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The construction of an infinite tensor product of the C*-algebra C_0(R) isnot obvious, because it is nonunital, and it has no nonzero projection. Basedon a choice of an approximate identity, we construct here an infinite tensorproduct of C_0(R), denoted L_V. We use this to construct (partial) groupalgebras for the full continuous unitary representation theory of the groupR^(N) = the infinite sequences with real entries, of which only finitely manyentries are nonzero. We obtain an interpretation of the Bochner-Minlos theoremin R^(N) as the pure state space decomposition of the partial group algebraswhich generate L_V. We analyze the representation theory of L_V, and show thatthere is a bijection between a natural set of representations of L_V and thecontinuous unitary representations of R^(N), but that there is an extra partwhich essentially consists of the representation theory of a multiplicativesemigroup which depends on the initial choice of approximate identity.

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