Vol. 49, No. 2, 1973

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Multipliers and the group Lp-algebras

John Griffin and Kelly Denis McKennon

Vol. 49 (1973), No. 2, 365–370

Let G be a locally compact group, p a number in [1,∞ [, and Lp the usual Lp-space with respect to left Haar measure on G. The space Lpt consists of those functions f in Lpt such that g f is well-defined and in Lp for each g in Lp. Since each function in Lpt may be identified with a linear operator on Lp which, as it turns out, is bounded; the operator norm may be super-imposed on Lpt and, under this norm ∥∥pt,Lpt 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 mr(A) and the family of left multipliers as m1(A). The family of all bounded linear operators on Lp which commute with left translations will be written as Mp.

It was shown in a previous issue of this journal that the Banach algebra Mp is linearly isomorphic to the normed algebra mr(Lpt), 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 ∥∥pt is defective in that, unless p = 1,(Lpt,∥∥pt) is never complete.

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

Mathematical Subject Classification 2000
Primary: 43A22
Received: 24 July 1972
Published: 1 December 1973
John Griffin
Kelly Denis McKennon