Algebra isomorphisms between standard operator algebras

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

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 .

Introduction

Question:

Definition 1.

σπ(f ) = σ(f ) ∩{

z ∈ C : |z| = r(f )}

Introduction

Question:

Definition 1.

σπ(f ) = σ(f ) ∩{

z ∈ C : |z| = r(f )}

Introduction

Question:

Definition 1.

σπ(f ) = σ(f ) ∩{

z ∈ C : |z| = r(f )}

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.

(λ(Tf ) + µ(Tg)

)= σπ(λf + µg), f ,g ∈ A, λ, µ ∈ C,

(λ(Tf ) + µ(Tg)

)= σπ(λf + µg), f ,g ∈ A, λ, µ ∈ C,

(λ(Tf ) + µ(Tg)

)= σπ(λf + µg), f ,g ∈ A, λ, µ ∈ C,

(λ(Tf ) + µ(Tg)

)= σπ(λf + µg), f ,g ∈ A, λ, µ ∈ C,

[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.

((Tf )(Tg)

)= σπ(fg), f ,g ∈ A, i.e. T –

((Tf )(Tg)

)∩ σπ(fg) 6= ∅, f ,g ∈ A, i.e. T –

((Tf )(Tg)

)= σπ(fg), f ,g ∈ A, i.e. T –

((Tf )(Tg)

)∩ σπ(fg) 6= ∅, f ,g ∈ A, i.e. T –

((Tf )(Tg)

)= σπ(fg), f ,g ∈ A, i.e. T –

((Tf )(Tg)

)∩ σπ(fg) 6= ∅, f ,g ∈ A, i.e. T –

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.

(φ(T )

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

((φ(A))(φ(B))

(φ(A) ◦ φ(B)

(φ(T )

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

((φ(A))(φ(B))

(φ(A) ◦ φ(B)

(φ(T )

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

((φ(A))(φ(B))

(φ(A) ◦ φ(B)

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: