Three spectral notions for representations of commutative Banach algebras

Yngve Domar; Lars-Ake Lindahl

Displaying similar documents to “Three spectral notions for representations of commutative Banach algebras”

The norm spectrum in certain classes of commutative Banach algebras

H. S. Mustafayev (2011)

Colloquium Mathematicae

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Let A be a commutative Banach algebra and let $Σ_{A}$ be its structure space. The norm spectrum σ(f) of the functional f ∈ A* is defined by $σ (f) = \bar{f \cdot a : a \in A} \cap Σ_{A}$ , where f·a is the functional on A defined by ⟨f·a,b⟩ = ⟨f,ab⟩, b ∈ A. We investigate basic properties of the norm spectrum in certain classes of commutative Banach algebras and present some applications.

On certain products of Banach algebras with applications to harmonic analysis

Mehdi Sangani Monfared (2007)

Studia Mathematica

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Given Banach algebras A and B with spectrum σ(B) ≠ ∅, and given θ ∈ σ(B), we define a product $A \times_{θ} B$ , which is a strongly splitting Banach algebra extension of B by A. We obtain characterizations of bounded approximate identities, spectrum, topological center, minimal idempotents, and study the ideal structure of these products. By assuming B to be a Banach algebra in ₀(X) whose spectrum can be identified with X, we apply our results to harmonic analysis, and study the question of spectral...

Spectrum of commutative Banach algebras and isomorphism of C*-algebras related to locally compact groups

Zhiguo Hu (1998)

Studia Mathematica

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Let A be a semisimple commutative regular tauberian Banach algebra with spectrum $Σ_{A}$ . In this paper, we study the norm spectra of elements of $\bar{s p a n} Σ_{A}$ and present some applications. In particular, we characterize the discreteness of $Σ_{A}$ in terms of norm spectra. The algebra A is said to have property (S) if, for all $φ \in \bar{} Σ_{A} 0$ , φ has a nonempty norm spectrum. For a locally compact group G, let $ℳ_{2}^{d} (Ĝ)$ denote the C*-algebra generated by left translation operators on $L^{2} (G)$ and $G_{d}$ denote the discrete group G. We prove...

A spectral theory for locally compact abelian groups of automorphisms of commutative Banach algebras

Sen Huang (1999)

Studia Mathematica

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Let A be a commutative Banach algebra with Gelfand space ∆ (A). Denote by Aut (A) the group of all continuous automorphisms of A. Consider a σ(A,∆(A))-continuous group representation α:G → Aut(A) of a locally compact abelian group G by automorphisms of A. For each a ∈ A and φ ∈ ∆(A), the function $φ_{a} (t) : = φ (α_{t} a)$ t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum $σ_{w} * (φ_{a})$ is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define $Ʌ_{φ}^{a}$ to be the...

Bounded elements and spectrum in Banach quasi *-algebras

Camillo Trapani (2006)

Studia Mathematica

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A normal Banach quasi *-algebra (,) has a distinguished Banach *-algebra $_{b}$ consisting of bounded elements of . The latter *-algebra is shown to coincide with the set of elements of having finite spectral radius. If the family () of bounded invariant positive sesquilinear forms on contains sufficiently many elements then the Banach *-algebra of bounded elements can be characterized via a C*-seminorm defined by the elements of ().

Isometries between groups of invertible elements in Banach algebras

Osamu Hatori (2009)

Studia Mathematica

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We show that if T is an isometry (as metric spaces) from an open subgroup of the group of invertible elements in a unital semisimple commutative Banach algebra A onto a open subgroup of the group of invertible elements in a unital Banach algebra B, then $T {(1)}^{- 1} T$ is an isometrical group isomorphism. In particular, $T {(1)}^{- 1} T$ extends to an isometrical real algebra isomorphism from A onto B.

Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras

Enrico Boasso

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Given two complex Banach spaces X₁ and X₂, a tensor product X₁ ⊗̃ X₂ of X₁ and X₂ in the sense of [14], two complex solvable finite-dimensional Lie algebras L₁ and L₂, and two representations $ϱ_{i} : L_{i} \to L (X_{i})$ of the algebras, i = 1,2, we consider the Lie algebra L = L₁ × L₂ and the tensor product representation of L, ϱ: L → L(X₁ ⊗̃ X₂), ϱ = ϱ₁ ⊗ I + I ⊗ ϱ₂. We study the Słodkowski and split joint spectra of the representation ϱ, and we describe them in terms of the corresponding joint spectra of ϱ₁...

Algebra isomorphisms between standard operator algebras

Thomas Tonev, Aaron Luttman (2009)

Studia Mathematica

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If X and Y are Banach spaces, then subalgebras ⊂ B(X) and ⊂ 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 ∈ is the set $σ_{π} (A) = λ \in σ (A) : | λ | = m a x_{z \in σ (A)} | z |$ of spectral values of A of maximum modulus, and a map φ: → is called peripherally-multiplicative if it satisfies the equation $σ_{π} (φ (A) \circ φ (B)) = σ_{π} (A B)$ for all A,B ∈ . We show that any peripherally-multiplicative and surjective map φ: → , neither assumed to be...

Non-commutative Gelfand-Naimark theorem

Janusz Migda (1993)

Commentationes Mathematicae Universitatis Carolinae

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We show that if Y is the Hausdorffization of the primitive spectrum of a $C^{*}$ -algebra $A$ then $A$ is $*$ -isomorphic to the $C^{*}$ -algebra of sections vanishing at infinity of the canonical $C^{*}$ -bundle over $Y$ .

On the defect spectrum of an extension of a Banach space operator

Vladimír Kordula (1998)

Czechoslovak Mathematical Journal

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Let $T$ be an operator acting on a Banach space $X$ . We show that between extensions of $T$ to some Banach space $Y \supset X$ which do not increase the defect spectrum (or the spectrum) it is possible to find an extension with the minimal possible defect spectrum.

Conditions equivalent to C* independence

Shuilin Jin, Li Xu, Qinghua Jiang, Li Li (2012)

Studia Mathematica

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Let and be mutually commuting unital C* subalgebras of (). It is shown that and are C* independent if and only if for all natural numbers n, m, for all n-tuples A = (A₁, ..., Aₙ) of doubly commuting nonzero operators of and m-tuples B = (B₁, ..., Bₘ) of doubly commuting nonzero operators of , $S p (A, B) = S p (A) \times S p (B)$ , where Sp denotes the joint Taylor spectrum.

On a class of inner maps

Edoardo Vesentini (2005)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let $f$ be a continuous map of the closure $\bar{Δ}$ of the open unit disc $Δ$ of $C$ into a unital associative Banach algebra $A$ , whose restriction to $Δ$ is holomorphic, and which satisfies the condition whereby $0 \notin σ (f (z)) \subset \bar{Δ}$ for all $z \in Δ$ and $σ (f (z)) \subset \partial Δ$ whenever $z \in \partial Δ$ (where $σ (x)$ is the spectrum of any $x \in A$ ). One of the basic results of the present paper is that $f$ is , that is to say, $σ (f (z))$ is then a compact subset of $\partial Δ$ that does not depend on $z$ for all $z \in \bar{Δ}$ . This fact will be applied to holomorphic self-maps of the open unit ball of some...