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

Zhiguo Hu

Studia Mathematica (1998)

  • Volume: 129, Issue: 3, page 207-223
  • ISSN: 0039-3223

Abstract

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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 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 ∈ ¯ 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 that the Fourier algebra A ( G ) has property (S) iff the canonical trace on ℳ 2 d ( Ĝ ) is faithful iff ℳ 2 d ( Ĝ ) ≅ ℳ 2 d ( Ĝ d ) . This provides an answer to the isomorphism problem of the two C*-algebras and generalizes the so-called “uniqueness theorem” on the group algebra L 1 ( G ) of a locally compact abelian group G. We also prove that G d is amenable iff G is amenable and the Figà-Talamanca-Herz algebra A p ( G ) has property (S) for all p.

How to cite

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Hu, Zhiguo. "Spectrum of commutative Banach algebras and isomorphism of C*-algebras related to locally compact groups." Studia Mathematica 129.3 (1998): 207-223. <http://eudml.org/doc/216501>.

@article{Hu1998,
abstract = {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 $\overline\{span\} Σ_A$ 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 $φ ∈ \overline\{\} Σ_A \ \{0\}$, 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 that the Fourier algebra $A(G)$ has property (S) iff the canonical trace on $ℳ_2^\{d\}(Ĝ)$ is faithful iff $ℳ_2^\{d\} (Ĝ)≅ ℳ_2^\{d\} (Ĝ_\{d\})$. This provides an answer to the isomorphism problem of the two C*-algebras and generalizes the so-called “uniqueness theorem” on the group algebra $L^1(G)$ of a locally compact abelian group G. We also prove that $G_\{d\}$ is amenable iff G is amenable and the Figà-Talamanca-Herz algebra $A_p(G)$ has property (S) for all p.},
author = {Hu, Zhiguo},
journal = {Studia Mathematica},
keywords = {spectrum; synthesizable ideal; locally compact group; Fourier algebra; Figà-Talamanca-Herz algebra; amenability; -algebra; semisimple commutative regular Tauberian Banach algebra; norm spectrum; Figá-Talamanca-Herz algebra; amenable group},
language = {eng},
number = {3},
pages = {207-223},
title = {Spectrum of commutative Banach algebras and isomorphism of C*-algebras related to locally compact groups},
url = {http://eudml.org/doc/216501},
volume = {129},
year = {1998},
}

TY - JOUR
AU - Hu, Zhiguo
TI - Spectrum of commutative Banach algebras and isomorphism of C*-algebras related to locally compact groups
JO - Studia Mathematica
PY - 1998
VL - 129
IS - 3
SP - 207
EP - 223
AB - 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 $\overline{span} Σ_A$ 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 $φ ∈ \overline{} Σ_A \ {0}$, 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 that the Fourier algebra $A(G)$ has property (S) iff the canonical trace on $ℳ_2^{d}(Ĝ)$ is faithful iff $ℳ_2^{d} (Ĝ)≅ ℳ_2^{d} (Ĝ_{d})$. This provides an answer to the isomorphism problem of the two C*-algebras and generalizes the so-called “uniqueness theorem” on the group algebra $L^1(G)$ of a locally compact abelian group G. We also prove that $G_{d}$ is amenable iff G is amenable and the Figà-Talamanca-Herz algebra $A_p(G)$ has property (S) for all p.
LA - eng
KW - spectrum; synthesizable ideal; locally compact group; Fourier algebra; Figà-Talamanca-Herz algebra; amenability; -algebra; semisimple commutative regular Tauberian Banach algebra; norm spectrum; Figá-Talamanca-Herz algebra; amenable group
UR - http://eudml.org/doc/216501
ER -

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