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Journal of Operator Theory

Volume 70, Issue 2, Autumn 2013  pp. 311-353.

Infinite tensor products of $C_0(\mathbb{R})$: towards a group algebra for $\mathbb{R}^{(\mathbb{N})}$

Authors Hendrik Grundling (1) and Karl-Hermann Neeb (2)
Author institution: (1) Department of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
(2) Department of Mathematics, FAU Erlangen-Nuernberg, Cauerstrasse 11, 91058 Erlangen, 91054 Germany

Summary:  The construction of an infinite tensor product of the $C^*$-algebra $C_0(\R)$ is not obvious, because it is nonunital, and it has no nonzero projection. Based on a choice of an approximate identity, we construct here an infinite tensor product of $C_0(\R),$ denoted $\al L.\s{\al V.}.,$ and use it to find (partial) group algebras for the full continuous representation theory of $\R^{(\N)}.$ We obtain an interpretation of the Bochner--Minlos theorem in $\R^{(\N)}$ as the pure state space decomposition of the partial group algebras which generate $\al L.\s{\al V.}..$ We analyze the representation theory of $\al L.\s{\al V.}.,$ and show that there is a bijection between a natural set of representations of $\al L.\s{\al V.}.$ and ${\rm Rep} (\R^{(\N)},\al H. )\,,$ but that there is an extra part which essentially consists of the representation theory of a multiplicative semigroup $\al Q.$ which depends on the initial choice of approximate identity.

Keywords:  $C^*$-algebra, group algebra, infinite tensor product, topological group, Bochner-Minlos theorem, state space decomposition, continuous representation

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