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| Paper IPM / M / 14973 |
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| Abstract: | |
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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 GL0(\g) of GL(\g) as an introverted subspaceâ
âof M(\g)*â. âIn the case where \g is non-compact we show thatâ
âany topological left invariant mean on GL(\g) lies inâ
âGL0(\g)⊥â. âWe then endow GL0(\g)* with an Arens-typeâ
âproduct which contains M(\g) as a closed subalgebra andâ
âMa(\g) as a closed ideal which is a solid set with respect toâ
âabsolute continuity in GL0(\g)*â. âAmong other thingsâ, âwe proveâ
âthat \g is compact if and only if GL0(\g)* has a non-zeroâ
âleft (weakly) completely continuous elementâ.
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