Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

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

H. Farhadia1 and F. Ghahramania2

a1 Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran (hfarhadi@sharif.edu)

a2 Department of Mathematics, University of Manitoba, Winnipeg R3T 2N2, Canada (fereidou@cc.umanitoba.ca)

Abstract

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.

(Online publication February 09 2007)

(Received May 06 2005)

Keywords

  • Primary 43A20;
  • 43A22;
  • Secondary 46K99;
  • amenable group;
  • Arens product;
  • involution;
  • multiplier;
  • left uniformly continuous function;
  • weakly almost periodic function