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Convolution Operators And Homomorphisms Of Locally Compact Groups

Let 1 < p < infinity, let G and H be locally compact groups and let c) be a continuous homomorphism of G into H. We prove, if G is amenable, the existence of a linear contraction of the Banach algebra CVp (G) of the p-convolution operators on G into CVp (H) which extends the usual definition of the image of a bounded measure by omega. We also discuss the uniqueness of this linear contraction onto important subalgebras of CVp(G). Even if G and H are abelian, we obtain new results. Let G(d) denote the group G provided with a discrete topology. As a corollary, we obtain, for every discrete measure, vertical bar parallel to mu vertical bar parallel to CVp(G) <= vertical bar parallel to mu vertical bar parallel to CVp(G(d)), for G(d) amenable. For arbitrary G, we also obtain vertical bar parallel to mu vertical bar parallel to CVp(G(d)) <= vertical bar parallel to mu vertical bar parallel to CVp(G). These inequalities were already known for p = 2. The proofs presented in this paper avoid the use of the Hilbertian techniques which are not applicable to p :A 2. Finally, for Gd amenable, we construct a natural map of CVp(G) into CVp (G(d)).

Published in:

Journal Of The Australian Mathematical Society, 84, 329-344

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Record created 2010-11-30, last modified 2018-09-13