ) if
∀ (hi )i∈I ⊆ Ball(A∗), |I | ≤ κ∃ (Xi )i∈I ⊆ Ball(A∗∗) ∃ h ∈ A∗
hi = Xi2h (i ∈ I )
2 A has Mazur’s property of level κ (Mκ) ifany X ∈ A∗∗ which is w∗-κ-continuous on A∗, lies in A.
Theorem (N. ’05)
A has Fκ and Mκ for some κ ≥ ℵ0 ⇒ A is SAI
Theorem (N. ’05; Hu–N. ’04)
M(G) has Fk(G) (for non-compact G) and M|G|.
Topological centre basics Topological centre problems Topological centres as a tool
Key technique: Factorization
Definition (N. ’05)
A Banach algebra, κ ≥ ℵ0.
1 A has factorization property of level κ (Fκ) if
∀ (hi )i∈I ⊆ Ball(A∗), |I | ≤ κ∃ (Xi )i∈I ⊆ Ball(A∗∗) ∃ h ∈ A∗
hi = Xi2h (i ∈ I )
2 A has Mazur’s property of level κ (Mκ) ifany X ∈ A∗∗ which is w∗-κ-continuous on A∗, lies in A.
Theorem (N. ’05)
A has Fκ and Mκ for some κ ≥ ℵ0 ⇒ A is SAI
Theorem (N. ’05; Hu–N. ’04)
M(G) has Fk(G) (for non-compact G) and M|G|.
Topological centre basics Topological centre problems Topological centres as a tool
Factorization works for any 2nd countable group!
Theorem (N.–Pachl–Steprans ’09)
M(G) is SAI for any locally compact group with |G| ≤ c.In particular, the conjecture holds for all 2nd countable groups.
Idea of proof: Factorization in the dual of singular measures!
Topological centre basics Topological centre problems Topological centres as a tool
Factorization works for any 2nd countable group!
Theorem (N.–Pachl–Steprans ’09)
M(G) is SAI for any locally compact group with |G| ≤ c.In particular, the conjecture holds for all 2nd countable groups.
Idea of proof: Factorization in the dual of singular measures!
Topological centre basics Topological centre problems Topological centres as a tool
Separation of singular measures
The following generalizes a result by Prokaj (’03) for G = R.We can assume that G is non-discrete.
Theorem (N.–Pachl–Steprans ’09)
For every µ ∈ Ms(G) there exist a Kσ set E ⊆ G and P ⊆ Gs.t. |µ|(G \ E ) = 0, |P| = c, (Ep) ∩ (Ep′) = ∅ if p 6= p′ in P.
Idea Choose compacta Ki with |µ|(G \ Ki ) ≤ 2−i and xi ∈ G“along” a Cantor set C; set E = ∪n ∩∞i=n Ki and construct P from xi and C
Corollary (Separation Lemma)
(Fα)α∈I family of finite subsets of Ms(G) with |I | ≤ c
⇒ ∃ (xα) ⊆ G s.t. (Fα ∗ xα) ⊥ (Fβ ∗ xβ) if α 6= β in I
Topological centre basics Topological centre problems Topological centres as a tool
Separation of singular measures
The following generalizes a result by Prokaj (’03) for G = R.We can assume that G is non-discrete.
Theorem (N.–Pachl–Steprans ’09)
For every µ ∈ Ms(G) there exist a Kσ set E ⊆ G and P ⊆ Gs.t. |µ|(G \ E ) = 0, |P| = c, (Ep) ∩ (Ep′) = ∅ if p 6= p′ in P.
Idea Choose compacta Ki with |µ|(G \ Ki ) ≤ 2−i and xi ∈ G“along” a Cantor set C; set E = ∪n ∩∞i=n Ki and construct P from xi and C
Corollary (Separation Lemma)
(Fα)α∈I family of finite subsets of Ms(G) with |I | ≤ c
⇒ ∃ (xα) ⊆ G s.t. (Fα ∗ xα) ⊥ (Fβ ∗ xβ) if α 6= β in I
Topological centre basics Topological centre problems Topological centres as a tool
Separation of singular measures
The following generalizes a result by Prokaj (’03) for G = R.We can assume that G is non-discrete.
Theorem (N.–Pachl–Steprans ’09)
For every µ ∈ Ms(G) there exist a Kσ set E ⊆ G and P ⊆ Gs.t. |µ|(G \ E ) = 0, |P| = c, (Ep) ∩ (Ep′) = ∅ if p 6= p′ in P.
Idea Choose compacta Ki with |µ|(G \ Ki ) ≤ 2−i and xi ∈ G“along” a Cantor set C; set E = ∪n ∩∞i=n Ki and construct P from xi and C
Corollary (Separation Lemma)
(Fα)α∈I family of finite subsets of Ms(G) with |I | ≤ c
⇒ ∃ (xα) ⊆ G s.t. (Fα ∗ xα) ⊥ (Fβ ∗ xβ) if α 6= β in I
Topological centre basics Topological centre problems Topological centres as a tool
Separation of singular measures
The following generalizes a result by Prokaj (’03) for G = R.We can assume that G is non-discrete.
Theorem (N.–Pachl–Steprans ’09)
For every µ ∈ Ms(G) there exist a Kσ set E ⊆ G and P ⊆ Gs.t. |µ|(G \ E ) = 0, |P| = c, (Ep) ∩ (Ep′) = ∅ if p 6= p′ in P.
Idea Choose compacta Ki with |µ|(G \ Ki ) ≤ 2−i and xi ∈ G“along” a Cantor set C; set E = ∪n ∩∞i=n Ki and construct P from xi and C
Corollary (Separation Lemma)
(Fα)α∈I family of finite subsets of Ms(G) with |I | ≤ c
⇒ ∃ (xα) ⊆ G s.t. (Fα ∗ xα) ⊥ (Fβ ∗ xβ) if α 6= β in I
Topological centre basics Topological centre problems Topological centres as a tool
Factorization in the dual of singular measures
Factorization theorem (N.–Pachl–Steprans ’09)
|G| ≤ c ⇒ ∃ h ∈ BallMs(G)∗ s.t. δGw∗
2h = BallMs(G)∗
Proof.
Sketch
BallMs(G)∗ has w∗-open nhd. basis O with |O| ≤ c (Hu–N. ’06)
O 3 U = { f ∈ BallMs(G)∗ | |〈f , µ〉 − 〈gU , µ〉| < εU ∀ µ ∈ FU }
where FU ⊆ Ms(G) finite, gU ∈ BallMs(G)∗ and εU > 0
Separation Lemma ; xU ∈ G s.t. (FU ∗ xU) ⊥ (FV ∗ xV ) if U 6= V
Define hU ∈ Ms(G)∗ by 〈hU , ν〉 = 〈gU , ν ∗ x−1U 〉
; h ∈ Ms(G)∗ s.t. xU2h ∈ U
Topological centre basics Topological centre problems Topological centres as a tool
Factorization in the dual of singular measures
Factorization theorem (N.–Pachl–Steprans ’09)
|G| ≤ c ⇒ ∃ h ∈ BallMs(G)∗ s.t. δGw∗
2h = BallMs(G)∗
Proof.
Sketch
BallMs(G)∗ has w∗-open nhd. basis O with |O| ≤ c (Hu–N. ’06)
O 3 U = { f ∈ BallMs(G)∗ | |〈f , µ〉 − 〈gU , µ〉| < εU ∀ µ ∈ FU }
where FU ⊆ Ms(G) finite, gU ∈ BallMs(G)∗ and εU > 0
Separation Lemma ; xU ∈ G s.t. (FU ∗ xU) ⊥ (FV ∗ xV ) if U 6= V
Define hU ∈ Ms(G)∗ by 〈hU , ν〉 = 〈gU , ν ∗ x−1U 〉
; h ∈ Ms(G)∗ s.t. xU2h ∈ U
Topological centre basics Topological centre problems Topological centres as a tool
Factorization in the dual of singular measures
Factorization theorem (N.–Pachl–Steprans ’09)
|G| ≤ c ⇒ ∃ h ∈ BallMs(G)∗ s.t. δGw∗
2h = BallMs(G)∗
Proof.
Sketch
BallMs(G)∗ has w∗-open nhd. basis O with |O| ≤ c (Hu–N. ’06)
O 3 U = { f ∈ BallMs(G)∗ | |〈f , µ〉 − 〈gU , µ〉| < εU ∀ µ ∈ FU }
where FU ⊆ Ms(G) finite, gU ∈ BallMs(G)∗ and εU > 0
Separation Lemma ; xU ∈ G s.t. (FU ∗ xU) ⊥ (FV ∗ xV ) if U 6= V
Define hU ∈ Ms(G)∗ by 〈hU , ν〉 = 〈gU , ν ∗ x−1U 〉
; h ∈ Ms(G)∗ s.t. xU2h ∈ U
Topological centre basics Topological centre problems Topological centres as a tool
Factorization in the dual of singular measures
Factorization theorem (N.–Pachl–Steprans ’09)
|G| ≤ c ⇒ ∃ h ∈ BallMs(G)∗ s.t. δGw∗
2h = BallMs(G)∗
Proof.
Sketch
BallMs(G)∗ has w∗-open nhd. basis O with |O| ≤ c (Hu–N. ’06)
O 3 U = { f ∈ BallMs(G)∗ | |〈f , µ〉 − 〈gU , µ〉| < εU ∀ µ ∈ FU }
where FU ⊆ Ms(G) finite, gU ∈ BallMs(G)∗ and εU > 0
Separation Lemma ; xU ∈ G s.t. (FU ∗ xU) ⊥ (FV ∗ xV ) if U 6= V
Define hU ∈ Ms(G)∗ by 〈hU , ν〉 = 〈gU , ν ∗ x−1U 〉
; h ∈ Ms(G)∗ s.t. xU2h ∈ U
Topological centre basics Topological centre problems Topological centres as a tool
Factorization in the dual of singular measures
Factorization theorem (N.–Pachl–Steprans ’09)
|G| ≤ c ⇒ ∃ h ∈ BallMs(G)∗ s.t. δGw∗
2h = BallMs(G)∗
Proof.
Sketch
BallMs(G)∗ has w∗-open nhd. basis O with |O| ≤ c (Hu–N. ’06)
O 3 U = { f ∈ BallMs(G)∗ | |〈f , µ〉 − 〈gU , µ〉| < εU ∀ µ ∈ FU }
where FU ⊆ Ms(G) finite, gU ∈ BallMs(G)∗ and εU > 0
Separation Lemma ; xU ∈ G s.t. (FU ∗ xU) ⊥ (FV ∗ xV ) if U 6= V
Define hU ∈ Ms(G)∗ by 〈hU , ν〉 = 〈gU , ν ∗ x−1U 〉
; h ∈ Ms(G)∗ s.t. xU2h ∈ U
Topological centre basics Topological centre problems Topological centres as a tool
Factorization in the dual of singular measures
Factorization theorem (N.–Pachl–Steprans ’09)
|G| ≤ c ⇒ ∃ h ∈ BallMs(G)∗ s.t. δGw∗
2h = BallMs(G)∗
Proof.
Sketch
BallMs(G)∗ has w∗-open nhd. basis O with |O| ≤ c (Hu–N. ’06)
O 3 U = { f ∈ BallMs(G)∗ | |〈f , µ〉 − 〈gU , µ〉| < εU ∀ µ ∈ FU }
where FU ⊆ Ms(G) finite, gU ∈ BallMs(G)∗ and εU > 0
Separation Lemma ; xU ∈ G s.t. (FU ∗ xU) ⊥ (FV ∗ xV ) if U 6= V
Define hU ∈ Ms(G)∗ by 〈hU , ν〉 = 〈gU , ν ∗ x−1U 〉
; h ∈ Ms(G)∗ s.t. xU2h ∈ U
Topological centre basics Topological centre problems Topological centres as a tool
Factorization in the dual of singular measures
Factorization theorem (N.–Pachl–Steprans ’09)
|G| ≤ c ⇒ ∃ h ∈ BallMs(G)∗ s.t. δGw∗
2h = BallMs(G)∗
Proof.
Sketch
BallMs(G)∗ has w∗-open nhd. basis O with |O| ≤ c (Hu–N. ’06)
O 3 U = { f ∈ BallMs(G)∗ | |〈f , µ〉 − 〈gU , µ〉| < εU ∀ µ ∈ FU }
where FU ⊆ Ms(G) finite, gU ∈ BallMs(G)∗ and εU > 0
Separation Lemma ; xU ∈ G s.t. (FU ∗ xU) ⊥ (FV ∗ xV ) if U 6= V
Define hU ∈ Ms(G)∗ by 〈hU , ν〉 = 〈gU , ν ∗ x−1U 〉
; h ∈ Ms(G)∗ s.t. xU2h ∈ U
Topological centre basics Topological centre problems Topological centres as a tool
Solution of the conjecture
Theorem (N.–Pachl–Steprans ’09)
Let G be a locally compact group with |G| ≤ c.Then M(G) is SAI.
Proof.
Let m ∈ Z`(M(G)∗∗)
⇒ ms := m |Ms(G)∗ is w∗-cont. on any set of the form
δGw∗
2h ⊆ Ms(G)∗ where h ∈ Ms(G)∗
Factorization theorem ⇒ ms is w∗-cont. on BallMs(G)∗
⇒ ms ∈ M(G)
ma := m |L1(G)∗∈ L1(G)∗∗ satisfies ma = m −ms ∈ Z`(M(G)∗∗)
⇒ ma ∈ Z`(L1(G)∗∗) = L1(G), and m = ma + ms ∈ M(G) .
Topological centre basics Topological centre problems Topological centres as a tool
Solution of the conjecture
Theorem (N.–Pachl–Steprans ’09)
Let G be a locally compact group with |G| ≤ c.Then M(G) is SAI.
Proof.
Let m ∈ Z`(M(G)∗∗)
⇒ ms := m |Ms(G)∗ is w∗-cont. on any set of the form
δGw∗
2h ⊆ Ms(G)∗ where h ∈ Ms(G)∗
Factorization theorem ⇒ ms is w∗-cont. on BallMs(G)∗
⇒ ms ∈ M(G)
ma := m |L1(G)∗∈ L1(G)∗∗ satisfies ma = m −ms ∈ Z`(M(G)∗∗)
⇒ ma ∈ Z`(L1(G)∗∗) = L1(G), and m = ma + ms ∈ M(G) .
Topological centre basics Topological centre problems Topological centres as a tool
Solution of the conjecture
Theorem (N.–Pachl–Steprans ’09)
Let G be a locally compact group with |G| ≤ c.Then M(G) is SAI.
Proof.
Let m ∈ Z`(M(G)∗∗)
⇒ ms := m |Ms(G)∗ is w∗-cont. on any set of the form
δGw∗
2h ⊆ Ms(G)∗ where h ∈ Ms(G)∗
Factorization theorem ⇒ ms is w∗-cont. on BallMs(G)∗
⇒ ms ∈ M(G)
ma := m |L1(G)∗∈ L1(G)∗∗ satisfies ma = m −ms ∈ Z`(M(G)∗∗)
⇒ ma ∈ Z`(L1(G)∗∗) = L1(G), and m = ma + ms ∈ M(G) .
Topological centre basics Topological centre problems Topological centres as a tool
Solution of the conjecture
Theorem (N.–Pachl–Steprans ’09)
Let G be a locally compact group with |G| ≤ c.Then M(G) is SAI.
Proof.
Let m ∈ Z`(M(G)∗∗)
⇒ ms := m |Ms(G)∗ is w∗-cont. on any set of the form
δGw∗
2h ⊆ Ms(G)∗ where h ∈ Ms(G)∗
Factorization theorem ⇒ ms is w∗-cont. on BallMs(G)∗
⇒ ms ∈ M(G)
ma := m |L1(G)∗∈ L1(G)∗∗ satisfies ma = m −ms ∈ Z`(M(G)∗∗)
⇒ ma ∈ Z`(L1(G)∗∗) = L1(G), and m = ma + ms ∈ M(G) .
Topological centre basics Topological centre problems Topological centres as a tool
Solution of the conjecture
Theorem (N.–Pachl–Steprans ’09)
Let G be a locally compact group with |G| ≤ c.Then M(G) is SAI.
Proof.
Let m ∈ Z`(M(G)∗∗)
⇒ ms := m |Ms(G)∗ is w∗-cont. on any set of the form
δGw∗
2h ⊆ Ms(G)∗ where h ∈ Ms(G)∗
Factorization theorem ⇒ ms is w∗-cont. on BallMs(G)∗
⇒ ms ∈ M(G)
ma := m |L1(G)∗∈ L1(G)∗∗ satisfies ma = m −ms ∈ Z`(M(G)∗∗)
⇒ ma ∈ Z`(L1(G)∗∗) = L1(G), and m = ma + ms ∈ M(G) .
Topological centre basics Topological centre problems Topological centres as a tool
Solution of the conjecture
Theorem (N.–Pachl–Steprans ’09)
Let G be a locally compact group with |G| ≤ c.Then M(G) is SAI.
Proof.
Let m ∈ Z`(M(G)∗∗)
⇒ ms := m |Ms(G)∗ is w∗-cont. on any set of the form
δGw∗
2h ⊆ Ms(G)∗ where h ∈ Ms(G)∗
Factorization theorem ⇒ ms is w∗-cont. on BallMs(G)∗
⇒ ms ∈ M(G)
ma := m |L1(G)∗∈ L1(G)∗∗ satisfies ma = m −ms ∈ Z`(M(G)∗∗)
⇒ ma ∈ Z`(L1(G)∗∗) = L1(G), and m = ma + ms ∈ M(G) .
Topological centre basics Topological centre problems Topological centres as a tool
Solution of the conjecture
Theorem (N.–Pachl–Steprans ’09)
Let G be a locally compact group with |G| ≤ c.Then M(G) is SAI.
Proof.
Let m ∈ Z`(M(G)∗∗)
⇒ ms := m |Ms(G)∗ is w∗-cont. on any set of the form
δGw∗
2h ⊆ Ms(G)∗ where h ∈ Ms(G)∗
Factorization theorem ⇒ ms is w∗-cont. on BallMs(G)∗
⇒ ms ∈ M(G)
ma := m |L1(G)∗∈ L1(G)∗∗ satisfies ma = m −ms ∈ Z`(M(G)∗∗)
⇒ ma ∈ Z`(L1(G)∗∗) = L1(G), and m = ma + ms ∈ M(G) .
Topological centre basics Topological centre problems Topological centres as a tool
Solution of the conjecture
Theorem (N.–Pachl–Steprans ’09)
Let G be a locally compact group with |G| ≤ c.Then M(G) is SAI.
Proof.
Let m ∈ Z`(M(G)∗∗)
⇒ ms := m |Ms(G)∗ is w∗-cont. on any set of the form
δGw∗
2h ⊆ Ms(G)∗ where h ∈ Ms(G)∗
Factorization theorem ⇒ ms is w∗-cont. on BallMs(G)∗
⇒ ms ∈ M(G)
ma := m |L1(G)∗∈ L1(G)∗∗ satisfies ma = m −ms ∈ Z`(M(G)∗∗)
⇒ ma ∈ Z`(L1(G)∗∗) = L1(G), and m = ma + ms ∈ M(G) .
Topological centre basics Topological centre problems Topological centres as a tool
Beyond second countability
Theorem (N.–Pachl–Steprans ’09)
Let G be a product of at most c many compact, metrizable groups.Then M(G) is SAI.
G. Dales, T. Lau & D. Strauss (’09)Second duals of measure algebras
Topological centre basics Topological centre problems Topological centres as a tool
Beyond second countability
Theorem (N.–Pachl–Steprans ’09)
Let G be a product of at most c many compact, metrizable groups.Then M(G) is SAI.
G. Dales, T. Lau & D. Strauss (’09)Second duals of measure algebras
Topological centre basics Topological centre problems Topological centres as a tool
Beyond second countability
Theorem (N.–Pachl–Steprans ’09)
Let G be a product of at most c many compact, metrizable groups.Then M(G) is SAI.
G. Dales, T. Lau & D. Strauss (’09)Second duals of measure algebras
Topological centre basics Topological centre problems Topological centres as a tool
The dual setting
Problem (Cechini–Zappa ’81)
Is A(G) SAI?
Theorem (Lau–Losert ’93)
Yes for large classes of amenable groups.
Theorem (Losert ’02 & ’04)
No for G = F2 and also for G = SU(3) !
Theorem (Filali–Monfared–N. ’08)
Yes for any compact group that is sufficiently non-metrizable(b(G) has uncountable cofinality); e.g., SU(3)ℵ1 and SU(3)c
Theorem (Lau–Losert ’05)
Yes for SU(3)ℵ0
Topological centre basics Topological centre problems Topological centres as a tool
The dual setting
Problem (Cechini–Zappa ’81)
Is A(G) SAI?
Theorem (Lau–Losert ’93)
Yes for large classes of amenable groups.
Theorem (Losert ’02 & ’04)
No for G = F2 and also for G = SU(3) !
Theorem (Filali–Monfared–N. ’08)
Yes for any compact group that is sufficiently non-metrizable(b(G) has uncountable cofinality); e.g., SU(3)ℵ1 and SU(3)c
Theorem (Lau–Losert ’05)
Yes for SU(3)ℵ0
Topological centre basics Topological centre problems Topological centres as a tool
The dual setting
Problem (Cechini–Zappa ’81)
Is A(G) SAI?
Theorem (Lau–Losert ’93)
Yes for large classes of amenable groups.
Theorem (Losert ’02 & ’04)
No for G = F2 and also for G = SU(3) !
Theorem (Filali–Monfared–N. ’08)
Yes for any compact group that is sufficiently non-metrizable(b(G) has uncountable cofinality); e.g., SU(3)ℵ1 and SU(3)c
Theorem (Lau–Losert ’05)
Yes for SU(3)ℵ0
Topological centre basics Topological centre problems Topological centres as a tool
The dual setting
Problem (Cechini–Zappa ’81)
Is A(G) SAI?
Theorem (Lau–Losert ’93)
Yes for large classes of amenable groups.
Theorem (Losert ’02 & ’04)
No for G = F2 and also for G = SU(3) !
Theorem (Filali–Monfared–N. ’08)
Yes for any compact group that is sufficiently non-metrizable(b(G) has uncountable cofinality); e.g., SU(3)ℵ1 and SU(3)c
Theorem (Lau–Losert ’05)
Yes for SU(3)ℵ0
Topological centre basics Topological centre problems Topological centres as a tool
The dual setting
Problem (Cechini–Zappa ’81)
Is A(G) SAI?
Theorem (Lau–Losert ’93)
Yes for large classes of amenable groups.
Theorem (Losert ’02 & ’04)
No for G = F2 and also for G = SU(3) !
Theorem (Filali–Monfared–N. ’08)
Yes for any compact group that is sufficiently non-metrizable(b(G) has uncountable cofinality); e.g., SU(3)ℵ1 and SU(3)c
Theorem (Lau–Losert ’05)
Yes for SU(3)ℵ0
Topological centre basics Topological centre problems Topological centres as a tool
The dual setting
Problem (Cechini–Zappa ’81)
Is A(G) SAI?
Theorem (Lau–Losert ’93)
Yes for large classes of amenable groups.
Theorem (Losert ’02 & ’04)
No for G = F2 and also for G = SU(3) !
Theorem (Filali–Monfared–N. ’08)
Yes for any compact group that is sufficiently non-metrizable(b(G) has uncountable cofinality); e.g., SU(3)ℵ1 and SU(3)c
Theorem (Lau–Losert ’05)
Yes for SU(3)ℵ0
Topological centre basics Topological centre problems Topological centres as a tool
Method of proof:Factorization & Mazur’s property for A(G)
Theorem (Filali–Monfared–N. ’08)
G compact s.t. b(G) has uncountable cofinality. Then:
∀ (Tα)α∈I ⊆ BallL(G) with |I | ≤ b(G)
∃ (X kα )α∈I ⊆ BallL(G)∗ (k = 1, . . . , n)
∃ T k ∈ L(G) (k = 1, . . . , n) s.t.
Tα =n∑
k=1
X kα 2 T k
So A(G) has (a slightly weakened form of) Fb(G).
Theorem (Hu–N. ’04)
A(G) has Mb(G)·ℵ0.
Topological centre basics Topological centre problems Topological centres as a tool
Method of proof:Factorization & Mazur’s property for A(G)
Theorem (Filali–Monfared–N. ’08)
G compact s.t. b(G) has uncountable cofinality.
Then:
∀ (Tα)α∈I ⊆ BallL(G) with |I | ≤ b(G)
∃ (X kα )α∈I ⊆ BallL(G)∗ (k = 1, . . . , n)
∃ T k ∈ L(G) (k = 1, . . . , n) s.t.
Tα =n∑
k=1
X kα 2 T k
So A(G) has (a slightly weakened form of) Fb(G).
Theorem (Hu–N. ’04)
A(G) has Mb(G)·ℵ0.
Topological centre basics Topological centre problems Topological centres as a tool
Method of proof:Factorization & Mazur’s property for A(G)
Theorem (Filali–Monfared–N. ’08)
G compact s.t. b(G) has uncountable cofinality. Then:
∀ (Tα)α∈I ⊆ BallL(G) with |I | ≤ b(G)
∃ (X kα )α∈I ⊆ BallL(G)∗ (k = 1, . . . , n)
∃ T k ∈ L(G) (k = 1, . . . , n) s.t.
Tα =n∑
k=1
X kα 2 T k
So A(G) has (a slightly weakened form of) Fb(G).
Theorem (Hu–N. ’04)
A(G) has Mb(G)·ℵ0.
Topological centre basics Topological centre problems Topological centres as a tool
Method of proof:Factorization & Mazur’s property for A(G)
Theorem (Filali–Monfared–N. ’08)
G compact s.t. b(G) has uncountable cofinality. Then:
∀ (Tα)α∈I ⊆ BallL(G) with |I | ≤ b(G)
∃ (X kα )α∈I ⊆ BallL(G)∗ (k = 1, . . . , n)
∃ T k ∈ L(G) (k = 1, . . . , n) s.t.
Tα =n∑
k=1
X kα 2 T k
So A(G) has (a slightly weakened form of) Fb(G).
Theorem (Hu–N. ’04)
A(G) has Mb(G)·ℵ0.
Topological centre basics Topological centre problems Topological centres as a tool
Method of proof:Factorization & Mazur’s property for A(G)
Theorem (Filali–Monfared–N. ’08)
G compact s.t. b(G) has uncountable cofinality. Then:
∀ (Tα)α∈I ⊆ BallL(G) with |I | ≤ b(G)
∃ (X kα )α∈I ⊆ BallL(G)∗ (k = 1, . . . , n)
∃ T k ∈ L(G) (k = 1, . . . , n) s.t.
Tα =n∑
k=1
X kα 2 T k
So A(G) has (a slightly weakened form of) Fb(G).
Theorem (Hu–N. ’04)
A(G) has Mb(G)·ℵ0.
Topological centre basics Topological centre problems Topological centres as a tool
Method of proof:Factorization & Mazur’s property for A(G)
Theorem (Filali–Monfared–N. ’08)
G compact s.t. b(G) has uncountable cofinality. Then:
∀ (Tα)α∈I ⊆ BallL(G) with |I | ≤ b(G)
∃ (X kα )α∈I ⊆ BallL(G)∗ (k = 1, . . . , n)
∃ T k ∈ L(G) (k = 1, . . . , n) s.t.
Tα =n∑
k=1
X kα 2 T k
So A(G) has (a slightly weakened form of) Fb(G).
Theorem (Hu–N. ’04)
A(G) has Mb(G)·ℵ0.
Topological centre basics Topological centre problems Topological centres as a tool
Method of proof:Factorization & Mazur’s property for A(G)
Theorem (Filali–Monfared–N. ’08)
G compact s.t. b(G) has uncountable cofinality. Then:
∀ (Tα)α∈I ⊆ BallL(G) with |I | ≤ b(G)
∃ (X kα )α∈I ⊆ BallL(G)∗ (k = 1, . . . , n)
∃ T k ∈ L(G) (k = 1, . . . , n) s.t.
Tα =n∑
k=1
X kα 2 T k
So A(G) has (a slightly weakened form of) Fb(G).
Theorem (Hu–N. ’04)
A(G) has Mb(G)·ℵ0.
Topological centre basics Topological centre problems Topological centres as a tool
Strategy of (technical) proof
Tα =n∑
k=1
X kα 2 T k
use special central projections in L(G) arising from family ofcompact normal subgroups of G decreasing to e, of sizeI = b(G)
double the index set I (“family breeding”): α ; (α, β)
“separate” projections by coordinate functions of ab(G)-family of irred. unitary rep.s of Gthis gives a family of pairwise orthogonal projections in L(G)
truncate Tα with these and sum up (w∗) ; operators T k
X kα are w∗-cluster points in A(G)∗∗ of rep.s along β
Topological centre basics Topological centre problems Topological centres as a tool
Strategy of (technical) proof
Tα =n∑
k=1
X kα 2 T k
use special central projections in L(G) arising from family ofcompact normal subgroups of G decreasing to e, of sizeI = b(G)
double the index set I (“family breeding”): α ; (α, β)
“separate” projections by coordinate functions of ab(G)-family of irred. unitary rep.s of Gthis gives a family of pairwise orthogonal projections in L(G)
truncate Tα with these and sum up (w∗) ; operators T k
X kα are w∗-cluster points in A(G)∗∗ of rep.s along β
Topological centre basics Topological centre problems Topological centres as a tool
Strategy of (technical) proof
Tα =n∑
k=1
X kα 2 T k
use special central projections in L(G) arising from family ofcompact normal subgroups of G decreasing to e, of sizeI = b(G)
double the index set I (“family breeding”): α ; (α, β)
“separate” projections by coordinate functions of ab(G)-family of irred. unitary rep.s of Gthis gives a family of pairwise orthogonal projections in L(G)
truncate Tα with these and sum up (w∗) ; operators T k
X kα are w∗-cluster points in A(G)∗∗ of rep.s along β
Topological centre basics Topological centre problems Topological centres as a tool
Strategy of (technical) proof
Tα =n∑
k=1
X kα 2 T k
use special central projections in L(G) arising from family ofcompact normal subgroups of G decreasing to e, of sizeI = b(G)
double the index set I (“family breeding”): α ; (α, β)
“separate” projections by coordinate functions of ab(G)-family of irred. unitary rep.s of G
this gives a family of pairwise orthogonal projections in L(G)
truncate Tα with these and sum up (w∗) ; operators T k
X kα are w∗-cluster points in A(G)∗∗ of rep.s along β
Topological centre basics Topological centre problems Topological centres as a tool
Strategy of (technical) proof
Tα =n∑
k=1
X kα 2 T k
use special central projections in L(G) arising from family ofcompact normal subgroups of G decreasing to e, of sizeI = b(G)
double the index set I (“family breeding”): α ; (α, β)
“separate” projections by coordinate functions of ab(G)-family of irred. unitary rep.s of Gthis gives a family of pairwise orthogonal projections in L(G)
truncate Tα with these and sum up (w∗) ; operators T k
X kα are w∗-cluster points in A(G)∗∗ of rep.s along β
Topological centre basics Topological centre problems Topological centres as a tool
Strategy of (technical) proof
Tα =n∑
k=1
X kα 2 T k
use special central projections in L(G) arising from family ofcompact normal subgroups of G decreasing to e, of sizeI = b(G)
double the index set I (“family breeding”): α ; (α, β)
“separate” projections by coordinate functions of ab(G)-family of irred. unitary rep.s of Gthis gives a family of pairwise orthogonal projections in L(G)
truncate Tα with these and sum up (w∗) ; operators T k
X kα are w∗-cluster points in A(G)∗∗ of rep.s along β
Topological centre basics Topological centre problems Topological centres as a tool
Strategy of (technical) proof
Tα =n∑
k=1
X kα 2 T k
use special central projections in L(G) arising from family ofcompact normal subgroups of G decreasing to e, of sizeI = b(G)
double the index set I (“family breeding”): α ; (α, β)
“separate” projections by coordinate functions of ab(G)-family of irred. unitary rep.s of Gthis gives a family of pairwise orthogonal projections in L(G)
truncate Tα with these and sum up (w∗) ; operators T k
X kα are w∗-cluster points in A(G)∗∗ of rep.s along β
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres and multipliers
Problem (Lau–Ulger ’96)
A Banach algebra with BAI s.t. A∗ vN algebra. Let X ∈ Zr (A∗∗) .Consider X2 : A∗ 3 h 7→ X2h ∈ A∗ .Are Ker(X2) and X2(BallA∗) w∗-closed?
Theorem (Hu–N.–Ruan ’09)
No for A = A(SU(3))
Proof: Combine Losert’s result Z (A∗∗) 6= A with the following
Theorem (Hu–N.–Ruan ’09)
Assume A separable. Then, for X ∈ Zr (A∗∗):
X ∈ A ⇔ Ker(X2) and X2(BallA∗) are w∗-closed
This relies on work by Godefroy–Talagrand ’89 & N. ’01
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres and multipliers
Problem (Lau–Ulger ’96)
A Banach algebra with BAI s.t. A∗ vN algebra. Let X ∈ Zr (A∗∗) .Consider X2 : A∗ 3 h 7→ X2h ∈ A∗ .
Are Ker(X2) and X2(BallA∗) w∗-closed?
Theorem (Hu–N.–Ruan ’09)
No for A = A(SU(3))
Proof: Combine Losert’s result Z (A∗∗) 6= A with the following
Theorem (Hu–N.–Ruan ’09)
Assume A separable. Then, for X ∈ Zr (A∗∗):
X ∈ A ⇔ Ker(X2) and X2(BallA∗) are w∗-closed
This relies on work by Godefroy–Talagrand ’89 & N. ’01
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres and multipliers
Problem (Lau–Ulger ’96)
A Banach algebra with BAI s.t. A∗ vN algebra. Let X ∈ Zr (A∗∗) .Consider X2 : A∗ 3 h 7→ X2h ∈ A∗ .Are Ker(X2) and X2(BallA∗) w∗-closed?
Theorem (Hu–N.–Ruan ’09)
No for A = A(SU(3))
Proof: Combine Losert’s result Z (A∗∗) 6= A with the following
Theorem (Hu–N.–Ruan ’09)
Assume A separable. Then, for X ∈ Zr (A∗∗):
X ∈ A ⇔ Ker(X2) and X2(BallA∗) are w∗-closed
This relies on work by Godefroy–Talagrand ’89 & N. ’01
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres and multipliers
Problem (Lau–Ulger ’96)
A Banach algebra with BAI s.t. A∗ vN algebra. Let X ∈ Zr (A∗∗) .Consider X2 : A∗ 3 h 7→ X2h ∈ A∗ .Are Ker(X2) and X2(BallA∗) w∗-closed?
Theorem (Hu–N.–Ruan ’09)
No for A = A(SU(3))
Proof: Combine Losert’s result Z (A∗∗) 6= A with the following
Theorem (Hu–N.–Ruan ’09)
Assume A separable. Then, for X ∈ Zr (A∗∗):
X ∈ A ⇔ Ker(X2) and X2(BallA∗) are w∗-closed
This relies on work by Godefroy–Talagrand ’89 & N. ’01
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres and multipliers
Problem (Lau–Ulger ’96)
A Banach algebra with BAI s.t. A∗ vN algebra. Let X ∈ Zr (A∗∗) .Consider X2 : A∗ 3 h 7→ X2h ∈ A∗ .Are Ker(X2) and X2(BallA∗) w∗-closed?
Theorem (Hu–N.–Ruan ’09)
No for A = A(SU(3))
Proof: Combine Losert’s result Z (A∗∗) 6= A with the following
Theorem (Hu–N.–Ruan ’09)
Assume A separable. Then, for X ∈ Zr (A∗∗):
X ∈ A ⇔ Ker(X2) and X2(BallA∗) are w∗-closed
This relies on work by Godefroy–Talagrand ’89 & N. ’01
Topological centre basics Topological centre problems Topological centres as a tool
1 Topological centre basics
2 Topological centre problems
3 Topological centres as a tool
Topological centre basics Topological centre problems Topological centres as a tool
Ghahramani–Farhadi’s multiplier problem
Problem (Duncan–Hosseiniun ’79)
G locally compact group. Does the involution on L1(G) extend toan involution on its bidual?
Proposition (Farhadi–Ghahramani ’07)
1 This fails for non-discrete groups.
2 It also fails for all groups with the following property (∗):Consider any Φ : L∞(G)∗∗ → L∞(G)∗∗ normal & surjective;if Φ commutes with L1(G), then also with L1(G)∗∗.
Problem (Farhadi–Ghahramani ’07)
Does every group G satisfy (∗) ?
Topological centre basics Topological centre problems Topological centres as a tool
Ghahramani–Farhadi’s multiplier problem
Problem (Duncan–Hosseiniun ’79)
G locally compact group. Does the involution on L1(G) extend toan involution on its bidual?
Proposition (Farhadi–Ghahramani ’07)
1 This fails for non-discrete groups.
2 It also fails for all groups with the following property (∗):Consider any Φ : L∞(G)∗∗ → L∞(G)∗∗ normal & surjective;if Φ commutes with L1(G), then also with L1(G)∗∗.
Problem (Farhadi–Ghahramani ’07)
Does every group G satisfy (∗) ?
Topological centre basics Topological centre problems Topological centres as a tool
Ghahramani–Farhadi’s multiplier problem
Problem (Duncan–Hosseiniun ’79)
G locally compact group. Does the involution on L1(G) extend toan involution on its bidual?
Proposition (Farhadi–Ghahramani ’07)
1 This fails for non-discrete groups.
2 It also fails for all groups with the following property (∗):Consider any Φ : L∞(G)∗∗ → L∞(G)∗∗ normal & surjective;if Φ commutes with L1(G), then also with L1(G)∗∗.
Problem (Farhadi–Ghahramani ’07)
Does every group G satisfy (∗) ?
Topological centre basics Topological centre problems Topological centres as a tool
Ghahramani–Farhadi’s multiplier problem
Problem (Duncan–Hosseiniun ’79)
G locally compact group. Does the involution on L1(G) extend toan involution on its bidual?
Proposition (Farhadi–Ghahramani ’07)
1 This fails for non-discrete groups.
2 It also fails for all groups with the following property (∗):Consider any Φ : L∞(G)∗∗ → L∞(G)∗∗ normal & surjective;if Φ commutes with L1(G), then also with L1(G)∗∗.
Problem (Farhadi–Ghahramani ’07)
Does every group G satisfy (∗) ?
Topological centre basics Topological centre problems Topological centres as a tool
Ghahramani–Farhadi’s multiplier problem
Problem (Duncan–Hosseiniun ’79)
G locally compact group. Does the involution on L1(G) extend toan involution on its bidual?
Proposition (Farhadi–Ghahramani ’07)
1 This fails for non-discrete groups.
2 It also fails for all groups with the following property (∗):Consider any Φ : L∞(G)∗∗ → L∞(G)∗∗ normal & surjective;if Φ commutes with L1(G), then also with L1(G)∗∗.
Problem (Farhadi–Ghahramani ’07)
Does every group G satisfy (∗) ?
Topological centre basics Topological centre problems Topological centres as a tool
Solution to the multiplier problem
Theorem (N. ’08)
The problem has a negative answer for all infinite countablediscrete abelian groups.
For the proof, consider βG ⊆ `1(G)∗∗ :
compact right topological semigroup with first Arens product
Growth/Remainder/Corona:
G∗ := βG \ G
⇒ G∗ compact right topological semigroup
Topological centre basics Topological centre problems Topological centres as a tool
Solution to the multiplier problem
Theorem (N. ’08)
The problem has a negative answer for all infinite countablediscrete abelian groups.
For the proof, consider βG ⊆ `1(G)∗∗ :
compact right topological semigroup with first Arens product
Growth/Remainder/Corona:
G∗ := βG \ G
⇒ G∗ compact right topological semigroup
Topological centre basics Topological centre problems Topological centres as a tool
Solution to the multiplier problem
Theorem (N. ’08)
The problem has a negative answer for all infinite countablediscrete abelian groups.
For the proof, consider βG ⊆ `1(G)∗∗ :
compact right topological semigroup with first Arens product
Growth/Remainder/Corona:
G∗ := βG \ G
⇒ G∗ compact right topological semigroup
Topological centre basics Topological centre problems Topological centres as a tool
Solution to the multiplier problem
Theorem (N. ’08)
The problem has a negative answer for all infinite countablediscrete abelian groups.
For the proof, consider βG ⊆ `1(G)∗∗ :
compact right topological semigroup with first Arens product
Growth/Remainder/Corona:
G∗ := βG \ G
⇒ G∗ compact right topological semigroup
Topological centre basics Topological centre problems Topological centres as a tool
Module maps
m ∈ βG is called left cancellable if λm is injective on βG
Proposition (Dales–Lau–Strauss ’08)
m ∈ βG left cancellable ⇒ λm : `1(G)∗∗ → `1(G)∗∗ isometry
Write A := `1(G)∗∗.
∃ m ∈ G∗ ⊆ A such that m is left cancellable in βGProposition ⇒ Φ := λ∗m : A∗ → A∗ (normal &) surjective
Need to show:
1 Φ is a right `1(G)-module map
2 Φ is not a right `1(G)∗∗-module map
Topological centre basics Topological centre problems Topological centres as a tool
Module maps
m ∈ βG is called left cancellable if λm is injective on βG
Proposition (Dales–Lau–Strauss ’08)
m ∈ βG left cancellable ⇒ λm : `1(G)∗∗ → `1(G)∗∗ isometry
Write A := `1(G)∗∗.
∃ m ∈ G∗ ⊆ A such that m is left cancellable in βGProposition ⇒ Φ := λ∗m : A∗ → A∗ (normal &) surjective
Need to show:
1 Φ is a right `1(G)-module map
2 Φ is not a right `1(G)∗∗-module map
Topological centre basics Topological centre problems Topological centres as a tool
Module maps
m ∈ βG is called left cancellable if λm is injective on βG
Proposition (Dales–Lau–Strauss ’08)
m ∈ βG left cancellable ⇒ λm : `1(G)∗∗ → `1(G)∗∗ isometry
Write A := `1(G)∗∗.
∃ m ∈ G∗ ⊆ A such that m is left cancellable in βG
Proposition ⇒ Φ := λ∗m : A∗ → A∗ (normal &) surjective
Need to show:
1 Φ is a right `1(G)-module map
2 Φ is not a right `1(G)∗∗-module map
Topological centre basics Topological centre problems Topological centres as a tool
Module maps
m ∈ βG is called left cancellable if λm is injective on βG
Proposition (Dales–Lau–Strauss ’08)
m ∈ βG left cancellable ⇒ λm : `1(G)∗∗ → `1(G)∗∗ isometry
Write A := `1(G)∗∗.
∃ m ∈ G∗ ⊆ A such that m is left cancellable in βGProposition ⇒ Φ := λ∗m : A∗ → A∗ (normal &) surjective
Need to show:
1 Φ is a right `1(G)-module map
2 Φ is not a right `1(G)∗∗-module map
Topological centre basics Topological centre problems Topological centres as a tool
Module maps
m ∈ βG is called left cancellable if λm is injective on βG
Proposition (Dales–Lau–Strauss ’08)
m ∈ βG left cancellable ⇒ λm : `1(G)∗∗ → `1(G)∗∗ isometry
Write A := `1(G)∗∗.
∃ m ∈ G∗ ⊆ A such that m is left cancellable in βGProposition ⇒ Φ := λ∗m : A∗ → A∗ (normal &) surjective
Need to show:
1 Φ is a right `1(G)-module map
2 Φ is not a right `1(G)∗∗-module map
Topological centre basics Topological centre problems Topological centres as a tool
Φ = λ∗m is a right `1(G)-module map
Recall: A = `1(G)∗∗
∀ H ∈ A∗, a ∈ `1(G) ⊆ A, b ∈ A
〈Φ(H2a), b〉 = 〈H, a ∗m ∗ b〉
But a ∈ `1(G) = Z (A), so a commutes with m ∈ G∗ ⊆ A :
〈Φ(H2a), b〉 = 〈H,m ∗ a ∗ b〉 = 〈Φ(H)2a, b〉
as desired.
Topological centre basics Topological centre problems Topological centres as a tool
Φ = λ∗m is a right `1(G)-module map
Recall: A = `1(G)∗∗
∀ H ∈ A∗, a ∈ `1(G) ⊆ A, b ∈ A
〈Φ(H2a), b〉 = 〈H, a ∗m ∗ b〉
But a ∈ `1(G) = Z (A), so a commutes with m ∈ G∗ ⊆ A :
〈Φ(H2a), b〉 = 〈H,m ∗ a ∗ b〉 = 〈Φ(H)2a, b〉
as desired.
Topological centre basics Topological centre problems Topological centres as a tool
Φ = λ∗m is a right `1(G)-module map
Recall: A = `1(G)∗∗
∀ H ∈ A∗, a ∈ `1(G) ⊆ A, b ∈ A
〈Φ(H2a), b〉 = 〈H, a ∗m ∗ b〉
But a ∈ `1(G) = Z (A), so a commutes with m ∈ G∗ ⊆ A :
〈Φ(H2a), b〉 = 〈H,m ∗ a ∗ b〉 = 〈Φ(H)2a, b〉
as desired.
Topological centre basics Topological centre problems Topological centres as a tool
Φ = λ∗m is not a right `1(G)∗∗-module map
Recall: A = `1(G)∗∗
Suppose Φ is a right A-module map
⇒ ∀ H ∈ A∗, a, b ∈ A
〈H, a ∗m ∗ b〉 = 〈Φ(H2a), b〉 = 〈Φ(H)2a, b〉 = 〈H,m ∗ a ∗ b〉
⇒ a ∗m ∗ b = m ∗ a ∗ b ∀ a, b ∈ A⇒ (with b = δe) m ∈ Z (A) = `1(G)
This contradicts m ∈ G∗.
Topological centre basics Topological centre problems Topological centres as a tool
Φ = λ∗m is not a right `1(G)∗∗-module map
Recall: A = `1(G)∗∗
Suppose Φ is a right A-module map
⇒ ∀ H ∈ A∗, a, b ∈ A
〈H, a ∗m ∗ b〉 = 〈Φ(H2a), b〉 = 〈Φ(H)2a, b〉 = 〈H,m ∗ a ∗ b〉
⇒ a ∗m ∗ b = m ∗ a ∗ b ∀ a, b ∈ A⇒ (with b = δe) m ∈ Z (A) = `1(G)
This contradicts m ∈ G∗.
Topological centre basics Topological centre problems Topological centres as a tool
Φ = λ∗m is not a right `1(G)∗∗-module map
Recall: A = `1(G)∗∗
Suppose Φ is a right A-module map
⇒ ∀ H ∈ A∗, a, b ∈ A
〈H, a ∗m ∗ b〉 = 〈Φ(H2a), b〉 = 〈Φ(H)2a, b〉 = 〈H,m ∗ a ∗ b〉
⇒ a ∗m ∗ b = m ∗ a ∗ b ∀ a, b ∈ A⇒ (with b = δe) m ∈ Z (A) = `1(G)
This contradicts m ∈ G∗.
Topological centre basics Topological centre problems Topological centres as a tool
Φ = λ∗m is not a right `1(G)∗∗-module map
Recall: A = `1(G)∗∗
Suppose Φ is a right A-module map
⇒ ∀ H ∈ A∗, a, b ∈ A
〈H, a ∗m ∗ b〉 = 〈Φ(H2a), b〉 = 〈Φ(H)2a, b〉 = 〈H,m ∗ a ∗ b〉
⇒ a ∗m ∗ b = m ∗ a ∗ b ∀ a, b ∈ A⇒ (with b = δe) m ∈ Z (A) = `1(G)
This contradicts m ∈ G∗.
Topological centre basics Topological centre problems Topological centres as a tool
Group actions and invariant means
Solution to the Banach–Ruziewicz Problem (Banach ’23;Margulis/Sullivan ’80/’81; Drinfeld ’84)
Except for n = 1, Lebesgue measure is the only invariant mean onL∞(Sn) for the O(n + 1)-action.
What about the discrete situation?
Of course, for G y G: ∃! inv. mean on `∞(G) ⇔ G finite
Quick proof using topological centres (Lau ’86):
unique inv. mean M
⇒ M ∈ Z`(`∞(G)∗) = `1(G)
⇒ M finite Haar measure, so G finite!
What about general actions G y X ?
Topological centre basics Topological centre problems Topological centres as a tool
Group actions and invariant means
Solution to the Banach–Ruziewicz Problem (Banach ’23;Margulis/Sullivan ’80/’81; Drinfeld ’84)
Except for n = 1, Lebesgue measure is the only invariant mean onL∞(Sn) for the O(n + 1)-action.
What about the discrete situation?
Of course, for G y G: ∃! inv. mean on `∞(G) ⇔ G finite
Quick proof using topological centres (Lau ’86):
unique inv. mean M
⇒ M ∈ Z`(`∞(G)∗) = `1(G)
⇒ M finite Haar measure, so G finite!
What about general actions G y X ?
Topological centre basics Topological centre problems Topological centres as a tool
Group actions and invariant means
Solution to the Banach–Ruziewicz Problem (Banach ’23;Margulis/Sullivan ’80/’81; Drinfeld ’84)
Except for n = 1, Lebesgue measure is the only invariant mean onL∞(Sn) for the O(n + 1)-action.
What about the discrete situation?
Of course, for G y G: ∃! inv. mean on `∞(G) ⇔ G finite
Quick proof using topological centres (Lau ’86):
unique inv. mean M
⇒ M ∈ Z`(`∞(G)∗) = `1(G)
⇒ M finite Haar measure, so G finite!
What about general actions G y X ?
Topological centre basics Topological centre problems Topological centres as a tool
Group actions and invariant means
Solution to the Banach–Ruziewicz Problem (Banach ’23;Margulis/Sullivan ’80/’81; Drinfeld ’84)
Except for n = 1, Lebesgue measure is the only invariant mean onL∞(Sn) for the O(n + 1)-action.
What about the discrete situation?
Of course, for G y G: ∃! inv. mean on `∞(G) ⇔ G finite
Quick proof using topological centres (Lau ’86):
unique inv. mean M
⇒ M ∈ Z`(`∞(G)∗) = `1(G)
⇒ M finite Haar measure, so G finite!
What about general actions G y X ?
Topological centre basics Topological centre problems Topological centres as a tool
Group actions and invariant means
Solution to the Banach–Ruziewicz Problem (Banach ’23;Margulis/Sullivan ’80/’81; Drinfeld ’84)
Except for n = 1, Lebesgue measure is the only invariant mean onL∞(Sn) for the O(n + 1)-action.
What about the discrete situation?
Of course, for G y G: ∃! inv. mean on `∞(G) ⇔ G finite
Quick proof using topological centres (Lau ’86):
unique inv. mean M
⇒ M ∈ Z`(`∞(G)∗) = `1(G)
⇒ M finite Haar measure, so G finite!
What about general actions G y X ?
Topological centre basics Topological centre problems Topological centres as a tool
Group actions and invariant means
Solution to the Banach–Ruziewicz Problem (Banach ’23;Margulis/Sullivan ’80/’81; Drinfeld ’84)
Except for n = 1, Lebesgue measure is the only invariant mean onL∞(Sn) for the O(n + 1)-action.
What about the discrete situation?
Of course, for G y G: ∃! inv. mean on `∞(G) ⇔ G finite
Quick proof using topological centres (Lau ’86):
unique inv. mean M
⇒ M ∈ Z`(`∞(G)∗) = `1(G)
⇒ M finite Haar measure, so G finite!
What about general actions G y X ?
Topological centre basics Topological centre problems Topological centres as a tool
Group actions and invariant means
Solution to the Banach–Ruziewicz Problem (Banach ’23;Margulis/Sullivan ’80/’81; Drinfeld ’84)
Except for n = 1, Lebesgue measure is the only invariant mean onL∞(Sn) for the O(n + 1)-action.
What about the discrete situation?
Of course, for G y G: ∃! inv. mean on `∞(G) ⇔ G finite
Quick proof using topological centres (Lau ’86):
unique inv. mean M
⇒ M ∈ Z`(`∞(G)∗) = `1(G)
⇒ M finite Haar measure, so G finite!
What about general actions G y X ?
Topological centre basics Topological centre problems Topological centres as a tool
Group actions and invariant means
Solution to the Banach–Ruziewicz Problem (Banach ’23;Margulis/Sullivan ’80/’81; Drinfeld ’84)
Except for n = 1, Lebesgue measure is the only invariant mean onL∞(Sn) for the O(n + 1)-action.
What about the discrete situation?
Of course, for G y G: ∃! inv. mean on `∞(G) ⇔ G finite
Quick proof using topological centres (Lau ’86):
unique inv. mean M
⇒ M ∈ Z`(`∞(G)∗) = `1(G)
⇒ M finite Haar measure, so G finite!
What about general actions G y X ?
Topological centre basics Topological centre problems Topological centres as a tool
Group actions and invariant means
Solution to the Banach–Ruziewicz Problem (Banach ’23;Margulis/Sullivan ’80/’81; Drinfeld ’84)
Except for n = 1, Lebesgue measure is the only invariant mean onL∞(Sn) for the O(n + 1)-action.
What about the discrete situation?
Of course, for G y G: ∃! inv. mean on `∞(G) ⇔ G finite
Quick proof using topological centres (Lau ’86):
unique inv. mean M
⇒ M ∈ Z`(`∞(G)∗) = `1(G)
⇒ M finite Haar measure, so G finite!
What about general actions G y X ?
Topological centre basics Topological centre problems Topological centres as a tool
An independence result for general actions
Theorem (Foreman ’94)
The statement “∃ locally finite group G of permutations of N witha unique invariant mean on `∞(N)” is independent of ZFC!
Theorem (Foreman ’94)
CH ⇒ ∃ locally finite group of permutations of N, of size c,with a unique invariant mean on `∞(N)
Theorem (Rosenblatt–Talagrand ’81)
Infinite countable groups never admit a unique invariant mean.
How many?
Theorem (N.–Pachl–Steprans ’09)
G y X with G,X infinite countable.G amenable ⇒ ∃ 2c many invariant means on `∞(X )
Topological centre basics Topological centre problems Topological centres as a tool
An independence result for general actions
Theorem (Foreman ’94)
The statement “∃ locally finite group G of permutations of N witha unique invariant mean on `∞(N)” is independent of ZFC!
Theorem (Foreman ’94)
CH ⇒ ∃ locally finite group of permutations of N, of size c,with a unique invariant mean on `∞(N)
Theorem (Rosenblatt–Talagrand ’81)
Infinite countable groups never admit a unique invariant mean.
How many?
Theorem (N.–Pachl–Steprans ’09)
G y X with G,X infinite countable.G amenable ⇒ ∃ 2c many invariant means on `∞(X )
Topological centre basics Topological centre problems Topological centres as a tool
An independence result for general actions
Theorem (Foreman ’94)
The statement “∃ locally finite group G of permutations of N witha unique invariant mean on `∞(N)” is independent of ZFC!
Theorem (Foreman ’94)
CH ⇒ ∃ locally finite group of permutations of N, of size c,with a unique invariant mean on `∞(N)
Theorem (Rosenblatt–Talagrand ’81)
Infinite countable groups never admit a unique invariant mean.
How many?
Theorem (N.–Pachl–Steprans ’09)
G y X with G,X infinite countable.G amenable ⇒ ∃ 2c many invariant means on `∞(X )
Topological centre basics Topological centre problems Topological centres as a tool
An independence result for general actions
Theorem (Foreman ’94)
The statement “∃ locally finite group G of permutations of N witha unique invariant mean on `∞(N)” is independent of ZFC!
Theorem (Foreman ’94)
CH ⇒ ∃ locally finite group of permutations of N, of size c,with a unique invariant mean on `∞(N)
Theorem (Rosenblatt–Talagrand ’81)
Infinite countable groups never admit a unique invariant mean.
How many?
Theorem (N.–Pachl–Steprans ’09)
G y X with G,X infinite countable.G amenable ⇒ ∃ 2c many invariant means on `∞(X )
Topological centre basics Topological centre problems Topological centres as a tool
An independence result for general actions
Theorem (Foreman ’94)
The statement “∃ locally finite group G of permutations of N witha unique invariant mean on `∞(N)” is independent of ZFC!
Theorem (Foreman ’94)
CH ⇒ ∃ locally finite group of permutations of N, of size c,with a unique invariant mean on `∞(N)
Theorem (Rosenblatt–Talagrand ’81)
Infinite countable groups never admit a unique invariant mean.
How many?
Theorem (N.–Pachl–Steprans ’09)
G y X with G,X infinite countable.G amenable ⇒ ∃ 2c many invariant means on `∞(X )
Topological centre basics Topological centre problems Topological centres as a tool
An independence result for general actions
Theorem (Foreman ’94)
The statement “∃ locally finite group G of permutations of N witha unique invariant mean on `∞(N)” is independent of ZFC!
Theorem (Foreman ’94)
CH ⇒ ∃ locally finite group of permutations of N, of size c,with a unique invariant mean on `∞(N)
Theorem (Rosenblatt–Talagrand ’81)
Infinite countable groups never admit a unique invariant mean.
How many?
Theorem (N.–Pachl–Steprans ’09)
G y X with G,X infinite countable.G amenable ⇒ ∃ 2c many invariant means on `∞(X )
Topological centre basics Topological centre problems Topological centres as a tool
Arens type product and topological centrefor group actions
Definition
G y X .
1 For x ∈ βX and h ∈ `∞(X ) = C (βX ) define x2h ∈ `∞(G) by
(x2h)(g) := h(gx)
2 Define a “convolution” `∞(G)∗ × βX → `∞(X )∗ by
〈m2x , h〉 := 〈m, x2h〉
Definition
Zt(G,X ) := { m ∈ `∞(G)∗ | βX 3 x 7→ m2x w∗-cont. }
Topological centre basics Topological centre problems Topological centres as a tool
Arens type product and topological centrefor group actions
Definition
G y X .
1 For x ∈ βX and h ∈ `∞(X ) = C (βX ) define x2h ∈ `∞(G) by
(x2h)(g) := h(gx)
2 Define a “convolution” `∞(G)∗ × βX → `∞(X )∗ by
〈m2x , h〉 := 〈m, x2h〉
Definition
Zt(G,X ) := { m ∈ `∞(G)∗ | βX 3 x 7→ m2x w∗-cont. }
Topological centre basics Topological centre problems Topological centres as a tool
Arens type product and topological centrefor group actions
Definition
G y X .
1 For x ∈ βX and h ∈ `∞(X ) = C (βX )
define x2h ∈ `∞(G) by
(x2h)(g) := h(gx)
2 Define a “convolution” `∞(G)∗ × βX → `∞(X )∗ by
〈m2x , h〉 := 〈m, x2h〉
Definition
Zt(G,X ) := { m ∈ `∞(G)∗ | βX 3 x 7→ m2x w∗-cont. }
Topological centre basics Topological centre problems Topological centres as a tool
Arens type product and topological centrefor group actions
Definition
G y X .
1 For x ∈ βX and h ∈ `∞(X ) = C (βX ) define x2h ∈ `∞(G) by
(x2h)(g) := h(gx)
2 Define a “convolution” `∞(G)∗ × βX → `∞(X )∗ by
〈m2x , h〉 := 〈m, x2h〉
Definition
Zt(G,X ) := { m ∈ `∞(G)∗ | βX 3 x 7→ m2x w∗-cont. }
Topological centre basics Topological centre problems Topological centres as a tool
Arens type product and topological centrefor group actions
Definition
G y X .
1 For x ∈ βX and h ∈ `∞(X ) = C (βX ) define x2h ∈ `∞(G) by
(x2h)(g) := h(gx)
2 Define a “convolution” `∞(G)∗ × βX → `∞(X )∗ by
〈m2x , h〉 := 〈m, x2h〉
Definition
Zt(G,X ) := { m ∈ `∞(G)∗ | βX 3 x 7→ m2x w∗-cont. }
Topological centre basics Topological centre problems Topological centres as a tool
Arens type product and topological centrefor group actions
Definition
G y X .
1 For x ∈ βX and h ∈ `∞(X ) = C (βX ) define x2h ∈ `∞(G) by
(x2h)(g) := h(gx)
2 Define a “convolution” `∞(G)∗ × βX → `∞(X )∗ by
〈m2x , h〉 := 〈m, x2h〉
Definition
Zt(G,X ) := { m ∈ `∞(G)∗ | βX 3 x 7→ m2x w∗-cont. }
Topological centre basics Topological centre problems Topological centres as a tool
Foreman’s group has non-trivial topological centre!
Theorem (N.–Pachl–Steprans ’09)
G y X with G amenable and Zt(G,X ) = `1(G).If the number of inv. means on `∞(X ) is finite, then G is finite.
Corollary (N.–Pachl–Steprans ’09)
CH ⇒ Zt(G,X ) 6= `1(G) for Foreman’s group!
But we (N.–P.–S.) have a class of infinite groups of size < c withZt(G,N) = `1(G)
Topological centre basics Topological centre problems Topological centres as a tool
Foreman’s group has non-trivial topological centre!
Theorem (N.–Pachl–Steprans ’09)
G y X with G amenable and Zt(G,X ) = `1(G).
If the number of inv. means on `∞(X ) is finite, then G is finite.
Corollary (N.–Pachl–Steprans ’09)
CH ⇒ Zt(G,X ) 6= `1(G) for Foreman’s group!
But we (N.–P.–S.) have a class of infinite groups of size < c withZt(G,N) = `1(G)
Topological centre basics Topological centre problems Topological centres as a tool
Foreman’s group has non-trivial topological centre!
Theorem (N.–Pachl–Steprans ’09)
G y X with G amenable and Zt(G,X ) = `1(G).If the number of inv. means on `∞(X ) is finite, then G is finite.
Corollary (N.–Pachl–Steprans ’09)
CH ⇒ Zt(G,X ) 6= `1(G) for Foreman’s group!
But we (N.–P.–S.) have a class of infinite groups of size < c withZt(G,N) = `1(G)
Topological centre basics Topological centre problems Topological centres as a tool
Foreman’s group has non-trivial topological centre!
Theorem (N.–Pachl–Steprans ’09)
G y X with G amenable and Zt(G,X ) = `1(G).If the number of inv. means on `∞(X ) is finite, then G is finite.
Corollary (N.–Pachl–Steprans ’09)
CH ⇒ Zt(G,X ) 6= `1(G) for Foreman’s group!
But we (N.–P.–S.) have a class of infinite groups of size < c withZt(G,N) = `1(G)
Topological centre basics Topological centre problems Topological centres as a tool
Foreman’s group has non-trivial topological centre!
Theorem (N.–Pachl–Steprans ’09)
G y X with G amenable and Zt(G,X ) = `1(G).If the number of inv. means on `∞(X ) is finite, then G is finite.
Corollary (N.–Pachl–Steprans ’09)
CH ⇒ Zt(G,X ) 6= `1(G) for Foreman’s group!
But we (N.–P.–S.) have a class of infinite groups of size < c withZt(G,N) = `1(G)
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres for quantum group algebras
Definition
Hopf–von Neumann algebra (M, Γ)
M von Neumann algebra
Γ : M → M⊗M co-multiplication
Examples
M = L∞(G) = L1(G)∗
Γ = adjoint of convolution product ∗
M = L(G) = A(G)∗
Γ = adjoint of pointwise product •
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres for quantum group algebras
Definition
Hopf–von Neumann algebra (M, Γ)
M von Neumann algebra
Γ : M → M⊗M co-multiplication
Examples
M = L∞(G) = L1(G)∗
Γ = adjoint of convolution product ∗
M = L(G) = A(G)∗
Γ = adjoint of pointwise product •
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres for quantum group algebras
Definition
Hopf–von Neumann algebra (M, Γ)
M von Neumann algebra
Γ : M → M⊗M co-multiplication
Examples
M = L∞(G) = L1(G)∗
Γ = adjoint of convolution product ∗
M = L(G) = A(G)∗
Γ = adjoint of pointwise product •
Topological centre basics Topological centre problems Topological centres as a tool
Topological centres for quantum group algebras
Definition
Hopf–von Neumann algebra (M, Γ)
M von Neumann algebra
Γ : M → M⊗M co-multiplication
Examples
M = L∞(G) = L1(G)∗
Γ = adjoint of convolution product ∗
M = L(G) = A(G)∗
Γ = adjoint of pointwise product •
Topological centre basics Topological centre problems Topological centres as a tool
Locally compact quantum groups
Non-commutative integration
N.s.f. weight λ : M+ → [0,∞]
Mλ := lin { x ∈ M+ | λ(x) <∞ }
Definition (Kustermans–Vaes ’00)
LC Quantum Group G = (M, Γ, λ, ρ)
λ left Haar weight on M:
λ((f ⊗ id)Γx) = 〈f , 1〉 λ(x) ∀ f ∈ M∗, x ∈ Mλ
ρ right Haar weight on M:
ρ((id⊗ f )Γx) = 〈f , 1〉 ρ(x) ∀ f ∈ M∗, x ∈ Mρ
Theorem (Kustermans–Vaes ’00)
“Pontryagin duality”G ∼= G
Topological centre basics Topological centre problems Topological centres as a tool
Locally compact quantum groups
Non-commutative integration
N.s.f. weight λ : M+ → [0,∞]
Mλ := lin { x ∈ M+ | λ(x) <∞ }
Definition (Kustermans–Vaes ’00)
LC Quantum Group G = (M, Γ, λ, ρ)
λ left Haar weight on M:
λ((f ⊗ id)Γx) = 〈f , 1〉 λ(x) ∀ f ∈ M∗, x ∈ Mλ
ρ right Haar weight on M:
ρ((id⊗ f )Γx) = 〈f , 1〉 ρ(x) ∀ f ∈ M∗, x ∈ Mρ
Theorem (Kustermans–Vaes ’00)
“Pontryagin duality”G ∼= G
Topological centre basics Topological centre problems Topological centres as a tool
Locally compact quantum groups
Non-commutative integration
N.s.f. weight λ : M+ → [0,∞]
Mλ := lin { x ∈ M+ | λ(x) <∞ }
Definition (Kustermans–Vaes ’00)
LC Quantum Group G = (M, Γ, λ, ρ)
λ left Haar weight on M:
λ((f ⊗ id)Γx) = 〈f , 1〉 λ(x) ∀ f ∈ M∗, x ∈ Mλ
ρ right Haar weight on M:
ρ((id⊗ f )Γx) = 〈f , 1〉 ρ(x) ∀ f ∈ M∗, x ∈ Mρ
Theorem (Kustermans–Vaes ’00)
“Pontryagin duality”G ∼= G
Topological centre basics Topological centre problems Topological centres as a tool
Locally compact quantum groups
Non-commutative integration
N.s.f. weight λ : M+ → [0,∞]
Mλ := lin { x ∈ M+ | λ(x) <∞ }
Definition (Kustermans–Vaes ’00)
LC Quantum Group G = (M, Γ, λ, ρ)
λ left Haar weight on M:
λ((f ⊗ id)Γx) = 〈f , 1〉 λ(x) ∀ f ∈ M∗, x ∈ Mλ
ρ right Haar weight on M:
ρ((id⊗ f )Γx) = 〈f , 1〉 ρ(x) ∀ f ∈ M∗, x ∈ Mρ
Theorem (Kustermans–Vaes ’00)
“Pontryagin duality”G ∼= G
Topological centre basics Topological centre problems Topological centres as a tool
Locally compact quantum groups
Non-commutative integration
N.s.f. weight λ : M+ → [0,∞]
Mλ := lin { x ∈ M+ | λ(x) <∞ }
Definition (Kustermans–Vaes ’00)
LC Quantum Group G = (M, Γ, λ, ρ)
λ left Haar weight on M:
λ((f ⊗ id)Γx) = 〈f , 1〉 λ(x) ∀ f ∈ M∗, x ∈ Mλ
ρ right Haar weight on M:
ρ((id⊗ f )Γx) = 〈f , 1〉 ρ(x) ∀ f ∈ M∗, x ∈ Mρ
Theorem (Kustermans–Vaes ’00)
“Pontryagin duality”G ∼= G
Topological centre basics Topological centre problems Topological centres as a tool
Algebras over quantum groups
L∞(G) := M L1(G) := M∗ L2(G) := L2(M, λ)
L1(G) Banach algebra via f ∗ g = Γ∗(f ⊗ g)
LUC(G) := lin L∞(G)2L1(G) ⊆ L∞(G)
WAP(G) := { T ∈ L∞(G) | L1(G) 3 f 7→ T2f weakly compact }
C0(G) := { (id⊗ τ)(W ) | τ ∈ T (L2(G)) }‖·‖
where W left fundamental unitary: Γ(x) = W ∗(1⊗ x)W
Topological centre basics Topological centre problems Topological centres as a tool
Algebras over quantum groups
L∞(G) := M L1(G) := M∗ L2(G) := L2(M, λ)
L1(G) Banach algebra via f ∗ g = Γ∗(f ⊗ g)
LUC(G) := lin L∞(G)2L1(G) ⊆ L∞(G)
WAP(G) := { T ∈ L∞(G) | L1(G) 3 f 7→ T2f weakly compact }
C0(G) := { (id⊗ τ)(W ) | τ ∈ T (L2(G)) }‖·‖
where W left fundamental unitary: Γ(x) = W ∗(1⊗ x)W
Topological centre basics Topological centre problems Topological centres as a tool
Algebras over quantum groups
L∞(G) := M L1(G) := M∗ L2(G) := L2(M, λ)
L1(G) Banach algebra via f ∗ g = Γ∗(f ⊗ g)
LUC(G) := lin L∞(G)2L1(G) ⊆ L∞(G)
WAP(G) := { T ∈ L∞(G) | L1(G) 3 f 7→ T2f weakly compact }
C0(G) := { (id⊗ τ)(W ) | τ ∈ T (L2(G)) }‖·‖
where W left fundamental unitary: Γ(x) = W ∗(1⊗ x)W
Topological centre basics Topological centre problems Topological centres as a tool
Algebras over quantum groups
L∞(G) := M L1(G) := M∗ L2(G) := L2(M, λ)
L1(G) Banach algebra via f ∗ g = Γ∗(f ⊗ g)
LUC(G) := lin L∞(G)2L1(G) ⊆ L∞(G)
WAP(G) := { T ∈ L∞(G) | L1(G) 3 f 7→ T2f weakly compact }
C0(G) := { (id⊗ τ)(W ) | τ ∈ T (L2(G)) }‖·‖
where W left fundamental unitary: Γ(x) = W ∗(1⊗ x)W
Topological centre basics Topological centre problems Topological centres as a tool
Algebras over quantum groups
L∞(G) := M L1(G) := M∗ L2(G) := L2(M, λ)
L1(G) Banach algebra via f ∗ g = Γ∗(f ⊗ g)
LUC(G) := lin L∞(G)2L1(G) ⊆ L∞(G)
WAP(G) := { T ∈ L∞(G) | L1(G) 3 f 7→ T2f weakly compact }
C0(G) := { (id⊗ τ)(W ) | τ ∈ T (L2(G)) }‖·‖
where W left fundamental unitary: Γ(x) = W ∗(1⊗ x)W
Topological centre basics Topological centre problems Topological centres as a tool
Algebras over quantum groups
L∞(G) := M L1(G) := M∗ L2(G) := L2(M, λ)
L1(G) Banach algebra via f ∗ g = Γ∗(f ⊗ g)
LUC(G) := lin L∞(G)2L1(G) ⊆ L∞(G)
WAP(G) := { T ∈ L∞(G) | L1(G) 3 f 7→ T2f weakly compact }
C0(G) := { (id⊗ τ)(W ) | τ ∈ T (L2(G)) }‖·‖
where W left fundamental unitary: Γ(x) = W ∗(1⊗ x)W
Topological centre basics Topological centre problems Topological centres as a tool
Characterization of discrete quantum groups
Theorem (Hu–N.–Ruan ’09)
Assume G co-amenable (i.e., L1(G) has BAI) s.t. L1(G) separable.TFAE:
G discrete (i.e., L1(G) unital)
LUC(G) = L∞(G)
Topological centre basics Topological centre problems Topological centres as a tool
Characterization of discrete quantum groups
Theorem (Hu–N.–Ruan ’09)
Assume G co-amenable (i.e., L1(G) has BAI) s.t. L1(G) separable.
TFAE:
G discrete (i.e., L1(G) unital)
LUC(G) = L∞(G)
Topological centre basics Topological centre problems Topological centres as a tool
Characterization of discrete quantum groups
Theorem (Hu–N.–Ruan ’09)
Assume G co-amenable (i.e., L1(G) has BAI) s.t. L1(G) separable.TFAE:
G discrete (i.e., L1(G) unital)
LUC(G) = L∞(G)
Topological centre basics Topological centre problems Topological centres as a tool
Characterization of compact quantum groups
Since LUC(G) ⊆ L∞(G) we have
LUC(G)∗ ←− L1(G)∗∗
; Transport of left Arens product ; LUC(G)∗ Banach algebra
Zt(LUC(G)∗) := { X ∈ LUC(G)∗ | Y 7→ X2Y w∗-cont. }
Theorem (Hu–N.–Ruan ’09)
TFAE:
G compact (i.e., C0(G) unital)
LUC(G) ⊆WAP(G) and Zt(LUC(G)∗) = M(G)
Question G = L(G) with G discrete?⇒ WAP(G) ⊆ LUC(G)
If yes, then there is NO infinite G with A(G) Arens regular!Open for Olshanskii group . . .
Topological centre basics Topological centre problems Topological centres as a tool
Characterization of compact quantum groups
Since LUC(G) ⊆ L∞(G) we have
LUC(G)∗ ←− L1(G)∗∗
; Transport of left Arens product ; LUC(G)∗ Banach algebra
Zt(LUC(G)∗) := { X ∈ LUC(G)∗ | Y 7→ X2Y w∗-cont. }
Theorem (Hu–N.–Ruan ’09)
TFAE:
G compact (i.e., C0(G) unital)
LUC(G) ⊆WAP(G) and Zt(LUC(G)∗) = M(G)
Question G = L(G) with G discrete?⇒ WAP(G) ⊆ LUC(G)
If yes, then there is NO infinite G with A(G) Arens regular!Open for Olshanskii group . . .
Topological centre basics Topological centre problems Topological centres as a tool
Characterization of compact quantum groups
Since LUC(G) ⊆ L∞(G) we have
LUC(G)∗ ←− L1(G)∗∗
; Transport of left Arens product ; LUC(G)∗ Banach algebra
Zt(LUC(G)∗) := { X ∈ LUC(G)∗ | Y 7→ X2Y w∗-cont. }
Theorem (Hu–N.–Ruan ’09)
TFAE:
G compact (i.e., C0(G) unital)
LUC(G) ⊆WAP(G) and Zt(LUC(G)∗) = M(G)
Question G = L(G) with G discrete?⇒ WAP(G) ⊆ LUC(G)
If yes, then there is NO infinite G with A(G) Arens regular!Open for Olshanskii group . . .
Topological centre basics Topological centre problems Topological centres as a tool
Characterization of compact quantum groups
Since LUC(G) ⊆ L∞(G) we have
LUC(G)∗ ←− L1(G)∗∗
; Transport of left Arens product ; LUC(G)∗ Banach algebra
Zt(LUC(G)∗) := { X ∈ LUC(G)∗ | Y 7→ X2Y w∗-cont. }
Theorem (Hu–N.–Ruan ’09)
TFAE:
G compact (i.e., C0(G) unital)
LUC(G) ⊆WAP(G) and Zt(LUC(G)∗) = M(G)
Question G = L(G) with G discrete?⇒ WAP(G) ⊆ LUC(G)
If yes, then there is NO infinite G with A(G) Arens regular!Open for Olshanskii group . . .
Topological centre basics Topological centre problems Topological centres as a tool
Characterization of compact quantum groups
Since LUC(G) ⊆ L∞(G) we have
LUC(G)∗ ←− L1(G)∗∗
; Transport of left Arens product ; LUC(G)∗ Banach algebra
Zt(LUC(G)∗) := { X ∈ LUC(G)∗ | Y 7→ X2Y w∗-cont. }
Theorem (Hu–N.–Ruan ’09)
TFAE:
G compact (i.e., C0(G) unital)
LUC(G) ⊆WAP(G) and Zt(LUC(G)∗) = M(G)
Question G = L(G) with G discrete?⇒ WAP(G) ⊆ LUC(G)
If yes, then there is NO infinite G with A(G) Arens regular!Open for Olshanskii group . . .
Topological centre basics Topological centre problems Topological centres as a tool
Characterization of compact quantum groups
Since LUC(G) ⊆ L∞(G) we have
LUC(G)∗ ←− L1(G)∗∗
; Transport of left Arens product ; LUC(G)∗ Banach algebra
Zt(LUC(G)∗) := { X ∈ LUC(G)∗ | Y 7→ X2Y w∗-cont. }
Theorem (Hu–N.–Ruan ’09)
TFAE:
G compact (i.e., C0(G) unital)
LUC(G) ⊆WAP(G) and Zt(LUC(G)∗) = M(G)
Question G = L(G) with G discrete?⇒ WAP(G) ⊆ LUC(G)
If yes, then there is NO infinite G with A(G) Arens regular!Open for Olshanskii group . . .
Topological centre basics Topological centre problems Topological centres as a tool
Characterization of compact quantum groups
Since LUC(G) ⊆ L∞(G) we have
LUC(G)∗ ←− L1(G)∗∗
; Transport of left Arens product ; LUC(G)∗ Banach algebra
Zt(LUC(G)∗) := { X ∈ LUC(G)∗ | Y 7→ X2Y w∗-cont. }
Theorem (Hu–N.–Ruan ’09)
TFAE:
G compact (i.e., C0(G) unital)
LUC(G) ⊆WAP(G) and Zt(LUC(G)∗) = M(G)
Question G = L(G) with G discrete?⇒ WAP(G) ⊆ LUC(G)
If yes, then there is NO infinite G with A(G) Arens regular!Open for Olshanskii group . . .
Topological centre basics Topological centre problems Topological centres as a tool
Characterizations using invariant meanson quantum groups
Definition
G amenable :⇔ ∃ mean on L∞(G) s.t.
f 2M = 〈f , 1〉 M ∀ f ∈ L1(G)
Theorem (Hu–N.–Ruan ’09)
Let G be amenable with L1(G) separable or SAI.Then: G uniquely amenable ⇔ G compact
Theorem (Hu–N.–Ruan ’09)
Let G be amenable & co-amenable, with L1(G) separable.Then: L1(G) Arens regular ⇔ G finite
Topological centre basics Topological centre problems Topological centres as a tool
Characterizations using invariant meanson quantum groups
Definition
G amenable :⇔ ∃ mean on L∞(G) s.t.
f 2M = 〈f , 1〉 M ∀ f ∈ L1(G)
Theorem (Hu–N.–Ruan ’09)
Let G be amenable with L1(G) separable or SAI.Then: G uniquely amenable ⇔ G compact
Theorem (Hu–N.–Ruan ’09)
Let G be amenable & co-amenable, with L1(G) separable.Then: L1(G) Arens regular ⇔ G finite
Topological centre basics Topological centre problems Topological centres as a tool
Characterizations using invariant meanson quantum groups
Definition
G amenable :⇔ ∃ mean on L∞(G) s.t.
f 2M = 〈f , 1〉 M ∀ f ∈ L1(G)
Theorem (Hu–N.–Ruan ’09)
Let G be amenable with L1(G) separable or SAI.
Then: G uniquely amenable ⇔ G compact
Theorem (Hu–N.–Ruan ’09)
Let G be amenable & co-amenable, with L1(G) separable.Then: L1(G) Arens regular ⇔ G finite
Topological centre basics Topological centre problems Topological centres as a tool
Characterizations using invariant meanson quantum groups
Definition
G amenable :⇔ ∃ mean on L∞(G) s.t.
f 2M = 〈f , 1〉 M ∀ f ∈ L1(G)
Theorem (Hu–N.–Ruan ’09)
Let G be amenable with L1(G) separable or SAI.Then: G uniquely amenable ⇔ G compact
Theorem (Hu–N.–Ruan ’09)
Let G be amenable & co-amenable, with L1(G) separable.Then: L1(G) Arens regular ⇔ G finite
Topological centre basics Topological centre problems Topological centres as a tool
Characterizations using invariant meanson quantum groups
Definition
G amenable :⇔ ∃ mean on L∞(G) s.t.
f 2M = 〈f , 1〉 M ∀ f ∈ L1(G)
Theorem (Hu–N.–Ruan ’09)
Let G be amenable with L1(G) separable or SAI.Then: G uniquely amenable ⇔ G compact
Theorem (Hu–N.–Ruan ’09)
Let G be amenable & co-amenable, with L1(G) separable.
Then: L1(G) Arens regular ⇔ G finite
Topological centre basics Topological centre problems Topological centres as a tool
Characterizations using invariant meanson quantum groups
Definition
G amenable :⇔ ∃ mean on L∞(G) s.t.
f 2M = 〈f , 1〉 M ∀ f ∈ L1(G)
Theorem (Hu–N.–Ruan ’09)
Let G be amenable with L1(G) separable or SAI.Then: G uniquely amenable ⇔ G compact
Theorem (Hu–N.–Ruan ’09)
Let G be amenable & co-amenable, with L1(G) separable.Then: L1(G) Arens regular ⇔ G finite
Topological centre basics Topological centre problems Topological centres as a tool
From quantum groups back to groups
McbL1(G) := { Φ : L1(G)→ L1(G) | Φ CB, Φ(a∗b) = a∗Φ(b) }
Consider m ∈McbL1(G) s.t.
m∗ UCP (= Markov operator)
m invertible and complete isometry on L1(G)
Denote by G the set of those multipliers.
Examples
L∞(G) ∼= G
L(G) ∼= G
Topological centre basics Topological centre problems Topological centres as a tool
From quantum groups back to groups
McbL1(G) := { Φ : L1(G)→ L1(G) | Φ CB, Φ(a∗b) = a∗Φ(b) }
Consider m ∈McbL1(G) s.t.
m∗ UCP (= Markov operator)
m invertible and complete isometry on L1(G)
Denote by G the set of those multipliers.
Examples
L∞(G) ∼= G
L(G) ∼= G
Topological centre basics Topological centre problems Topological centres as a tool
From quantum groups back to groups
McbL1(G) := { Φ : L1(G)→ L1(G) | Φ CB, Φ(a∗b) = a∗Φ(b) }
Consider m ∈McbL1(G) s.t.
m∗ UCP (= Markov operator)
m invertible and complete isometry on L1(G)
Denote by G the set of those multipliers.
Examples
L∞(G) ∼= G
L(G) ∼= G
Topological centre basics Topological centre problems Topological centres as a tool
From quantum groups back to groups
McbL1(G) := { Φ : L1(G)→ L1(G) | Φ CB, Φ(a∗b) = a∗Φ(b) }
Consider m ∈McbL1(G) s.t.
m∗ UCP (= Markov operator)
m invertible and complete isometry on L1(G)
Denote by G the set of those multipliers.
Examples
L∞(G) ∼= G
L(G) ∼= G
Topological centre basics Topological centre problems Topological centres as a tool
From quantum groups back to groups
McbL1(G) := { Φ : L1(G)→ L1(G) | Φ CB, Φ(a∗b) = a∗Φ(b) }
Consider m ∈McbL1(G) s.t.
m∗ UCP (= Markov operator)
m invertible and complete isometry on L1(G)
Denote by G the set of those multipliers.
Examples
L∞(G) ∼= G
L(G) ∼= G
Topological centre basics Topological centre problems Topological centres as a tool
From quantum groups back to groups
McbL1(G) := { Φ : L1(G)→ L1(G) | Φ CB, Φ(a∗b) = a∗Φ(b) }
Consider m ∈McbL1(G) s.t.
m∗ UCP (= Markov operator)
m invertible and complete isometry on L1(G)
Denote by G the set of those multipliers.
Examples
L∞(G) ∼= G
L(G) ∼= G
Topological centre basics Topological centre problems Topological centres as a tool
From quantum groups back to groups
McbL1(G) := { Φ : L1(G)→ L1(G) | Φ CB, Φ(a∗b) = a∗Φ(b) }
Consider m ∈McbL1(G) s.t.
m∗ UCP (= Markov operator)
m invertible and complete isometry on L1(G)
Denote by G the set of those multipliers.
Examples
L∞(G) ∼= G
L(G) ∼= G
Topological centre basics Topological centre problems Topological centres as a tool
A functor LC quantum groups → LC groups
Theorem (Kalantar–N. ’09)
G is a LC group w.r.t. point weak topology on L1(G)
G ∼= Sp(L1(G))
Theorem (Kalantar–N. ’09)
The functor G→ G preserves
local compactness
compactness
discreteness
(hence) finiteness
The functor also “almost commutes” with quantum group duality.
Topological centre basics Topological centre problems Topological centres as a tool
A functor LC quantum groups → LC groups
Theorem (Kalantar–N. ’09)
G is a LC group w.r.t. point weak topology on L1(G)
G ∼= Sp(L1(G))
Theorem (Kalantar–N. ’09)
The functor G→ G preserves
local compactness
compactness
discreteness
(hence) finiteness
The functor also “almost commutes” with quantum group duality.
Topological centre basics Topological centre problems Topological centres as a tool
A functor LC quantum groups → LC groups
Theorem (Kalantar–N. ’09)
G is a LC group w.r.t. point weak topology on L1(G)
G ∼= Sp(L1(G))
Theorem (Kalantar–N. ’09)
The functor G→ G preserves
local compactness
compactness
discreteness
(hence) finiteness
The functor also “almost commutes” with quantum group duality.
Topological centre basics Topological centre problems Topological centres as a tool
A functor LC quantum groups → LC groups
Theorem (Kalantar–N. ’09)
G is a LC group w.r.t. point weak topology on L1(G)
G ∼= Sp(L1(G))
Theorem (Kalantar–N. ’09)
The functor G→ G preserves
local compactness
compactness
discreteness
(hence) finiteness
The functor also “almost commutes” with quantum group duality.
Topological centre basics Topological centre problems Topological centres as a tool
A functor LC quantum groups → LC groups
Theorem (Kalantar–N. ’09)
G is a LC group w.r.t. point weak topology on L1(G)
G ∼= Sp(L1(G))
Theorem (Kalantar–N. ’09)
The functor G→ G preserves
local compactness
compactness
discreteness
(hence) finiteness
The functor also “almost commutes” with quantum group duality.
Topological centre basics Topological centre problems Topological centres as a tool
Structure of G
Theorem (Kalantar–N. ’09)
G compact matrix pseudogroup (Woronowicz ’87)⇒ G is a compact Lie group
Example Woronowicz’s SUq(2) with deformation parameterq ∈ (0, 1]
SUq(2) = C (SU(2)) for q = 1
Non-commutative C ∗-algebra for q ∈ (0, 1)
SUq(2) ∼= SU(2) for q = 1
SUq(2) ∼= T for q 6= 1
Theorem (Kalantar–N. ’09)
G compact, non-Kac with L1(G) separable ⇒ G uncountable
Topological centre basics Topological centre problems Topological centres as a tool
Structure of G
Theorem (Kalantar–N. ’09)
G compact matrix pseudogroup (Woronowicz ’87)⇒ G is a compact Lie group
Example Woronowicz’s SUq(2) with deformation parameterq ∈ (0, 1]
SUq(2) = C (SU(2)) for q = 1
Non-commutative C ∗-algebra for q ∈ (0, 1)
SUq(2) ∼= SU(2) for q = 1
SUq(2) ∼= T for q 6= 1
Theorem (Kalantar–N. ’09)
G compact, non-Kac with L1(G) separable ⇒ G uncountable
Topological centre basics Topological centre problems Topological centres as a tool
Structure of G
Theorem (Kalantar–N. ’09)
G compact matrix pseudogroup (Woronowicz ’87)⇒ G is a compact Lie group
Example Woronowicz’s SUq(2) with deformation parameterq ∈ (0, 1]
SUq(2) = C (SU(2)) for q = 1
Non-commutative C ∗-algebra for q ∈ (0, 1)
SUq(2) ∼= SU(2) for q = 1
SUq(2) ∼= T for q 6= 1
Theorem (Kalantar–N. ’09)
G compact, non-Kac with L1(G) separable ⇒ G uncountable
Topological centre basics Topological centre problems Topological centres as a tool
Structure of G
Theorem (Kalantar–N. ’09)
G compact matrix pseudogroup (Woronowicz ’87)⇒ G is a compact Lie group
Example Woronowicz’s SUq(2) with deformation parameterq ∈ (0, 1]
SUq(2) = C (SU(2)) for q = 1
Non-commutative C ∗-algebra for q ∈ (0, 1)
SUq(2) ∼= SU(2) for q = 1
SUq(2) ∼= T for q 6= 1
Theorem (Kalantar–N. ’09)
G compact, non-Kac with L1(G) separable ⇒ G uncountable
Topological centre basics Topological centre problems Topological centres as a tool
Structure of G
Theorem (Kalantar–N. ’09)
G compact matrix pseudogroup (Woronowicz ’87)⇒ G is a compact Lie group
Example Woronowicz’s SUq(2) with deformation parameterq ∈ (0, 1]
SUq(2) = C (SU(2)) for q = 1
Non-commutative C ∗-algebra for q ∈ (0, 1)
SUq(2) ∼= SU(2) for q = 1
SUq(2) ∼= T for q 6= 1
Theorem (Kalantar–N. ’09)
G compact, non-Kac with L1(G) separable ⇒ G uncountable
Topological centre basics Topological centre problems Topological centres as a tool
Structure of G
Theorem (Kalantar–N. ’09)
G compact matrix pseudogroup (Woronowicz ’87)⇒ G is a compact Lie group
Example Woronowicz’s SUq(2) with deformation parameterq ∈ (0, 1]
SUq(2) = C (SU(2)) for q = 1
Non-commutative C ∗-algebra for q ∈ (0, 1)
SUq(2) ∼= SU(2) for q = 1
SUq(2) ∼= T for q 6= 1
Theorem (Kalantar–N. ’09)
G compact, non-Kac with L1(G) separable ⇒ G uncountable
Topological centre basics Topological centre problems Topological centres as a tool
Heisenberg relation for quantum groups
G abelian. For s ∈ G and γ ∈ G
Ls Mγ = 〈γ, s〉︸ ︷︷ ︸∈T
Mγ Ls
We obtain a generalization to quantum groups, using acommutation result by Junge–N.–Ruan (’09):
Theorem: Non-commutative Torus (Kalantar–N. ’09)
g ∈ G, g ∈ ˜G ⇒ ∃ 〈g , g〉 ∈ T s.t.
g g = 〈g , g〉 g g⟨˜G, G
⟩=: G0 is a subgroup of T .
Example SUq(2)0 = T (q 6= 1)
Topological centre basics Topological centre problems Topological centres as a tool
Heisenberg relation for quantum groups
G abelian. For s ∈ G and γ ∈ G
Ls Mγ = 〈γ, s〉︸ ︷︷ ︸∈T
Mγ Ls
We obtain a generalization to quantum groups, using acommutation result by Junge–N.–Ruan (’09):
Theorem: Non-commutative Torus (Kalantar–N. ’09)
g ∈ G, g ∈ ˜G ⇒ ∃ 〈g , g〉 ∈ T s.t.
g g = 〈g , g〉 g g⟨˜G, G
⟩=: G0 is a subgroup of T .
Example SUq(2)0 = T (q 6= 1)
Topological centre basics Topological centre problems Topological centres as a tool
Heisenberg relation for quantum groups
G abelian. For s ∈ G and γ ∈ G
Ls Mγ = 〈γ, s〉︸ ︷︷ ︸∈T
Mγ Ls
We obtain a generalization to quantum groups, using acommutation result by Junge–N.–Ruan (’09):
Theorem: Non-commutative Torus (Kalantar–N. ’09)
g ∈ G, g ∈ ˜G ⇒ ∃ 〈g , g〉 ∈ T s.t.
g g = 〈g , g〉 g g⟨˜G, G
⟩=: G0 is a subgroup of T .
Example SUq(2)0 = T (q 6= 1)
Topological centre basics Topological centre problems Topological centres as a tool
Heisenberg relation for quantum groups
G abelian. For s ∈ G and γ ∈ G
Ls Mγ = 〈γ, s〉︸ ︷︷ ︸∈T
Mγ Ls
We obtain a generalization to quantum groups, using acommutation result by Junge–N.–Ruan (’09):
Theorem: Non-commutative Torus (Kalantar–N. ’09)
g ∈ G, g ∈ ˜G ⇒ ∃ 〈g , g〉 ∈ T s.t.
g g = 〈g , g〉 g g
⟨˜G, G
⟩=: G0 is a subgroup of T .
Example SUq(2)0 = T (q 6= 1)
Topological centre basics Topological centre problems Topological centres as a tool
Heisenberg relation for quantum groups
G abelian. For s ∈ G and γ ∈ G
Ls Mγ = 〈γ, s〉︸ ︷︷ ︸∈T
Mγ Ls
We obtain a generalization to quantum groups, using acommutation result by Junge–N.–Ruan (’09):
Theorem: Non-commutative Torus (Kalantar–N. ’09)
g ∈ G, g ∈ ˜G ⇒ ∃ 〈g , g〉 ∈ T s.t.
g g = 〈g , g〉 g g⟨˜G, G
⟩=: G0 is a subgroup of T .
Example SUq(2)0 = T (q 6= 1)
Topological centre basics Topological centre problems Topological centres as a tool
Heisenberg relation for quantum groups
G abelian. For s ∈ G and γ ∈ G
Ls Mγ = 〈γ, s〉︸ ︷︷ ︸∈T
Mγ Ls
We obtain a generalization to quantum groups, using acommutation result by Junge–N.–Ruan (’09):
Theorem: Non-commutative Torus (Kalantar–N. ’09)
g ∈ G, g ∈ ˜G ⇒ ∃ 〈g , g〉 ∈ T s.t.
g g = 〈g , g〉 g g⟨˜G, G
⟩=: G0 is a subgroup of T .
Example SUq(2)0 = T (q 6= 1)
Topological centre basics Topological centre problems Topological centres as a tool
Semigroup compactifications from quantum groups
Assume G co-amenable.
Theorem (Hu–N.–Ruan ’08)
McbL1(G) ∼= M(G) ↪→ Zt(LUC(G)∗)
Theorem (Kalantar–N. ’08)
The embedding G ⊆McbL1(G) in LUC(G)∗ gives rise to
GLUC := Gw∗
Then GLUC is a compact right topological semigroup.
Note: G = L∞(G) for a LC group G ⇒ GLUC = GLUC
Topological centre basics Topological centre problems Topological centres as a tool
Semigroup compactifications from quantum groups
Assume G co-amenable.
Theorem (Hu–N.–Ruan ’08)
McbL1(G) ∼= M(G) ↪→ Zt(LUC(G)∗)
Theorem (Kalantar–N. ’08)
The embedding G ⊆McbL1(G) in LUC(G)∗ gives rise to
GLUC := Gw∗
Then GLUC is a compact right topological semigroup.
Note: G = L∞(G) for a LC group G ⇒ GLUC = GLUC
Topological centre basics Topological centre problems Topological centres as a tool
Semigroup compactifications from quantum groups
Assume G co-amenable.
Theorem (Hu–N.–Ruan ’08)
McbL1(G) ∼= M(G) ↪→ Zt(LUC(G)∗)
Theorem (Kalantar–N. ’08)
The embedding G ⊆McbL1(G) in LUC(G)∗ gives rise to
GLUC := Gw∗
Then GLUC is a compact right topological semigroup.
Note: G = L∞(G) for a LC group G ⇒ GLUC = GLUC
Topological centre basics Topological centre problems Topological centres as a tool
Semigroup compactifications from quantum groups
Assume G co-amenable.
Theorem (Hu–N.–Ruan ’08)
McbL1(G) ∼= M(G) ↪→ Zt(LUC(G)∗)
Theorem (Kalantar–N. ’08)
The embedding G ⊆McbL1(G) in LUC(G)∗ gives rise to
GLUC := Gw∗
Then GLUC is a compact right topological semigroup.
Note: G = L∞(G) for a LC group G ⇒ GLUC = GLUC
Topological centre basics Topological centre problems Topological centres as a tool
Semigroup compactifications from quantum groups
Assume G co-amenable.
Theorem (Hu–N.–Ruan ’08)
McbL1(G) ∼= M(G) ↪→ Zt(LUC(G)∗)
Theorem (Kalantar–N. ’08)
The embedding G ⊆McbL1(G) in LUC(G)∗ gives rise to
GLUC := Gw∗
Then GLUC is a compact right topological semigroup.
Note: G = L∞(G) for a LC group G ⇒ GLUC = GLUC
Topological centre basics Topological centre problems Topological centres as a tool
Unification via T (L2(G))
Co-multiplications Γ and Γ extend to
B(L2(G)) → B(L2(G)) ⊗ B(L2(G))
⇒ Γ∗ = m and Γ∗ = m yield 2 dual products
T (L2(G)) ⊗ T (L2(G)) → T (L2(G))
Theorem (Kalantar–N. ’08)
m ◦ (m⊗ id) = m ◦ (m⊗ id) ◦ (id⊗ σ)
Here, σ(ϕ⊗ τ) = τ ⊗ ϕ is the flip.
Duality = Anti-Commutation Relation on tensor level
Topological centre basics Topological centre problems Topological centres as a tool
Unification via T (L2(G))
Co-multiplications Γ and Γ extend to
B(L2(G)) → B(L2(G)) ⊗ B(L2(G))
⇒ Γ∗ = m and Γ∗ = m yield 2 dual products
T (L2(G)) ⊗ T (L2(G)) → T (L2(G))
Theorem (Kalantar–N. ’08)
m ◦ (m⊗ id) = m ◦ (m⊗ id) ◦ (id⊗ σ)
Here, σ(ϕ⊗ τ) = τ ⊗ ϕ is the flip.
Duality = Anti-Commutation Relation on tensor level
Topological centre basics Topological centre problems Topological centres as a tool
Unification via T (L2(G))
Co-multiplications Γ and Γ extend to
B(L2(G)) → B(L2(G)) ⊗ B(L2(G))
⇒ Γ∗ = m and Γ∗ = m yield 2 dual products
T (L2(G)) ⊗ T (L2(G)) → T (L2(G))
Theorem (Kalantar–N. ’08)
m ◦ (m⊗ id) = m ◦ (m⊗ id) ◦ (id⊗ σ)
Here, σ(ϕ⊗ τ) = τ ⊗ ϕ is the flip.
Duality = Anti-Commutation Relation on tensor level
Topological centre basics Topological centre problems Topological centres as a tool
Unification via T (L2(G))
Co-multiplications Γ and Γ extend to
B(L2(G)) → B(L2(G)) ⊗ B(L2(G))
⇒ Γ∗ = m and Γ∗ = m yield 2 dual products
T (L2(G)) ⊗ T (L2(G)) → T (L2(G))
Theorem (Kalantar–N. ’08)
m ◦ (m⊗ id) = m ◦ (m⊗ id) ◦ (id⊗ σ)
Here, σ(ϕ⊗ τ) = τ ⊗ ϕ is the flip.
Duality = Anti-Commutation Relation on tensor level
Topological centre basics Topological centre problems Topological centres as a tool
T (L2(G))as a home for convolution and pointwise product
T (L2(G)) ∼= T (L2(G))
L1(G) L1(G)
∗ •
On T (L2(G)) we can compare“convolution” and “pointwise product”!
(ϕ ∗ τ) • ψ = (ϕ • ψ) ∗ τ(Kalantar–N. ’08)
Topological centre basics Topological centre problems Topological centres as a tool
T (L2(G))as a home for convolution and pointwise product
T (L2(G)) ∼= T (L2(G))
L1(G) L1(G)
∗ •
On T (L2(G)) we can compare“convolution” and “pointwise product”!
(ϕ ∗ τ) • ψ = (ϕ • ψ) ∗ τ(Kalantar–N. ’08)
Topological centre basics Topological centre problems Topological centres as a tool
T (L2(G))as a home for convolution and pointwise product
T (L2(G)) ∼= T (L2(G))
L1(G) L1(G)
∗ •
On T (L2(G)) we can compare“convolution” and “pointwise product”!
(ϕ ∗ τ) • ψ = (ϕ • ψ) ∗ τ(Kalantar–N. ’08)
Topological centre basics Topological centre problems Topological centres as a tool
Some cohomology for LC quantum groups
The following generalizes results by Pirkovskii on (T (L2(G)), ∗) toquantum groups.
Theorem (Kalantar–N. ’08)
L1(G) is projective in mod–T (L2(G))
⇔ L1(G) has the Radon–Nikodym Property
Theorem (Kalantar–N. ’08)
C is projective in mod–T (L2(G))
⇔ G is compact
Topological centre basics Topological centre problems Topological centres as a tool
Some cohomology for LC quantum groups
The following generalizes results by Pirkovskii on (T (L2(G)), ∗) toquantum groups.
Theorem (Kalantar–N. ’08)
L1(G) is projective in mod–T (L2(G))
⇔ L1(G) has the Radon–Nikodym Property
Theorem (Kalantar–N. ’08)
C is projective in mod–T (L2(G))
⇔ G is compact
Topological centre basics Topological centre problems Topological centres as a tool
Some cohomology for LC quantum groups
The following generalizes results by Pirkovskii on (T (L2(G)), ∗) toquantum groups.
Theorem (Kalantar–N. ’08)
L1(G) is projective in mod–T (L2(G))
⇔ L1(G) has the Radon–Nikodym Property
Theorem (Kalantar–N. ’08)
C is projective in mod–T (L2(G))
⇔ G is compact
Topological centre basics Topological centre problems Topological centres as a tool
Some cohomology for LC quantum groups
The following generalizes results by Pirkovskii on (T (L2(G)), ∗) toquantum groups.
Theorem (Kalantar–N. ’08)
L1(G) is projective in mod–T (L2(G))
⇔ L1(G) has the Radon–Nikodym Property
Theorem (Kalantar–N. ’08)
C is projective in mod–T (L2(G))
⇔ G is compact