Vol. 75, No. 2, 1978

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Convolution and separate continuity

James Chin-Sze Wong

Vol. 75 (1978), No. 2, 601–611
Abstract

In this paper, we extend the convolution formula

f dμ ν = f τ dμ × ν = f(xy)(x)(y)
= f(xy)(y)(x),
where f L1(μν), μ,ν M(S) and τ(x,y) = xy, x,y S for a locally compact group S to locally compact semigroups with separately continuous multiplication. More precisely, we show that for such semigroups, the same convolution formula is valid if the measure μ × ν is replaced by a suitable measure on S × S (closely related to μ × ν), thus improving a result of I. Glickberg and complementing results of B. E. Johnson. Some important consequences of this convolution formula in abstract harmonic analysis on separately continuous semigroups are discussed.
Mathematical Subject Classification 2000
Primary: 43A05
Milestones
Received: 16 May 1977
Published: 1 April 1978
Authors
James Chin-Sze Wong