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

Zhiguo Hu

ArticleOriginal scientific text

Title

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

Authors ¹

Affiliations

Department of Mathematics and Statistics, University of Windsor, Windsor, Ontario, Canada N9B 3P4

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

\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{s p a n} 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.

Keywords

ENG

spectrum, synthesizable ideal, locally compact group, Fourier algebra, Figà-Talamanca-Herz algebra, amenability

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Title

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

Affiliations

Abstract

Keywords

Bibliography