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2013 Noncommutative spectral synthesis for the involutive Banach algebra associated with a topological dynamical system
Marcel de Jeu , Jun Tomiyama
Banach J. Math. Anal. 7(2): 103-135 (2013). DOI: 10.15352/bjma/1363784226

Abstract

If $\Sigma=(X,\sigma)$ is a topological dynamical system, where $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$, then a crossed product involutive Banach algebra $\ell^1$ is naturally associated with these data. If $X$ consists of one point, then $\ell^1$ is the group algebra of the integers, hence the general$\ell^1$could be regarded as a noncommutative $\ell^1$-algebra. In this paper, we study spectral synthesis for the closed ideals of $\ell^1$ in two versions, one modeled after $C(X)$and one modeled after $\ell^1(\mathbb{Z})$. We identify the closed ideals which are equal to (what is the analogue of) the kernel of their hull, and determine when this holds for all closed ideals, i.e., when spectral synthesis holds. In both models, this is the case precisely when $\Sigma$ is free.

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Marcel de Jeu . Jun Tomiyama . "Noncommutative spectral synthesis for the involutive Banach algebra associated with a topological dynamical system." Banach J. Math. Anal. 7 (2) 103 - 135, 2013. https://doi.org/10.15352/bjma/1363784226

Information

Published: 2013
First available in Project Euclid: 20 March 2013

zbMATH: 1275.46039
MathSciNet: MR3039942
Digital Object Identifier: 10.15352/bjma/1363784226

Subjects:
Primary: 46K99
Secondary: 46H10 , 47L65

Keywords: crossed product , Involutive Banach algebra , spectral synthesis , structure of ideals , topological dynamical system

Rights: Copyright © 2013 Tusi Mathematical Research Group

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Vol.7 • No. 2 • 2013
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