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Paper IPM / M / 14973  


Abstract:  
Let \g be a locally compact groupâ. âIn continuation of ourâ
âstudies on the first and second duals of measure algebras by theâ
âuse of the theory of generalised functionsâ, âhere we study theâ
âC^{*}subalgebra GL_{0}(\g) of GL(\g) as an introverted subspaceâ
âof M(\g)^{*}â. âIn the case where \g is noncompact we show thatâ
âany topological left invariant mean on GL(\g) lies inâ
âGL_{0}(\g)^{⊥}â. âWe then endow GL_{0}(\g)^{*} with an Arenstypeâ
âproduct which contains M(\g) as a closed subalgebra andâ
âM_{a}(\g) as a closed ideal which is a solid set with respect toâ
âabsolute continuity in GL_{0}(\g)^{*}â. âAmong other thingsâ, âwe proveâ
âthat \g is compact if and only if GL_{0}(\g)^{*} has a nonzeroâ
âleft (weakly) completely continuous elementâ.
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