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Involutions on the second duals of group algebras and a multiplier problem. (English) Zbl 1121.43001

Summary: We show that if a locally compact group \(G\) is non-discrete or has an infinite amenable subgroup, then the second dual algebra \(L^{1}(G)^{**}\) does not admit an involution extending the natural involution of \(L^{1}(G)\). Thus, for the above classes of groups we answer in the negative a question raised by J. Duncan and S. A. R. Hosseiniun [Proc. R. Soc. Edinb., Sect. A 84, 309–325 (1979; Zbl 0427.46028)]. We also find necessary and sufficient conditions for the dual of certain left-introverted subspaces of the space \(C_{b}(G)\) (of bounded continuous functions on \(G\)) to admit involutions. We show that the involution problem is related to a multiplier problem. Finally, we show that certain non-trivial quotients of \(L^{1}(G)^{**}\) admit involutions.

MSC:

43A20 \(L^1\)-algebras on groups, semigroups, etc.
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
46K99 Topological (rings and) algebras with an involution

Citations:

Zbl 0427.46028
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