Let G be a locally compact
group and let B(G) be the algebra of linear combinations of positive definite
continuous functions. We let B0(G) = B(G) ∩C0(G) be the subalgebra consisting of
the elements of B(G) which vanish at infinity. A complex valued function F, defined
on an open interval of the real line containing zero, is said to operate on B0(G) if for
every u ∈ B0(G) whose range is contained in the domain of F, the composition F ∘u
is an element of B0(G). In this paper we prove that, if G is a separable group with
noncompact center or a separable nilpotent group, then every function which
operates on B0(G) can be extended to an entire function. This result follows
directly from the corresponding theorem for noncompact commutative groups,
which is well known, via a lemma which states that every function on the
center Z of G which belongs to B0(Z) can be extended to an element of
B0(G).
