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Paper   IPM / M / 14973
School of Mathematics
  Title:   Introverted subspaces of the duals of measure algebras
  Author(s):  Rasoul Nasr-Isfahani (Joint with H. Javanshiri)
  Status:   To Appear
  Journal: Rocky Mountain J. Math.
  Supported by:  IPM
Let \g be a locally compact group‎. ‎In continuation of our‎ ‎studies on the first and second duals of measure algebras by the‎ ‎use of the theory of generalised functions‎, ‎here we study the‎ ‎C*-subalgebra GL0(\g) of GL(\g) as an introverted subspace‎ ‎of M(\g)*‎. ‎In the case where \g is non-compact we show that‎ ‎any topological left invariant mean on GL(\g) lies in‎ ‎GL0(\g)‎. ‎We then endow GL0(\g)* with an Arens-type‎ ‎product which contains M(\g) as a closed subalgebra and‎ ‎Ma(\g) as a closed ideal which is a solid set with respect to‎ ‎absolute continuity in GL0(\g)*‎. ‎Among other things‎, ‎we prove‎ ‎that \g is compact if and only if GL0(\g)* has a non-zero‎ ‎left (weakly) completely continuous element‎.

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