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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 $GL_0(\g)$ 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
â$GL_0(\g)^\perp$â. âWe then endow $GL_0(\g)^*$ with an Arens-typeâ
âproduct which contains $M(\g)$ as a closed subalgebra andâ
â$M_a(\g)$ 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 non-zeroâ
âleft (weakly) completely continuous elementâ.
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