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Invariant means on weakly almost periodic functionals with application to quantum groups
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 66 / Issue 3 / September 2023
- Published online by Cambridge University Press:
- 16 January 2023, pp. 927-936
- Print publication:
- September 2023
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- Article
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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}))$.
