Let M(G) denote the
convolution algebra of finite regular Borel measures on a locally compact abelian
group G, and let Δ denote the maximal ideal space of M(G). It is well-known that
on certain subsets of Δ the Gelfand transforms μ of members μ of M(G)
behave like holomorphic functions. The simplest way to exhibit this is to use
Taylor’s description of Δ as the semigroup of all continuous semicharacters
of a compact semigroup S — the structure semigroup of M(G) (see [10]).
If f ∈ Δ(= S) and f(s) ≧ 0 for all s ∈ S, then fz∈ Δ for Re(z) > 0.
Thus, provided f2≠f, there is an analytic disc around f in the sense that
μ(fz) is holomorphic on Re(z) > 0 for all μ ∈ M(G). Using this fact, Taylor
(loc. cit.) has shown that if f is a strong boundary point of M(G), then
|f|2= |f|.
