Vol. 74, No. 1, 1978

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Functions that operate on the algebra B0(G)

Alessandro Figà-Talamanca and Massimo A. Picardello

Vol. 74 (1978), No. 1, 57–61

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).

Mathematical Subject Classification 2000
Primary: 43A35
Received: 15 February 1977
Published: 1 January 1978
Alessandro Figà-Talamanca
Massimo A. Picardello