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Algebra isomorphisms between standard operatoralgebras

T. Tonev

The University of Montana, Missoula, USA

Bedlewo, Poland, 2009

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 1 / 18

Introduction

Introduction

This is a joint work with A. Luttman (Studia Math., 2009)

Let A,B be Banach algebras.

Question:

When is a surjective operator T : A→ B an algebraic isomorphism, i.e.a linear and multiplicative bijection?

Significant róle in answering this question is played by the spectrumand the peripheral spectrum of f ∈ A.

Definition 1.

The spectrum of f ∈ A is the set σ(f ) = {ζ ∈ C : z − f /∈ A−1}.The peripheral spectrum of f ∈ A is the set

σπ(f ) = σ(f ) ∩{

z ∈ C : |z| = r(f )}

,where r(f ) is the spectral radius of f .

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 2 / 18

Introduction

Introduction

This is a joint work with A. Luttman (Studia Math., 2009)

Let A,B be Banach algebras.

Question:

When is a surjective operator T : A→ B an algebraic isomorphism, i.e.a linear and multiplicative bijection?

Significant róle in answering this question is played by the spectrumand the peripheral spectrum of f ∈ A.

Definition 1.

The spectrum of f ∈ A is the set σ(f ) = {ζ ∈ C : z − f /∈ A−1}.The peripheral spectrum of f ∈ A is the set

σπ(f ) = σ(f ) ∩{

z ∈ C : |z| = r(f )}

,where r(f ) is the spectral radius of f .

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 2 / 18

Introduction

Introduction

This is a joint work with A. Luttman (Studia Math., 2009)

Let A,B be Banach algebras.

Question:

When is a surjective operator T : A→ B an algebraic isomorphism, i.e.a linear and multiplicative bijection?

Significant róle in answering this question is played by the spectrumand the peripheral spectrum of f ∈ A.

Definition 1.

The spectrum of f ∈ A is the set σ(f ) = {ζ ∈ C : z − f /∈ A−1}.The peripheral spectrum of f ∈ A is the set

σπ(f ) = σ(f ) ∩{

z ∈ C : |z| = r(f )}

,where r(f ) is the spectral radius of f .

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 2 / 18

Introduction

Introduction

This is a joint work with A. Luttman (Studia Math., 2009)

Let A,B be Banach algebras.

Question:

Definition 1.

σπ(f ) = σ(f ) ∩{

z ∈ C : |z| = r(f )}

,where r(f ) is the spectral radius of f .

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 2 / 18

Conditions for Algebra Isomorphisms Isomorphisms between uniform algebras

Isomorphisms between uniform algebras

Uniform algebra: A ⊂ C(X ), X - compact Hausdorff space

[Gleason-Kahane-Zelazko, 1968]: A,B – uniform algebras,T : A→ B – linear surjection, σ(Tf ) = σ(f ), f ∈ A =⇒T is multiplicative.

[Banach-Stone]: T : C(X )→ C(Y ) – linear isometry, T1 = 1 =⇒T is an algebra isomorphism.

Extensions: Kaplansky, Nagasawa, Kowalski-Słodkowski, Choi,Rosenthal, Omladic, Šemrl, Brešar, etc.

[Rao-Toneva-T., 2006]: A,B – uniform algebras, T : A→ B –surjection, σπ

(λ(Tf ) + µ(Tg)

)= σπ(λf + µg), f ,g ∈ A, λ, µ ∈ C,

i.e. T – peripherally-linear =⇒ T is an isometric isomorphism.

[Yates-T., 2009]: A,B – uniform algebras, T : A→ B – surjection,‖λ(Tf ) + µ(Tg)‖ = ‖λf + µg‖, f ,g ∈ A, λ, µ ∈ C, i.e. T – norm-linear, T1 = 1, Ti = i =⇒ T is an isometric algebra isomorphism.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 3 / 18

Conditions for Algebra Isomorphisms Isomorphisms between uniform algebras

Isomorphisms between uniform algebras

Uniform algebra: A ⊂ C(X ), X - compact Hausdorff space

[Gleason-Kahane-Zelazko, 1968]: A,B – uniform algebras,T : A→ B – linear surjection, σ(Tf ) = σ(f ), f ∈ A =⇒T is multiplicative.

[Banach-Stone]: T : C(X )→ C(Y ) – linear isometry, T1 = 1 =⇒T is an algebra isomorphism.

Extensions: Kaplansky, Nagasawa, Kowalski-Słodkowski, Choi,Rosenthal, Omladic, Šemrl, Brešar, etc.

[Rao-Toneva-T., 2006]: A,B – uniform algebras, T : A→ B –surjection, σπ

(λ(Tf ) + µ(Tg)

)= σπ(λf + µg), f ,g ∈ A, λ, µ ∈ C,

i.e. T – peripherally-linear =⇒ T is an isometric isomorphism.

[Yates-T., 2009]: A,B – uniform algebras, T : A→ B – surjection,‖λ(Tf ) + µ(Tg)‖ = ‖λf + µg‖, f ,g ∈ A, λ, µ ∈ C, i.e. T – norm-linear, T1 = 1, Ti = i =⇒ T is an isometric algebra isomorphism.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 3 / 18

Conditions for Algebra Isomorphisms Isomorphisms between uniform algebras

Isomorphisms between uniform algebras

Uniform algebra: A ⊂ C(X ), X - compact Hausdorff space

[Gleason-Kahane-Zelazko, 1968]: A,B – uniform algebras,T : A→ B – linear surjection, σ(Tf ) = σ(f ), f ∈ A =⇒T is multiplicative.

[Banach-Stone]: T : C(X )→ C(Y ) – linear isometry, T1 = 1 =⇒T is an algebra isomorphism.

Extensions: Kaplansky, Nagasawa, Kowalski-Słodkowski, Choi,Rosenthal, Omladic, Šemrl, Brešar, etc.

[Rao-Toneva-T., 2006]: A,B – uniform algebras, T : A→ B –surjection, σπ

(λ(Tf ) + µ(Tg)

)= σπ(λf + µg), f ,g ∈ A, λ, µ ∈ C,

i.e. T – peripherally-linear =⇒ T is an isometric isomorphism.

[Yates-T., 2009]: A,B – uniform algebras, T : A→ B – surjection,‖λ(Tf ) + µ(Tg)‖ = ‖λf + µg‖, f ,g ∈ A, λ, µ ∈ C, i.e. T – norm-linear, T1 = 1, Ti = i =⇒ T is an isometric algebra isomorphism.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 3 / 18

Conditions for Algebra Isomorphisms Isomorphisms between uniform algebras

Isomorphisms between uniform algebras

Uniform algebra: A ⊂ C(X ), X - compact Hausdorff space

[Banach-Stone]: T : C(X )→ C(Y ) – linear isometry, T1 = 1 =⇒T is an algebra isomorphism.

Extensions: Kaplansky, Nagasawa, Kowalski-Słodkowski, Choi,Rosenthal, Omladic, Šemrl, Brešar, etc.

[Rao-Toneva-T., 2006]: A,B – uniform algebras, T : A→ B –surjection, σπ

(λ(Tf ) + µ(Tg)

)= σπ(λf + µg), f ,g ∈ A, λ, µ ∈ C,

i.e. T – peripherally-linear =⇒ T is an isometric isomorphism.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 3 / 18

Conditions for Algebra Isomorphisms Isomorphisms between uniform algebras

Isomorphisms between uniform algebras

Uniform algebra: A ⊂ C(X ), X - compact Hausdorff space

[Banach-Stone]: T : C(X )→ C(Y ) – linear isometry, T1 = 1 =⇒T is an algebra isomorphism.

Extensions: Kaplansky, Nagasawa, Kowalski-Słodkowski, Choi,Rosenthal, Omladic, Šemrl, Brešar, etc.

[Rao-Toneva-T., 2006]: A,B – uniform algebras, T : A→ B –surjection, σπ

(λ(Tf ) + µ(Tg)

)= σπ(λf + µg), f ,g ∈ A, λ, µ ∈ C,

i.e. T – peripherally-linear =⇒ T is an isometric isomorphism.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 3 / 18

Conditions for Algebra Isomorphisms Isomorphisms between uniform algebras

[Molnár, 2001]: T : C(X )→ C(X ) – surjection,σ((Tf )(Tg)

)= σ(fg), f ,g ∈ A, i.e. T – σ-multiplicative, T1 = 1

=⇒ T is an isometric algebra isomorphism.

Extentions: Rao-Roy, Hatori, Miura, Takagi

[Luttman-T., 2006]: A,B – uniform algebras, T : A→ B –surjection, σπ

((Tf )(Tg)

)= σπ(fg), f ,g ∈ A, i.e. T –

peripherally-multiplicative, T1 = 1 =⇒T is an isometric algebra isomorphism.

[Lambert-Luttman-T., 2007]: A,B – uniform algebras, T : A→ B –surjection, σπ

((Tf )(Tg)

)∩ σπ(fg) 6= ∅, f ,g ∈ A, i.e. T –

weakly peripherally-multiplicative, σπ(Tf ) = σπ(f ), T1 = 1 =⇒T is an isometric algebra isomorphism.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 4 / 18

Conditions for Algebra Isomorphisms Isomorphisms between uniform algebras

[Molnár, 2001]: T : C(X )→ C(X ) – surjection,σ((Tf )(Tg)

)= σ(fg), f ,g ∈ A, i.e. T – σ-multiplicative, T1 = 1

=⇒ T is an isometric algebra isomorphism.

Extentions: Rao-Roy, Hatori, Miura, Takagi

[Luttman-T., 2006]: A,B – uniform algebras, T : A→ B –surjection, σπ

((Tf )(Tg)

)= σπ(fg), f ,g ∈ A, i.e. T –

peripherally-multiplicative, T1 = 1 =⇒T is an isometric algebra isomorphism.

[Lambert-Luttman-T., 2007]: A,B – uniform algebras, T : A→ B –surjection, σπ

((Tf )(Tg)

)∩ σπ(fg) 6= ∅, f ,g ∈ A, i.e. T –

weakly peripherally-multiplicative, σπ(Tf ) = σπ(f ), T1 = 1 =⇒T is an isometric algebra isomorphism.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 4 / 18

Conditions for Algebra Isomorphisms Isomorphisms between uniform algebras

[Molnár, 2001]: T : C(X )→ C(X ) – surjection,σ((Tf )(Tg)

)= σ(fg), f ,g ∈ A, i.e. T – σ-multiplicative, T1 = 1

=⇒ T is an isometric algebra isomorphism.

Extentions: Rao-Roy, Hatori, Miura, Takagi

[Luttman-T., 2006]: A,B – uniform algebras, T : A→ B –surjection, σπ

((Tf )(Tg)

)= σπ(fg), f ,g ∈ A, i.e. T –

peripherally-multiplicative, T1 = 1 =⇒T is an isometric algebra isomorphism.

[Lambert-Luttman-T., 2007]: A,B – uniform algebras, T : A→ B –surjection, σπ

((Tf )(Tg)

)∩ σπ(fg) 6= ∅, f ,g ∈ A, i.e. T –

weakly peripherally-multiplicative, σπ(Tf ) = σπ(f ), T1 = 1 =⇒T is an isometric algebra isomorphism.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 4 / 18

Conditions for Algebra Isomorphisms Isomorphisms between uniform algebras

[Molnár, 2001]: T : C(X )→ C(X ) – surjection,σ((Tf )(Tg)

)= σ(fg), f ,g ∈ A, i.e. T – σ-multiplicative, T1 = 1

=⇒ T is an isometric algebra isomorphism.

Extentions: Rao-Roy, Hatori, Miura, Takagi

[Luttman-T., 2006]: A,B – uniform algebras, T : A→ B –surjection, σπ

((Tf )(Tg)

)= σπ(fg), f ,g ∈ A, i.e. T –

peripherally-multiplicative, T1 = 1 =⇒T is an isometric algebra isomorphism.

[Lambert-Luttman-T., 2007]: A,B – uniform algebras, T : A→ B –surjection, σπ

((Tf )(Tg)

)∩ σπ(fg) 6= ∅, f ,g ∈ A, i.e. T –

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 4 / 18

Conditions for Algebra Isomorphisms Isomorphisms between operator algebras

Isomorphisms between operator algebras

Operator algebra: A ⊂ B(X ), X - Banach space, dim X =∞[Jafarian-Sourour, 1986]: A = B(X ), B = B(Y ), φ : A→ B –linear surjection σ

(φ(T )

)= σ(T ), T ∈ A, =⇒ φ(T ) = ATA−1 for a

linear bijection A : X → Y , or, φ(T ) = BT ∗B−1 for a linear bijectionB : X ∗ → Y , i.e. φ is an isomorphism/anti-isomorphism.[Molnár, 2001]: φ : B(X )→ B(X ) – surjection, σp

((φ(A))(φ(B))

)=

σp(fg), A,B ∈ B(X ) =⇒ either φ or −φ is an algebra isomorphism.Here σp(A) is the point spectrum of A ∈ B(X ).[Molnár, 2001]: H – Hilbert space, φ : B(H)→ B(H) – surjection,σs

(φ(A) ◦ φ(B)

)= σs(AB) for all A,B ∈ A =⇒ there is a bijective

linear operator C ∈ B(H) such that ±φ(A) = CAC−1, i.e. either φor −φ is an algebraic isomorphism.Here σs(A) is the surjective spectrum of A ∈ B(H).Extensions: Hou, Šemrl, Bai, Xu, Di.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 5 / 18

Conditions for Algebra Isomorphisms Isomorphisms between operator algebras

Isomorphisms between operator algebras

Operator algebra: A ⊂ B(X ), X - Banach space, dim X =∞[Jafarian-Sourour, 1986]: A = B(X ), B = B(Y ), φ : A→ B –linear surjection σ

(φ(T )

)= σ(T ), T ∈ A, =⇒ φ(T ) = ATA−1 for a

linear bijection A : X → Y , or, φ(T ) = BT ∗B−1 for a linear bijectionB : X ∗ → Y , i.e. φ is an isomorphism/anti-isomorphism.[Molnár, 2001]: φ : B(X )→ B(X ) – surjection, σp

((φ(A))(φ(B))

)=

σp(fg), A,B ∈ B(X ) =⇒ either φ or −φ is an algebra isomorphism.Here σp(A) is the point spectrum of A ∈ B(X ).[Molnár, 2001]: H – Hilbert space, φ : B(H)→ B(H) – surjection,σs

(φ(A) ◦ φ(B)

)= σs(AB) for all A,B ∈ A =⇒ there is a bijective

linear operator C ∈ B(H) such that ±φ(A) = CAC−1, i.e. either φor −φ is an algebraic isomorphism.Here σs(A) is the surjective spectrum of A ∈ B(H).Extensions: Hou, Šemrl, Bai, Xu, Di.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 5 / 18

Conditions for Algebra Isomorphisms Isomorphisms between operator algebras

Isomorphisms between operator algebras

Operator algebra: A ⊂ B(X ), X - Banach space, dim X =∞[Jafarian-Sourour, 1986]: A = B(X ), B = B(Y ), φ : A→ B –linear surjection σ

(φ(T )

)= σ(T ), T ∈ A, =⇒ φ(T ) = ATA−1 for a

linear bijection A : X → Y , or, φ(T ) = BT ∗B−1 for a linear bijectionB : X ∗ → Y , i.e. φ is an isomorphism/anti-isomorphism.[Molnár, 2001]: φ : B(X )→ B(X ) – surjection, σp

((φ(A))(φ(B))

)=

σp(fg), A,B ∈ B(X ) =⇒ either φ or −φ is an algebra isomorphism.Here σp(A) is the point spectrum of A ∈ B(X ).[Molnár, 2001]: H – Hilbert space, φ : B(H)→ B(H) – surjection,σs

(φ(A) ◦ φ(B)

)= σs(AB) for all A,B ∈ A =⇒ there is a bijective

linear operator C ∈ B(H) such that ±φ(A) = CAC−1, i.e. either φor −φ is an algebraic isomorphism.Here σs(A) is the surjective spectrum of A ∈ B(H).Extensions: Hou, Šemrl, Bai, Xu, Di.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 5 / 18

Conditions for Algebra Isomorphisms Isomorphisms between operator algebras

Isomorphisms between operator algebras

(φ(T )

)= σ(T ), T ∈ A, =⇒ φ(T ) = ATA−1 for a

((φ(A))(φ(B))

)=

(φ(A) ◦ φ(B)

)= σs(AB) for all A,B ∈ A =⇒ there is a bijective

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 5 / 18

Standard Operator Algebras

Rank-one operators

Definition 2

A subalgebra A of B(X ), not necessarily complete nor unital, is calleda standard operator algebra if it contains all rank-one operators.

The algebra of finite-rank operators, its closures, and the algebra ofcompact operators on a Banach space – standard operator algebras.

An operator T ∈ B(X ) is of rank at most one if the dimension of itsrange is ≤ 1. Every such operator has the form x ⊗ f for some x ∈ Xand f ∈ X ∗, the dual space of X , where (x ⊗ f ) y = f (y) x .B1(X ) – the set of all operators in B(X ) of rank at most one. Note:

σ(x ⊗ f ) ⊂ {0, f (x)} and σπ(x ⊗ f ) = {f (x)}.

A ◦ (x ⊗ f ) = (Ax)⊗ f , thus, A ◦ (x ⊗ f ) ∈ B1(X ) for all A ∈ B(X ).

σπ

(A ◦ (x ⊗ f )

)=

{f (Ax)

}.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 6 / 18

Standard Operator Algebras

Rank-one operators

Definition 2

A subalgebra A of B(X ), not necessarily complete nor unital, is calleda standard operator algebra if it contains all rank-one operators.

The algebra of finite-rank operators, its closures, and the algebra ofcompact operators on a Banach space – standard operator algebras.

An operator T ∈ B(X ) is of rank at most one if the dimension of itsrange is ≤ 1. Every such operator has the form x ⊗ f for some x ∈ Xand f ∈ X ∗, the dual space of X , where (x ⊗ f ) y = f (y) x .B1(X ) – the set of all operators in B(X ) of rank at most one. Note:

σ(x ⊗ f ) ⊂ {0, f (x)} and σπ(x ⊗ f ) = {f (x)}.

A ◦ (x ⊗ f ) = (Ax)⊗ f , thus, A ◦ (x ⊗ f ) ∈ B1(X ) for all A ∈ B(X ).

σπ

(A ◦ (x ⊗ f )

)=

{f (Ax)

}.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 6 / 18

Standard Operator Algebras

Rank-one operators

Definition 2

A subalgebra A of B(X ), not necessarily complete nor unital, is calleda standard operator algebra if it contains all rank-one operators.

The algebra of finite-rank operators, its closures, and the algebra ofcompact operators on a Banach space – standard operator algebras.

An operator T ∈ B(X ) is of rank at most one if the dimension of itsrange is ≤ 1. Every such operator has the form x ⊗ f for some x ∈ Xand f ∈ X ∗, the dual space of X , where (x ⊗ f ) y = f (y) x .B1(X ) – the set of all operators in B(X ) of rank at most one. Note:

σ(x ⊗ f ) ⊂ {0, f (x)} and σπ(x ⊗ f ) = {f (x)}.

A ◦ (x ⊗ f ) = (Ax)⊗ f , thus, A ◦ (x ⊗ f ) ∈ B1(X ) for all A ∈ B(X ).

σπ

(A ◦ (x ⊗ f )

)=

{f (Ax)

}.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 6 / 18

Standard Operator Algebras

Rank-one operators

Definition 2

σ(x ⊗ f ) ⊂ {0, f (x)} and σπ(x ⊗ f ) = {f (x)}.

A ◦ (x ⊗ f ) = (Ax)⊗ f , thus, A ◦ (x ⊗ f ) ∈ B1(X ) for all A ∈ B(X ).

σπ

(A ◦ (x ⊗ f )

)=

{f (Ax)

}.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 6 / 18

Standard Operator Algebras

Rank-one operators

Definition 2

σ(x ⊗ f ) ⊂ {0, f (x)} and σπ(x ⊗ f ) = {f (x)}.

A ◦ (x ⊗ f ) = (Ax)⊗ f , thus, A ◦ (x ⊗ f ) ∈ B1(X ) for all A ∈ B(X ).

σπ

(A ◦ (x ⊗ f )

)=

{f (Ax)

}.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 6 / 18

Standard Operator Algebras

Rank-one operators

Definition 2

σ(x ⊗ f ) ⊂ {0, f (x)} and σπ(x ⊗ f ) = {f (x)}.

A ◦ (x ⊗ f ) = (Ax)⊗ f , thus, A ◦ (x ⊗ f ) ∈ B1(X ) for all A ∈ B(X ).

σπ

(A ◦ (x ⊗ f )

)=

{f (Ax)

}.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 6 / 18

Standard Operator Algebras

Identification Lemma.

Let X be a Banach space and A,B ∈ B(X ).If σπ(A T ) = σπ(B T ) for every rank-one operator T , then A = B.

Proof.

Let T = x ⊗ f ∈ B1(X ). If σπ(A T ) = σπ(B T ) for all T ∈ B1(X ), then{f (Ax)} = σπ(A ◦ (x ⊗ f )) = σπ(B ◦ (x ⊗ f )) = {f (Bx)}. Since f ∈ X ∗ isarbitrary, Ax = Bx for any x ∈ X , and thus A = B. �

Definition 3.

An operator φ : A→ B between Banach algebras is said to beperipherally-multiplicative if σπ

(φ(A) ◦ φ(B)

)= σπ(AB) for all A,B ∈ A

[Luttman-T., 2005].

Note: φ is not assumed to be linear, nor continuous, nor preservingoperators’ injectivity or surjectivity.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 7 / 18

Standard Operator Algebras

Identification Lemma.

Let X be a Banach space and A,B ∈ B(X ).If σπ(A T ) = σπ(B T ) for every rank-one operator T , then A = B.

Proof.

Let T = x ⊗ f ∈ B1(X ). If σπ(A T ) = σπ(B T ) for all T ∈ B1(X ), then{f (Ax)} = σπ(A ◦ (x ⊗ f )) = σπ(B ◦ (x ⊗ f )) = {f (Bx)}. Since f ∈ X ∗ isarbitrary, Ax = Bx for any x ∈ X , and thus A = B. �

Definition 3.

An operator φ : A→ B between Banach algebras is said to beperipherally-multiplicative if σπ

(φ(A) ◦ φ(B)

)= σπ(AB) for all A,B ∈ A

[Luttman-T., 2005].

Note: φ is not assumed to be linear, nor continuous, nor preservingoperators’ injectivity or surjectivity.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 7 / 18

Standard Operator Algebras

Identification Lemma.

Let X be a Banach space and A,B ∈ B(X ).If σπ(A T ) = σπ(B T ) for every rank-one operator T , then A = B.

Proof.

Let T = x ⊗ f ∈ B1(X ). If σπ(A T ) = σπ(B T ) for all T ∈ B1(X ), then{f (Ax)} = σπ(A ◦ (x ⊗ f )) = σπ(B ◦ (x ⊗ f )) = {f (Bx)}. Since f ∈ X ∗ isarbitrary, Ax = Bx for any x ∈ X , and thus A = B. �

Definition 3.

An operator φ : A→ B between Banach algebras is said to beperipherally-multiplicative if σπ

(φ(A) ◦ φ(B)

)= σπ(AB) for all A,B ∈ A

[Luttman-T., 2005].

Note: φ is not assumed to be linear, nor continuous, nor preservingoperators’ injectivity or surjectivity.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 7 / 18

Standard Operator Algebras

Identification Lemma.

Proof.

Definition 3.

An operator φ : A→ B between Banach algebras is said to beperipherally-multiplicative if σπ

(φ(A) ◦ φ(B)

)= σπ(AB) for all A,B ∈ A

[Luttman-T., 2005].

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 7 / 18

Standard Operator Algebras Peripherally-multiplicative operators

Peripherally-multiplicative operators

Lemma 1.

A peripherally-multiplicative operator φ : A→ B(Y ) on a standardoperator algebra A is injective.

Proof.

Indeed, if φ(A) = φ(B) for A,B ∈ A, then the peripheral multiplicativityof φ yields that for every T ∈ B1(X ),

σπ(A T ) = σπ

(φ(A) ◦ φ(T )

)= σπ

(φ(B) ◦ φ(T )

)= σπ(B T ).

The Identification Lemma yields A = B, hence, φ is injective. �

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 8 / 18

Standard Operator Algebras Peripherally-multiplicative operators

Peripherally-multiplicative operators

Lemma 1.

A peripherally-multiplicative operator φ : A→ B(Y ) on a standardoperator algebra A is injective.

Proof.

Indeed, if φ(A) = φ(B) for A,B ∈ A, then the peripheral multiplicativityof φ yields that for every T ∈ B1(X ),

σπ(A T ) = σπ

(φ(A) ◦ φ(T )

)= σπ

(φ(B) ◦ φ(T )

)= σπ(B T ).

The Identification Lemma yields A = B, hence, φ is injective. �

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 8 / 18

Standard Operator Algebras Peripherally-multiplicative operators

Lemma 2.

A peripherally-multiplicative surjective operator φ : A→ B is linear.

Sketch of the proof.

If T = u ⊗ g ∈ B1(Y ) for some u ∈ Y and g ∈ Y ∗, then T = φ(S) forsome S = x ⊗ f ∈ B1(X ) with x ∈ X and f ∈ X ∗. Then

σπ

(φ(λA+µB)◦T

)= σπ

(φ(λA+µB)◦φ(S)

)= σπ

((λA+µB)◦S

)= . . .

= σπ

((λφ(A) + µφ(B)) ◦ φ(S)

)= σπ

((λφ(A) + µφ(B)) ◦ T

).

The Identification Lemma yields φ(λA + µB) = λφ(A) + µφ(B). �

Lemma 3.

A peripherally-multiplicative operator φ : A→ B between standardoperator algebras preserves the rank-one operators, i.e.

φ(A) ∈ B1(Y )⇐⇒ A ∈ B1(X ).

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 9 / 18

Standard Operator Algebras Peripherally-multiplicative operators

Lemma 2.

A peripherally-multiplicative surjective operator φ : A→ B is linear.

Sketch of the proof.

If T = u ⊗ g ∈ B1(Y ) for some u ∈ Y and g ∈ Y ∗, then T = φ(S) forsome S = x ⊗ f ∈ B1(X ) with x ∈ X and f ∈ X ∗. Then

σπ

(φ(λA+µB)◦T

)= σπ

(φ(λA+µB)◦φ(S)

)= σπ

((λA+µB)◦S

)= . . .

= σπ

((λφ(A) + µφ(B)) ◦ φ(S)

)= σπ

((λφ(A) + µφ(B)) ◦ T

).

The Identification Lemma yields φ(λA + µB) = λφ(A) + µφ(B). �

Lemma 3.

A peripherally-multiplicative operator φ : A→ B between standardoperator algebras preserves the rank-one operators, i.e.

φ(A) ∈ B1(Y )⇐⇒ A ∈ B1(X ).

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 9 / 18

Standard Operator Algebras Peripherally-multiplicative operators

Lemma 2.

A peripherally-multiplicative surjective operator φ : A→ B is linear.

Sketch of the proof.

If T = u ⊗ g ∈ B1(Y ) for some u ∈ Y and g ∈ Y ∗, then T = φ(S) forsome S = x ⊗ f ∈ B1(X ) with x ∈ X and f ∈ X ∗. Then

σπ

(φ(λA+µB)◦T

)= σπ

(φ(λA+µB)◦φ(S)

)= σπ

((λA+µB)◦S

)= . . .

= σπ

((λφ(A) + µφ(B)) ◦ φ(S)

)= σπ

((λφ(A) + µφ(B)) ◦ T

).

The Identification Lemma yields φ(λA + µB) = λφ(A) + µφ(B). �

Lemma 3.

A peripherally-multiplicative operator φ : A→ B between standardoperator algebras preserves the rank-one operators, i.e.

φ(A) ∈ B1(Y )⇐⇒ A ∈ B1(X ).

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 9 / 18

Standard Operator Algebras Peripherally-multiplicative operators

Proposition 1.

If φ : A→ B is a linear surjective operator between two standardoperator algebras which preserves rank-one operators, then one of thefollowing holds:

1 There are bijective linear operators C : X → Y and D : X ∗ → Y ∗

so that φ(x ⊗ f ) = Cx ⊗ Df for every x ∈ X and f ∈ X ∗, or,2 There are bijective linear operators E : X ∗ → Y and F : X → Y ∗

so that φ(x ⊗ f ) = Ef ⊗ Fx for all x ∈ X and f ∈ X ∗.

The proof makes use of Jafarian-Sourour’s arguments for the case ofspectrum-preserving linear operators φ : B(X )→ B(X ).

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 10 / 18

Standard Operator Algebras Peripherally-multiplicative operators

Proposition 1.

If φ : A→ B is a linear surjective operator between two standardoperator algebras which preserves rank-one operators, then one of thefollowing holds:

1 There are bijective linear operators C : X → Y and D : X ∗ → Y ∗

so that φ(x ⊗ f ) = Cx ⊗ Df for every x ∈ X and f ∈ X ∗, or,2 There are bijective linear operators E : X ∗ → Y and F : X → Y ∗

so that φ(x ⊗ f ) = Ef ⊗ Fx for all x ∈ X and f ∈ X ∗.

The proof makes use of Jafarian-Sourour’s arguments for the case ofspectrum-preserving linear operators φ : B(X )→ B(X ).

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 10 / 18

Standard Operator Algebras Peripherally-multiplicative operators

Proposition 2.

If φ : A→ B is peripherally-multiplicative surjective operator betweentwo standard operator algebras, then one of the following holds:

1 There are bijective linear operators C : X → Y and D : X ∗ → Y ∗

so that φ(x ⊗ f ) = Cx ⊗ Df for every x ∈ X and f ∈ X ∗, or,2 There are bijective linear operators E : X ∗ → Y and F : X → Y ∗

so that φ(x ⊗ f ) = Ef ⊗ Fx for all x ∈ X and f ∈ X ∗.

This follows from Proposition 1, since any peripherally-multiplicativeoperator is linear and preserves rank-one operators.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 11 / 18

Standard Operator Algebras Peripherally-multiplicative operators

Proposition 2.

If φ : A→ B is peripherally-multiplicative surjective operator betweentwo standard operator algebras, then one of the following holds:

1 There are bijective linear operators C : X → Y and D : X ∗ → Y ∗

so that φ(x ⊗ f ) = Ef ⊗ Fx for all x ∈ X and f ∈ X ∗.

This follows from Proposition 1, since any peripherally-multiplicativeoperator is linear and preserves rank-one operators.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 11 / 18

The Main Theorem

Theorem [Luttman - T., 2009]

Let X ,Y be Banach spaces, A ⊂ B(X ), B ⊂ B(Y ) be standardoperator algebras, and let φ : A −→ B be a surjective operator. If φ isperipherally-multiplicative, i.e. σπ

(φ(A) ◦ φ(B)

)= σπ(A B) for every

A,B ∈ A, then φ is a bounded linear operator and1 there exists a bijective linear operator C : X → Y such that±φ(A) = CAC−1 for every A ∈ A, or,

2 there exists a bijective linear operator E : X ∗ → Y such that±φ(A) = EA∗E−1 for every A ∈ A.

Therefore, either φ or −φ is multiplicative/anti-multiplicative, thus eitherφ or −φ is an algebra isomorphism/anti-isomorphism.

Sketch of the proof.

If φ is of type (1) then φ(x ⊗ f ) = Cx ⊗ Df for any x ∈ X and f ∈ X ∗,where C : X → Y and D : X ∗ → Y ∗ are bijective linear operators. Nowσπ

((x ⊗ f ) ◦ (x ⊗ f )

)= σπ

{(f (x) x)⊗ f

}=

{(f (x))2}, and

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 12 / 18

The Main Theorem

Theorem [Luttman - T., 2009]

Let X ,Y be Banach spaces, A ⊂ B(X ), B ⊂ B(Y ) be standardoperator algebras, and let φ : A −→ B be a surjective operator. If φ isperipherally-multiplicative, i.e. σπ

(φ(A) ◦ φ(B)

)= σπ(A B) for every

A,B ∈ A, then φ is a bounded linear operator and1 there exists a bijective linear operator C : X → Y such that±φ(A) = CAC−1 for every A ∈ A, or,

2 there exists a bijective linear operator E : X ∗ → Y such that±φ(A) = EA∗E−1 for every A ∈ A.

Therefore, either φ or −φ is multiplicative/anti-multiplicative, thus eitherφ or −φ is an algebra isomorphism/anti-isomorphism.

Sketch of the proof.

If φ is of type (1) then φ(x ⊗ f ) = Cx ⊗ Df for any x ∈ X and f ∈ X ∗,where C : X → Y and D : X ∗ → Y ∗ are bijective linear operators. Nowσπ

((x ⊗ f ) ◦ (x ⊗ f )

)= σπ

{(f (x) x)⊗ f

}=

{(f (x))2}, and

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 12 / 18

The Main Theorem

Continuation of the proof:

σπ

(φ(x ⊗ f ) ◦ φ(x ⊗ f )

)= σπ

((Cx ⊗ Df ) ◦ (Cx ⊗ Df )

)=

σπ

((((Df )(Cx)) Cx

)⊗ Df

)=

{((Df )(Cx)

)2}. The peripheralmultiplicativity implies that (f (x))2 =

((Df )(Cx)

)2, and hence(Df )(Cx) = ± f (x). For any A ∈ A and x ⊗ f ∈ B1(X ) we have

{f (Ax)} = σπ(Ax ⊗ f ) = σπ

(A ◦ (x ⊗ f )

)= σπ

(φ(A) ◦ φ(x ⊗ f )

)=

σπ

(φ(A) ◦ (Cx ⊗ Df )

)= σπ

((φ(A)Cx)⊗ Df

)=

={(Df )

(CC−1(φ(A)Cx)

)}= (± f

((C−1φ(A)

)Cx

)}.

Since this holds for every f ∈ X ∗, it follows that Ax =(± C−1φ(A) C

)x ,

i.e. ±φ(A) Cx = CA x . Hence, φ(A) y = ± (C A C−1) y for any y ∈ Y .Therefore, φ(A) = ±C A C−1, thus ±φ is an algebra isomorphism.Similarly, if φ is of type (2), then φ(x ⊗ f ) = Ef ⊗ Fx for any x ∈ X andf ∈ X ∗, where E : Y ∗ → X and F : X → Y ∗ are bijective linearoperators, and φ(A) = ±E A∗ E−1. therefore, ±φ is an algebraanti-isomorphism. In both cases φ extends to a linear bijectionbetween B(X ) and B(Y ). �

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 13 / 18

The Main Theorem

If, in addition, φ preserves the peripheral spectra of operators in B1(X ),then the "–" case of Theorem 1 is ruled out.

Theorem [Luttman - T., 2009].

Let X ,Y be Banach spaces, A ⊂ B(X ), B ⊂ B(Y ) be standardoperator algebras, and let φ : A→ B be a surjective operator, notassumed to be linear or continuous. If σπ

(φ(A) ◦ φ(B)

)= σπ(AB), i.e.

φ is peripherally-multiplicative, and σπ

(φ(A)

)= σπ(A) for all A,B ∈ A,

then φ is a bijective and bounded linear operator and1 there exists a bijective linear operator C : X → Y such thatφ(A) = CAC−1 for every A ∈ A, i.e. φ is multiplicative, or,

2 there exists a bijective linear operator E : X ∗ → Y such thatφ(A) = EA∗E−1 for every A ∈ A, i.e. φ is anti-multiplicative.

Indeed, if T = x ⊗ f ∈ B1(X ) with f (x) 6= 0, then

σπ(−CTC−1) = {−f (x)} 6= {f (x)} = σπ(T ).

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 14 / 18

The Main Theorem

If, in addition, φ preserves the peripheral spectra of operators in B1(X ),then the "–" case of Theorem 1 is ruled out.

Theorem [Luttman - T., 2009].

Let X ,Y be Banach spaces, A ⊂ B(X ), B ⊂ B(Y ) be standardoperator algebras, and let φ : A→ B be a surjective operator, notassumed to be linear or continuous. If σπ

(φ(A) ◦ φ(B)

)= σπ(AB), i.e.

φ is peripherally-multiplicative, and σπ

(φ(A)

)= σπ(A) for all A,B ∈ A,

then φ is a bijective and bounded linear operator and1 there exists a bijective linear operator C : X → Y such thatφ(A) = CAC−1 for every A ∈ A, i.e. φ is multiplicative, or,

2 there exists a bijective linear operator E : X ∗ → Y such thatφ(A) = EA∗E−1 for every A ∈ A, i.e. φ is anti-multiplicative.

Indeed, if T = x ⊗ f ∈ B1(X ) with f (x) 6= 0, then

σπ(−CTC−1) = {−f (x)} 6= {f (x)} = σπ(T ).

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 14 / 18

The Main Theorem

Note that if one of the algebras A,B is the algebra of compactoperators (or, of finite rank operators), then so is the other.

Corollary 1.

A surjective peripherally-multiplicative operator φ : K(X )→ K(Y ), i.e.σπ

(φ(A) ◦ φ(B)

)= σπ(AB), A,B ∈ A, with σπ

(φ(A)

)= σπ(A), A ∈ A, is

continuous algebra isomorphism/anti-isomorphism.

Corollary 2.

A surjective peripherally-multiplicative operator φ : BF (X )→ BF (Y ), i.e.σπ

(φ(A) ◦ φ(B)

)= σπ(AB), A,B ∈ A, with σπ

(φ(A)

)= σπ(A), A ∈ A, is

continuous algebra isomorphism/anti-isomorphism.

Since a unital peripherally-multiplicative operator φ preserves theperipheral spectra σπ(A) for all A ∈ B1(X ), Theorem 3 implies:

Corollary 3.

Let A ⊂ B(X ) and B ⊂ B(Y ) be unital standard operator algebras. Asurjective unital peripherally-multiplicative operator φ : A→ B is, i.e.σπ

(φ(A) ◦ φ(B)

)= σπ(AB), A,B ∈ A, is an algebra

isomorphism/anti-isomorphism.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 15 / 18

The Main Theorem

Note that if one of the algebras A,B is the algebra of compactoperators (or, of finite rank operators), then so is the other.

Corollary 1.

A surjective peripherally-multiplicative operator φ : K(X )→ K(Y ), i.e.σπ

(φ(A) ◦ φ(B)

)= σπ(AB), A,B ∈ A, with σπ

(φ(A)

)= σπ(A), A ∈ A, is

continuous algebra isomorphism/anti-isomorphism.

Corollary 2.

A surjective peripherally-multiplicative operator φ : BF (X )→ BF (Y ), i.e.σπ

(φ(A) ◦ φ(B)

)= σπ(AB), A,B ∈ A, with σπ

(φ(A)

)= σπ(A), A ∈ A, is

continuous algebra isomorphism/anti-isomorphism.

Since a unital peripherally-multiplicative operator φ preserves theperipheral spectra σπ(A) for all A ∈ B1(X ), Theorem 3 implies:

Corollary 3.

Let A ⊂ B(X ) and B ⊂ B(Y ) be unital standard operator algebras. Asurjective unital peripherally-multiplicative operator φ : A→ B is, i.e.σπ

(φ(A) ◦ φ(B)

)= σπ(AB), A,B ∈ A, is an algebra

isomorphism/anti-isomorphism.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 15 / 18

The Main Theorem

Note that if one of the algebras A,B is the algebra of compactoperators (or, of finite rank operators), then so is the other.

Corollary 1.

A surjective peripherally-multiplicative operator φ : K(X )→ K(Y ), i.e.σπ

(φ(A) ◦ φ(B)

)= σπ(AB), A,B ∈ A, with σπ

(φ(A)

)= σπ(A), A ∈ A, is

continuous algebra isomorphism/anti-isomorphism.

Corollary 2.

A surjective peripherally-multiplicative operator φ : BF (X )→ BF (Y ), i.e.σπ

(φ(A) ◦ φ(B)

)= σπ(AB), A,B ∈ A, with σπ

(φ(A)

)= σπ(A), A ∈ A, is

continuous algebra isomorphism/anti-isomorphism.

Since a unital peripherally-multiplicative operator φ preserves theperipheral spectra σπ(A) for all A ∈ B1(X ), Theorem 3 implies:

Corollary 3.

Let A ⊂ B(X ) and B ⊂ B(Y ) be unital standard operator algebras. Asurjective unital peripherally-multiplicative operator φ : A→ B is, i.e.σπ

(φ(A) ◦ φ(B)

)= σπ(AB), A,B ∈ A, is an algebra

isomorphism/anti-isomorphism.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 15 / 18

The Main Theorem

Corollary 1.

A surjective peripherally-multiplicative operator φ : K(X )→ K(Y ), i.e.σπ

(φ(A) ◦ φ(B)

)= σπ(AB), A,B ∈ A, with σπ

(φ(A)

)= σπ(A), A ∈ A, is

continuous algebra isomorphism/anti-isomorphism.

Corollary 2.

A surjective peripherally-multiplicative operator φ : BF (X )→ BF (Y ), i.e.σπ

(φ(A) ◦ φ(B)

)= σπ(AB), A,B ∈ A, with σπ

(φ(A)

)= σπ(A), A ∈ A, is

continuous algebra isomorphism/anti-isomorphism.

Corollary 3.

(φ(A) ◦ φ(B)

)= σπ(AB), A,B ∈ A, is an algebra

isomorphism/anti-isomorphism.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 15 / 18

The Main Theorem

Theorem [Miura-Honma, 2009]

Let A and B be standard operator algebras on X and Y . If twosurjective (not necessarily linear nor continuous) operatorsφ, ψ : A→ B satisfy σπ

(φ(S)ψ(T )

)= σπ(ST ) for all S,T ∈ A, then one

of the following holds:1 there exist bijective bounded linear operators A1,A2 : X → Y such

that φ(T ) = A1TA−12 and ψ(T ) = A2TA−1

1 , T ∈ A, or2 there exist bijective bounded linear operators B1,B2 : X ∗ → Y

such that φ(T ) = B1T ∗B−12 and ψ(T ) = B2T ∗B−1

1 , T ∈ A. In thiscase, both X and Y are necessarily reflexive.

If, in addition, both A and B have unit I and φ(I) = I, then φ = ψ is analgebra isomorphism/anti-isomorphism.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 16 / 18

The Main Theorem

Theorem [Miura-Honma, 2009]

Let A and B be standard operator algebras on X and Y . If twosurjective (not necessarily linear nor continuous) operatorsφ, ψ : A→ B satisfy σπ

(φ(S)ψ(T )

)= σπ(ST ) for all S,T ∈ A, then one

of the following holds:1 there exist bijective bounded linear operators A1,A2 : X → Y such

that φ(T ) = A1TA−12 and ψ(T ) = A2TA−1

1 , T ∈ A, or2 there exist bijective bounded linear operators B1,B2 : X ∗ → Y

such that φ(T ) = B1T ∗B−12 and ψ(T ) = B2T ∗B−1

1 , T ∈ A. In thiscase, both X and Y are necessarily reflexive.

If, in addition, both A and B have unit I and φ(I) = I, then φ = ψ is analgebra isomorphism/anti-isomorphism.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 16 / 18

Symmetric spectral conditions

Theorem [Molnár, 2001]

Let H, dim H =∞, be a Hilbert space and φ : B(H)→ B(H) be asurjective operator such that σ

(φ(A)∗φ(B)

)= σ(A∗B) for all

A,B ∈ B(H). Then there are unitary operators U,V ∈ B(H) such thatφ(A) = UAV for all A ∈ B(H).

Here A∗ the Banach space adjoint of A ∈ B(H).

Theorem [Honma-Miura, 2009]

Let H be a Hilbert space and let A and B be unital ∗-standard operatoralgebras on H. If a surjective operator φ : A→ B is such thatσπ

(φ(A)∗φ(B)

)= σπ(A∗B) for all A,B ∈ A, then there exist unitary

operators U,V ∈ B(H) such that1 φ(A) = UAV , A ∈ A, or2 φ(A) = UAtr V , A ∈ A.

Atr – transpose of A with respect to a fixed orthonormal basis of H.T. Tonev (UM) Standard operator algebras Bedlewo, 2009 17 / 18

Symmetric spectral conditions

Theorem [Molnár, 2001]

Let H, dim H =∞, be a Hilbert space and φ : B(H)→ B(H) be asurjective operator such that σ

(φ(A)∗φ(B)

)= σ(A∗B) for all

A,B ∈ B(H). Then there are unitary operators U,V ∈ B(H) such thatφ(A) = UAV for all A ∈ B(H).

Here A∗ the Banach space adjoint of A ∈ B(H).

Theorem [Honma-Miura, 2009]

Let H be a Hilbert space and let A and B be unital ∗-standard operatoralgebras on H. If a surjective operator φ : A→ B is such thatσπ

(φ(A)∗φ(B)

)= σπ(A∗B) for all A,B ∈ A, then there exist unitary

operators U,V ∈ B(H) such that1 φ(A) = UAV , A ∈ A, or2 φ(A) = UAtr V , A ∈ A.

Atr – transpose of A with respect to a fixed orthonormal basis of H.T. Tonev (UM) Standard operator algebras Bedlewo, 2009 17 / 18

References

References

[GT] S. Grigoryan and T. Tonev, Shift-Invariant Uniform Algebras onGroups, Monografie Matematyczne 68, New Series, BirkhauserVerlag, Basel-Boston-Berlin, 2006.

[LT-2] A. Luttman and T. Tonev, Algebra isomorphisms betweenstandard operator algebras, Studia Math., 191(2009), 163-170.

[M-1] L. Molnár, Selected preserver problems on algebraic structuresof linear operators and on function spaces, Lecture Notes inMathematics, 1895, Springer-Verlag, Berlin, 2007.

T. Tonev (UM) Standard operator algebras Bedlewo, 2009 18 / 18

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