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Invariant means on weakly almost periodic functionals with application to quantum groups

Part of: Locally compact groups and their algebras Topological algebras, normed rings and algebras, Banach algebras Abstract harmonic analysis

Published online by Cambridge University Press:  16 January 2023

Ali Ebrahimzadeh Esfahani ,
Mehdi Nemati  and
Mohammad Reza Ghanei
Ali Ebrahimzadeh Esfahani
Affiliation:
Department of Mathematical Sciences, Isfahan Uinversity of Technology, Isfahan 84156-83111, Iran e-mail: ali.ebrahimzadeh@math.iut.ac.ir
Mehdi Nemati*
Affiliation:
Department of Mathematical Sciences, Isfahan Uinversity of Technology, Isfahan 84156-83111, Iran e-mail: ali.ebrahimzadeh@math.iut.ac.ir
Mohammad Reza Ghanei
Affiliation:
Department of Mathematics, Khansar Campus, University of Isfahan, Isfahan, Iran e-mail: mrg.ghanei@gmail.com m.r.ghanei@khc.ui.ac.ir
*
e-mail: m.nemati@iut.ac.ir
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Abstract

Let ${\mathcal A}$ be a Banach algebra, and let $\varphi $ be a nonzero character on ${\mathcal A}$. For a closed ideal I of ${\mathcal A}$ with $I\not \subseteq \ker \varphi $ such that I has a bounded approximate identity, we show that $\operatorname {WAP}(\mathcal {A})$, the space of weakly almost periodic functionals on ${\mathcal A}$, admits a right (left) invariant $\varphi $-mean if and only if $\operatorname {WAP}(I)$ admits a right (left) invariant $\varphi |_I$-mean. This generalizes a result due to Neufang for the group algebra $L^1(G)$ as an ideal in the measure algebra $M(G)$, for a locally compact group G. Then we apply this result to the quantum group algebra $L^1({\mathbb G})$ of a locally compact quantum group ${\mathbb G}$. Finally, we study the existence of left and right invariant $1$-means on $ \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$.

Keywords

Closed ideallocally compact quantum groupinvariant meanweakly almost periodic functional

MSC classification

Primary: 22D15: Group algebras of locally compact groups 43A07: Means on groups, semigroups, etc.; amenable groups 46H10: Ideals and subalgebras
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 3 , September 2023 , pp. 927 - 936
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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