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

Zhiguo Hu

Displaying similar documents to “Spectrum of commutative Banach algebras and isomorphism of C*-algebras related to locally compact groups”

Three spectral notions for representations of commutative Banach algebras

Yngve Domar, Lars-Ake Lindahl (1975)

Annales de l'institut Fourier

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Let $T$ be a bounded representation of a commutative Banach algebra $B$ . The following spectral sets are studied. $Λ_{1} (T)$ : the Gelfand space of the quotient algebra $B / Ker T$ . $Λ_{2} (T)$ : the Gelfand space of the operator algebra $Im T$ . $Λ_{3} (T)$ : those characters $φ$ of $B$ for which the inequalities $∥ T_{b} x - \hat{b} (φ) x ∥ < ϵ ∥ x ∥$ , $b \in F$ , have a common solution $x \neq 0$ , for any $ϵ > 0$ and any finite subset $F$ of $B$ . A theorem of Beurling on the spectrum of $L^{\infty}$ -functions and results of Slodkowski and Zelazko on joint topological divisors of zero appear as special cases of...

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.

$C (X)$ can sometimes determine $X$ without $X$ being realcompact

Melvin Henriksen, Biswajit Mitra (2005)

Commentationes Mathematicae Universitatis Carolinae

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As usual $C (X)$ will denote the ring of real-valued continuous functions on a Tychonoff space $X$ . It is well-known that if $X$ and $Y$ are realcompact spaces such that $C (X)$ and $C (Y)$ are isomorphic, then $X$ and $Y$ are homeomorphic; that is $C (X)$ $X$ . The restriction to realcompact spaces stems from the fact that $C (X)$ and $C (υ X)$ are isomorphic, where $υ X$ is the (Hewitt) realcompactification of $X$ . In this note, a class of locally compact spaces $X$ that includes properly the class of locally...

Shift-invariant functionals on Banach sequence spaces

Albrecht Pietsch (2013)

Studia Mathematica

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The present paper is a continuation of [23], from which we know that the theory of traces on the Marcinkiewicz operator ideal $(H) : = T \in (H) : s u p_{1 \leq m < \infty} 1 / (l o g m + 1) \sum_{n = 1}^{m} a ₙ (T) < \infty$ can be reduced to the theory of shift-invariant functionals on the Banach sequence space $(ℕ ₀) : = c = (γ_{l}) : s u p_{0 \leq k < \infty} 1 / (k + 1) \sum_{l = 0}^{k} | γ_{l} | < \infty$ . The final purpose of my studies, which will be finished in [24], is the following. Using the density character as a measure, I want to determine the size of some subspaces of the dual *(H). Of particular interest are the sets formed by the Dixmier traces and the Connes-Dixmier...

Haar measure and continuous representations of locally compact abelian groups

Jean-Christophe Tomasi (2011)

Studia Mathematica

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Let (X) be the algebra of all bounded operators on a Banach space X, and let θ: G → (X) be a strongly continuous representation of a locally compact and second countable abelian group G on X. Set σ¹(θ(g)): = λ/|λ| | λ ∈ σ(θ(g)), where σ(θ(g)) is the spectrum of θ(g), and let $Σ_{θ}$ be the set of all g ∈ G such that σ¹(θ(g)) does not contain any regular polygon of (by a regular polygon we mean the image under a rotation of a closed subgroup of the unit circle different from 1). We prove that...

The Banach algebra of continuous bounded functions with separable support

M. R. Koushesh (2012)

Studia Mathematica

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We prove a commutative Gelfand-Naimark type theorem, by showing that the set $C_{s} (X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space X (provided with the supremum norm) is a Banach algebra, isometrically isomorphic to C₀(Y) for some unique (up to homeomorphism) locally compact Hausdorff space Y. The space Y, which we explicitly construct as a subspace of the Stone-Čech compactification of X, is countably compact, and if...

Beurling-Figà-Talamanca-Herz algebras

Serap Öztop, Volker Runde, Nico Spronk (2012)

Studia Mathematica

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For a locally compact group G and p ∈ (1,∞), we define and study the Beurling-Figà-Talamanca-Herz algebras $A_{p} (G, ω)$ . For p = 2 and abelian G, these are precisely the Beurling algebras on the dual group Ĝ. For p = 2 and compact G, our approach subsumes an earlier one by H. H. Lee and E. Samei. The key to our approach is not to define Beurling algebras through weights, i.e., possibly unbounded continuous functions, but rather through their inverses, which are bounded continuous functions. We...

Operator Segal algebras in Fourier algebras

Brian E. Forrest, Nico Spronk, Peter J. Wood (2007)

Studia Mathematica

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Let G be a locally compact group, A(G) its Fourier algebra and L¹(G) the space of Haar integrable functions on G. We study the Segal algebra S¹A(G) = A(G) ∩ L¹(G) in A(G). It admits an operator space structure which makes it a completely contractive Banach algebra. We compute the dual space of S¹A(G). We use it to show that the restriction operator ${u \mapsto u |}_{H} : S ¹ A (G) \to A (H)$ , for some non-open closed subgroups H, is a surjective complete quotient map. We also show that if N is a non-compact closed subgroup,...

Haar null and non-dominating sets

Sławomir Solecki (2001)

Fundamenta Mathematicae

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We study the σ-ideal of Haar null sets on Polish groups. It is shown that on a non-locally compact Polish group with an invariant metric this σ-ideal is closely related, in a precise sense, to the σ-ideal of non-dominating subsets of $ω^{ω}$ . Among other consequences, this result implies that the family of closed Haar null sets on a Polish group with an invariant metric is Borel in the Effros Borel structure if, and only if, the group is locally compact. This answers a question of Kechris....

Amenability properties of Fourier algebras and Fourier-Stieltjes algebras: a survey

Nico Spronk (2010)

Banach Center Publications

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Let G be a locally compact group, and let A(G) and B(G) denote its Fourier and Fourier-Stieltjes algebras. These algebras are dual objects of the group and measure algebras, $L^{- 1} (G)$ and M(G), in a sense which generalizes the Pontryagin duality theorem on abelian groups. We wish to consider the amenability properties of A(G) and B(G) and compare them to such properties for $L^{- 1} (G)$ and M(G). For us, “amenability properties” refers to amenability, weak amenability, and biflatness, as well as some properties...

The ideal (a) is not $G_{δ}$ generated

Marta Frankowska, Andrzej Nowik (2011)

Colloquium Mathematicae

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We prove that the ideal (a) defined by the density topology is not $G_{δ}$ generated. This answers a question of Z. Grande and E. Strońska.

Second duals of measure algebras

H. G. Dales, A. T.-M. Lau, D. Strauss

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Let G be a locally compact group. We shall study the Banach algebras which are the group algebra L¹(G) and the measure algebra M(G) on G, concentrating on their second dual algebras. As a preliminary we shall study the second dual C₀(Ω)” of the C*-algebra C₀(Ω) for a locally compact space Ω, recognizing this space as C(Ω̃), where Ω̃ is the hyper-Stonean envelope of Ω. We shall study the C*-algebra $B^{b} (Ω)$ of bounded Borel functions on Ω, and we shall determine the exact cardinality of a variety...

Normal restrictions of the noncofinal ideal on $P_{κ} (λ)$

Pierre Matet (2013)

Fundamenta Mathematicae

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We discuss the problem of whether there exists a restriction of the noncofinal ideal on $P_{κ} (λ)$ that is normal.

An upper bound for the distance to finitely generated ideals in Douglas algebras

Pamela Gorkin, Raymond Mortini, Daniel Suárez (2001)

Studia Mathematica

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Let f be a function in the Douglas algebra A and let I be a finitely generated ideal in A. We give an estimate for the distance from f to I that allows us to generalize a result obtained by Bourgain for $H^{\infty}$ to arbitrary Douglas algebras.

A spectral mapping theorem for Banach modules

H. Seferoğlu (2003)

Studia Mathematica

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Let G be a locally compact abelian group, M(G) the convolution measure algebra, and X a Banach M(G)-module under the module multiplication μ ∘ x, μ ∈ M(G), x ∈ X. We show that if X is an essential L¹(G)-module, then $σ (T_{μ}) = \bar{μ ̂ (s p (X))}$ for each measure μ in reg(M(G)), where $T_{μ}$ denotes the operator in B(X) defined by $T_{μ} x = μ \circ x$ , σ(·) the usual spectrum in B(X), sp(X) the hull in L¹(G) of the ideal $I_{X} = f \in L ¹ (G) | T_{f} = 0$ , μ̂ the Fourier-Stieltjes transform of μ, and reg(M(G)) the largest closed regular subalgebra of M(G); reg(M(G)) contains...

Ideal amenability of module extensions of Banach algebras

Eshaghi M. Gordji, F. Habibian, B. Hayati (2007)

Archivum Mathematicum

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Let $𝒜$ be a Banach algebra. $𝒜$ is called ideally amenable if for every closed ideal $I$ of $𝒜$ , the first cohomology group of $𝒜$ with coefficients in $I^{*}$ is zero, i.e. $H^{1} (𝒜, I^{*}) = {0}$ . Some examples show that ideal amenability is different from weak amenability and amenability. Also for $n \in N$ , $𝒜$ is called $n$ -ideally amenable if for every closed ideal $I$ of $𝒜$ , $H^{1} (𝒜, I^{(n)}) = {0}$ . In this paper we find the necessary and sufficient conditions for a module extension Banach algebra to be 2-ideally amenable.

On some spaces of linear functionals on the algebras $A_{p} (G)$ for locally compact groups

E. E. Granirer (1987)

Colloquium Mathematicae

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