Multipliers and Modulus on Banach Algebras Related to Locally Compact Groups

✍ Scribed by Fereidoun Ghahramani; Anthony To-Ming Lau

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 359 KB
• Volume: 150
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3133

✦ Synopsis

Let G be a locally compact group. In this paper we study moduli of products of elements and of multipliers of Banach algebras which are related to locally compact groups and which admit lattice structure. As a consequence, we obtain a characterization of operators on L (G) which commute with convolutions when G is amenable as discrete.

1997 Academic Press 0. INTRODUCTION Many of the Banach algebras related to locally compact groups admit a Banach lattice structure. For example the group algebra L 1 (G) is a complete lattice whose positive cone consists of all the positive functions in the algebra. This lattice structure extends to the second dual space L 1 (G)**. Recall that a linear operator on a Banach algebra A is a left (resp. right) multiplier if for every a, b # A, T(ab)=T(a) b (resp. T(ab)=aT(b).) Existence of compact (or weakly compact) positive multipliers on certain Banach algebras related to locally compact groups can result in the group being amenable or compact (see [7] and [8]). It is natural to ask whether these results hold without the assumption of positivity. Thus, for a multiplier T we are led to consider the moduli |T |, |T *| and |T **| of T and its adjoints. We show that when A=L 1 (G), and + # M(G), the modulus of the left multiplier * + : f [ + V f is the left multiplier * |+| (Theorem 3.1). Furthermore, |* + *| =** |+| (Theorem 3.5). We use this to show that if G is a article no. FU973133 478 0022-1236Â97 25.00