Vol. 75, No. 1, 1978

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Analytic discs in the maximal ideal space of M(G)

Gavin Brown and William Moran

Vol. 75 (1978), No. 1, 45–57

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 f2f, 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|.

Mathematical Subject Classification 2000
Primary: 43A10
Received: 6 April 1977
Published: 1 March 1978
Gavin Brown
William Moran