  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.
