Rocky Mountain Journal of Mathematics

Introverted subspaces of the duals of measure algebras

Hossein Javanshiri and Rasoul Nasr-Isfahani

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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.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 4 (2018), 1171-1189.

Dates
First available in Project Euclid: 30 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1538272828

Digital Object Identifier
doi:10.1216/RMJ-2018-48-4-1171

Mathematical Reviews number (MathSciNet)
MR3859753

Zentralblatt MATH identifier
06958774

Subjects
Primary: 43A10: Measure algebras on groups, semigroups, etc. 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc. 43A20: $L^1$-algebras on groups, semigroups, etc. 47B07: Operators defined by compactness properties

Keywords
Measure algebra generalized functions vanishing at infinity introverted subspace topological invariant mean completely continuous element

Citation

Javanshiri, Hossein; Nasr-Isfahani, Rasoul. Introverted subspaces of the duals of measure algebras. Rocky Mountain J. Math. 48 (2018), no. 4, 1171--1189. doi:10.1216/RMJ-2018-48-4-1171. https://projecteuclid.org/euclid.rmjm/1538272828


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