Multipliers and the group L_p-algebras

John Griffin and Kelly Denis McKennon

Vol. 49 (1973), No. 2, 365–370

DOI: 10.2140/pjm.1973.49.365

Abstract

Let G be a locally compact group, p a number in , and L_p the usual L_p-space with respect to left Haar measure on G. The space L_p^t consists of those functions f in L_p^t such that g ∗ f is well-defined and in L_p for each g in L_p. Since each function in L_p^t may be identified with a linear operator on L_p which, as it turns out, is bounded; the operator norm may be super-imposed on L_p^t and, under this norm ∥∥_p^t,L_p^t is a normed algebra. The family of right multipliers (i.e., bounded linear operators which commute with left multiplication operators) on any normed algebra A will be written as m_r(A) and the family of left multipliers as m₁(A). The family of all bounded linear operators on L_p which commute with left translations will be written as M_p.

It was shown in a previous issue of this journal that the Banach algebra M_p is linearly isomorphic to the normed algebra m_r(L_p^t), whenever the group G is either Abelian or compact. This fact is shown, in the present paper, to hold for general locally compact G. The norm ∥∥_p^t is defective in that, unless p = 1,(L_p^t,∥∥_p^t) is never complete.

An attempt will be made in the sequel to supply this deficiency by the introduction of a second norm ∥|∥|_p^t on L_p^t under which L_p^t is always a Banach algebra. It will be seen that, for p = 2 (and of course for p = 1), the Banach algebra m_r(L_p^t,∥|∥|_p^t) is linearly isometric to Rl_p.

Mathematical Subject Classification 2000

Primary: 43A22

Milestones

Received: 24 July 1972

Published: 1 December 1973