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INVOLUTIONS ON THE SECOND DUALS OF GROUP ALGEBRAS AND A MULTIPLIER PROBLEM

Published online by Cambridge University Press:  09 February 2007

H. Farhadi
Affiliation:
Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran (hfarhadi@sharif.edu)
F. Ghahramani
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg R3T 2N2, Canada (fereidou@cc.umanitoba.ca)
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Abstract

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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 Duncan and Hosseiniun in 1979. 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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2007