Grundling H, Neeb KH (2013)
Publication Type: Journal article, Original article
Publication year: 2013
Book Volume: 70
Pages Range: 311 - 353
Journal Issue: 2
DOI: 10.7900/jot.2011aug22.1930
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.
APA:
Grundling, H., & Neeb, K.H. (2013). Infinite Tensor Products of C₀(R): Towards a Group Algebra for R∞. Journal of Operator Theory, 70(2), 311 - 353. https://doi.org/10.7900/jot.2011aug22.1930
MLA:
Grundling, Hendrik, and Karl Hermann Neeb. "Infinite Tensor Products of C₀(R): Towards a Group Algebra for R∞." Journal of Operator Theory 70.2 (2013): 311 - 353.
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