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Weakly compact multipliers on group algebras. (English) Zbl 1069.43001
For any infinite locally compact Hausdorff group \(G\), the group algebra \(\text{L}^1(G)\) ist not Arens regular, that is, the two Arens products on its bidual \(\text{L}^1(G)^{**}\) (each one extending the product of \(\text{L}^1(G)\)) do not coincide. Due to this asymmetry, left resp. right multipliers on \(\text{L}^1(G)^{**}\) behave differently in an essential way, requiring rather different methods of proof for their treatment. The same is true for \(\text{M}(G)^{**}\), the bidual of the algebra of complex Radon measures on \(G\). As F. Ghahramani and A. T. Lau have shown [Math. Proc. Camb. Philos. Soc. 111, No. 1, 161–168 (1992; Zbl 0818.46050); J. Funct. Anal. 132, No. 1, 170–191 (1995; Zbl 0832.22007)], the existence of non-zero compact resp. weakly compact right multipliers is equivalent to \(G\) being amenable. For the existence of non-zero compact resp. weakly compact left multipliers, compactness of \(G\) is sufficient, yet as to the converse, only partial results have been obtained by these authors.
The paper under review continues the study of the “left” case. In particular, it is shown in the first part that for non-compact \(G\), neither \(\text{L}^1(G)^{**}\) nor \(\text{M}(G)^{**}\) admit weakly compact (let alone compact) left multipliers different from zero. In the second part, a sufficient condition on \(G\) is given (namely, that \(\text{L}^\infty(G)\) has a unique right invariant mean) implying that any weakly compact left multiplier on \(\text{L}^1(G)^{**}\) is given by multiplying from the left by a suitable element of \(\text{L}^1(G)\). Both these results arise as corollaries of more general theorems; the first of those actually states that the existence of non-zero elements in \(\text{M}(G)^{**}\) acting as compact resp. weakly compact operators by left multiplication is equivalent to the compactness of \(G\).
Altogether, these results answer several of the problems listed by F. Ghahramani and A. T. Lau [see above and also J. Funct. Anal. 150, No. 2, 478–497 (1997; Zbl 0891.22007)]. Due to the concise style of writing of the author, the proofs of the main results – notwithstanding their considerable length and complexity – look deceptively simple, yet rely heavily on deep results from measure theory and functional analysis.

MSC:
43A10 Measure algebras on groups, semigroups, etc.
47B07 Linear operators defined by compactness properties
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46B42 Banach lattices
43A07 Means on groups, semigroups, etc.; amenable groups
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