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 $C_0(\mathbb{R})$ 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 $C_0(\mathbb{R})$, denoted $\mathcal{L}_{\mathcal{V}}$, and use it to find (partial) group algebras for the full continuous representation theory of $\mathbb{R}^{(\mathbb{N})}$. We obtain an interpretation of the Bochner-Minlos theorem in $\mathbb{R}^{(\mathbb{N})}$ as the pure state space decomposition of the partial group algebras which generate $\mathcal{L}_{\mathcal{V}}$. We analyze the representation theory of $\mathcal{L}_{\mathcal{V}}$ and show that there is a bijection between a natural set of representations of $\mathcal{L}_{\mathcal{V}}$ and ${\rm Rep} (\mathcal{R}^{(\mathcal{N})},\mathcal{H})$, but that there is an extra part which essentially consists of the representation theory of a multiplicative semigroup $\mathcal{Q}$ which depends on the initial choice of approximate identity.
DOI: http://dx.doi.org/10.7900/jot.2011aug22.1930
Keywords: $C^*$-algebra, group algebra, infinite tensor product, topological group, Bochner-Minlos theorem, state space decomposition, continuous representation
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