Semisimple Banach Algebras Generated by Strongly Continuous Representations of Locally Compact Abelian Groups

Muraz, Gilbert; Phóng, Vũ Qúôc

doi:10.1006/jfan.1994.1138

Cited by 4 publications

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“…Both results are a consequence of the semisimplicity of the algebra generated by a finite set of commutative hermitian elements with countable spectrum. This result is known, see . However, with ideas from (the case of one generator) and we give a different and short proof.…”

Section: Introductionmentioning

confidence: 76%

Fuglede Putnam theorem in Banach algebras

Turnšek

2017

Bull. London Math. Soc.

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Section: Introductionmentioning

confidence: 76%

Fuglede Putnam theorem in Banach algebras

Turnšek

2017

Bull. London Math. Soc.

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“…This has established the following Theorem 2.2 which refines [20,Theorem 7] where the corresponding result is given for weakly continuous representations. Note that our proof is completely different from that of [20].…”

Section: Completeness Of Eigenvectors Of Dual Representationsmentioning

confidence: 85%

Completeness of eigenvectors of group representations of operators whose Arveson spectrum is scattered

Huang

1999

Proc. Amer. Math. Soc.

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Abstract. We establish the following result.Theorem. Let α : G → L(X) be a σ(X, X * ) integrable bounded group representation whose Arveson spectrum Sp(α) is scattered. Then the subspace generated by all eigenvectors of the dual representation α * is w * dense in X * . Moreover, the σ(X, X * ) closed subalgebra Wα generated by the operators αt (t ∈ G) is semisimple.If, in addition, X does not contain any copy of c 0 , then the subspace spanned by all eigenvectors of α is σ(X, X * ) dense in X. Hence, the representation α is almost periodic whenever it is strongly continuous. Spectral theory for integrable bounded group representationsThroughout this paper G will denote a locally compact abelian (LCA) group with identity e and G will denote the dual group of G. The multiplication on LCA groups will be written by addition. Let L 1 (G) (resp. M(G)) be the usual group algebra (resp. measure algebra) with convolution as product operation. We refer to [11] or [21] for basic knowledge of Harmonic Analysis on LCA groups.Given a complex Banach space X, let L(X) be the Banach algebra of all bounded linear operators on X. Take a LCA group G. A bounded group representation α of G on X is a mapping α : G → L(X) satisfying the following properties:(a) Group property: α e = I X the identity operator on X and α s+t = α s α t for all s, t ∈ G; (b) Boundedness: α := sup t∈G α t < ∞. Moreover, α is called strongly (resp. weakly) continuous if for each x ∈ X the mapping t → α t x is norm (resp. weakly) continuous. We need a further notion. Definition 1.1. A bounded group representation α : G → L(X) of G on X is called integrable if there exists a subspace X * ⊂ X * satisfying the following

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“…An analogue result for a single isometric operator with countable spectrum was obtained by Feldman [1]. Muraz and Vũ [5] generalized both results by considering a strongly continuous representation τ of a locally compact abelian group G on a Banach space X, and established the semisimplicity of the Banach algebra generated by the Fourier transforms G f (t)τ (t) dt, with f ∈ L 1 (G), in the case when the Arveson spectrum of τ is scattered. Huang [4] refined all those results by means of the w * -denseness in X * of the eigenvectors of the dual representation of τ .…”

Section: Introductionmentioning

confidence: 94%