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Register a new userIntegration over the quantum diagonal subgroup and associated Fourier-like algebras
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By analogy with the classical construction due to Forrest, Samei and Spronk we associate to every compact quantum group $mathbb{G}$ a completely contractive Banach algebra $A_Delta(mathbb{G})$, which can be viewed as a deformed Fourier algebra of $mathbb{G}$. To motivate the construction we first analyse in detail the quantum version of the integration over the diagonal subgroup, showing that although the quantum diagonal subgroups in fact never exist, as noted earlier by Kasprzak and So{l}tan, the corresponding integration represented by a certain idempotent state on $C(mathbb{G})$ makes sense as long as $mathbb{G}$ is of Kac type. Finally we analyse as an explicit example the algebras $A_Delta(O_N^+)$, $Nge 2$, associated to Wangs free orthogonal groups, and show that they are not operator weakly amenable.
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We study Fourier multipliers on free group $mathbb{F}_infty$ associated with the first segment of the reduced words, and prove that they are completely bounded on the noncommutative $L^p$ spaces $L^p(hat{mathbb{F}}_infty)$ iff their restriction on $L^p(hat{mathbb{F}}_1)=L^p(mathbb{T})$ are completely bounded. As a consequence, every classical Mikhlin multiplier extends to a $L^p$ Fourier multiplier on free groups for all $1<p<infty$.
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In a recent paper on exotic crossed products, we included a lemma concerning ideals of the Fourier-Stieltjes algebra. Buss, Echterhoff, and Willett have pointed out to us that our proof of this lemma contains an error. In fact, it remains an open question whether the lemma is true as stated. In this note we indicate how to contain the resulting damage. Our investigation of the above question leads us to define two properties emph{ordered} and emph{weakly ordered} for invariant ideals of Fourier-Stieltjes algebras, and we initiate a study of these properties.
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