algebra isomorphisms between standard operator algebras
TRANSCRIPT
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:
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
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
[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
[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 –
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 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
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
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
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
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.
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 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 ) = 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
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
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
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