Infinite Tensor Products of C_0(R): Towards a Group Algebra for R^\infty

TL;DR

This paper constructs an infinite tensor product of the nonunital C*-algebra C_0(R) to develop a group algebra for R^∞, linking it to the continuous unitary representation theory of the infinite real sequences group.

Contribution

It introduces a novel construction of an infinite tensor product of C_0(R) and explores its application to the representation theory of R^∞, providing new insights into partial group algebras.

Findings

Established a bijection between representations of L_V and unitary representations of R^∞

Interpreted the Bochner-Minlos theorem as a state space decomposition of the partial group algebra

Identified an additional representation component related to a multiplicative semigroup

Abstract

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 L_V. We use this to construct (partial) group algebras for the full continuous unitary representation theory of the group R^(N) = the infinite sequences with real entries, of which only finitely many entries are nonzero. 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 L_V. We analyze the representation theory of L_V, and show that there is a bijection between a natural set of representations of L_V and the continuous unitary representations of R^(N), but that there is an extra part which essentially consists of the representation theory of a multiplicative semigroup which depends on the initial choice of approximate identity.