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2018 Introverted subspaces of the duals of measure algebras
Hossein Javanshiri, Rasoul Nasr-Isfahani
Rocky Mountain J. Math. 48(4): 1171-1189 (2018). DOI: 10.1216/RMJ-2018-48-4-1171

Abstract

Let $\mathcal{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 generalized functions, here we study the C$^*$-subalgebra $GL_0(\mathcal{G})$ of $GL(\mathcal{G})$ as an introverted subspace of $M(\mathcal{G} )^*$. In the case where $\mathcal{G} $ is non-compact, we show that any topological left invariant mean on $GL(\mathcal{G} )$ lies in $GL_0(\mathcal{G} )^\perp $. We then endow $GL_0(\mathcal{G} )^*$ with an Arens-type product, which contains $M(\mathcal{G} )$ as a closed subalgebra and $M_a(\mathcal{G} )$ as a closed ideal, which is a solid set with respect to absolute continuity in $GL_0(\mathcal{G} )^*$. Among other things, we prove that $\mathcal{G} $ is compact if and only if $GL_0(\mathcal{G} )^*$ has a non-zero left (weakly) completely continuous element.

Citation

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Hossein Javanshiri. Rasoul Nasr-Isfahani. "Introverted subspaces of the duals of measure algebras." Rocky Mountain J. Math. 48 (4) 1171 - 1189, 2018. https://doi.org/10.1216/RMJ-2018-48-4-1171

Information

Published: 2018
First available in Project Euclid: 30 September 2018

zbMATH: 06958774
MathSciNet: MR3859753
Digital Object Identifier: 10.1216/RMJ-2018-48-4-1171

Subjects:
Primary: 43A10, 43A15, 43A20, 47B07

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 4 • 2018
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