Shift-invariant algebras on groups

Abstract. Let ϕ : A → B be a surjective operator between two uniform algebras with ϕ(1) = 1. We show that if ϕ satisfies the peripheral multiplicativity conditionwhere σ π (f ) is the peripheral spectrum of f , then ϕ is an isometric algebra isomorphism from A onto B. One of the consequences of this result is that any surjective, unital, and multiplicative operator that preserves the peripheral ranges of algebra elements is an isometric algebra isomorphism. We describe also the structure of general, not necessarily unital, surjective and peripherally multiplicative operators between uniform algebras.An important question in Banach algebra theory, which still lacks a satisfactory answer, is to find criteria for an operator between two Banach algebras to be linear and multiplicative. For linear operators between semisimple algebras an answer is suggested by the theorem of Gleason-Kahane-Żelazko (e.g., [11]) in terms of spectra of algebra elements. A theorem by Kowalski and S lodkowski [6] considers alternative spectral conditions for not necessarily linear operators. N. V. Rao and A. K. Roy [9] have introduced an interesting spectral multiplicativity condition that contributes to the matter. In particular for unital, that is Φ(1) = 1, operators Φ they have proven the following:If A is a uniform algebra on X and Φ : A → A is a surjective unital operator such that where Ran (f ) = f (X) is the range of f . N. V. Rao, T. Tonev and E. Toneva [10] (see also [2]) considered a spectral additivity condition related to the peripheral

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“…N. V. Rao, T. Tonev and E. Toneva [10] (see also [2]) considered a spectral additivity condition related to the peripheral…”

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confidence: 99%

“…N. V. Rao, T. Tonev and E. Toneva [10] (see also [2]) considered a spectral additivity condition related to the peripheral spectra of algebra elements and gave alternative conditions for an operator to be an algebra isomorphism.…”

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confidence: 99%

Uniform algebra isomorphisms and peripheral multiplicativity

Luttman

Tonev

2007

Proc. Amer. Math. Soc.

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“…[3,4,5,8,10,13]). Rather than requiring that such a map multiplicatively preserves the entire spectrum, however, it is also natural to ask whether preserving particular subsets of the spectrum will suffice.…”

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confidence: 99%

“…The peripheral spectrum σ π (A) = {λ ∈ σ(A) : |λ| = max z∈σ(A) |z|} of A is the set of spectral values of A of maximum modulus. In [3,10] Molnár's result is extended to surjective maps ϕ : A → B between two uniform algebras that instead of (1) satisfy the equation…”

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confidence: 99%

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Algebra isomorphisms between standard operator algebras

Tonev

Luttman

2009

Studia Math.

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Abstract. If X and Y are Banach spaces, then subalgebras A ⊂ B(X) and B ⊂ B(Y ), not necessarily unital nor complete, are called standard operator algebras if they contain all finite rank operators on X and Y respectively. The peripheral spectrum of A ∈ A is the set σπ(A) = {λ ∈ σ(A) : |λ| = max z∈σ(A) |z|} of spectral values of A of maximum modulus, and a map ϕ : A → B is called peripherally-multiplicative if it satisfies the equation σπ(ϕ(A) • ϕ(B)) = σπ(AB) for all A, B ∈ A. We show that any peripherally-multiplicative and surjective map ϕ : A → B, neither assumed to be linear nor continuous, is a bijective bounded linear operator such that either ϕ or −ϕ is multiplicative or anti-multiplicative. This holds in particular for the algebras of finite rank operators or of compact operators on X and Y and extends earlier results of Molnár. If, in addition, σπ(ϕ(A0)) = −σπ(A0) for some A0 ∈ A then ϕ is either multiplicative, in which case X is isomorphic to Y , or anti-multiplicative, in which case X is isomorphic to Y * . Therefore, if X ∼ = Y * then ϕ is multiplicative, hence an algebra isomorphism, while if X ∼ = Y , then ϕ is anti-multiplicative, hence an algebra anti-isomorphism.

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Norm-linear and norm-additive operators between uniform algebras

Tonev

Yates

2009

Journal of Mathematical Analysis and Applications

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Generalized peak point Choquet boundary Shilov boundary Peripheral spectrum Homeomorphism Algebra isomorphism Norm-additive operator Norm-linear operatorLet A ⊂ C (X) and B ⊂ C (Y ) be uniform algebras with Choquet boundaries δ A and δ B.λ, μ ∈ C and all algebra elements f and g. We show that for any norm-linear surjection (y))| for every f ∈ A and y ∈ δ B. Sufficient conditions for norm-additive and norm-linear surjections, not assumed a priori to be linear, or continuous, to be unital isometric algebra isomorphisms are given. We prove that any unital norm-linear surjection T for which T (i) = i, or which preserves the peripheral spectra of C-peaking functions of A, is a unital isometric algebra isomorphism. In particular, we show that if a linear operator between two uniform algebras, which is surjective and norm-preserving, is unital, or preserves the peripheral spectra of C-peaking functions, then it is automatically multiplicative and, in fact, an algebra isomorphism.

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Shift-invariant algebras on groups

Cited by 9 publications

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Uniform algebra isomorphisms and peripheral multiplicativity

Uniform algebra isomorphisms and peripheral multiplicativity

Algebra isomorphisms between standard operator algebras

Norm-linear and norm-additive operators between uniform algebras

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