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Register a new userFree Fourier Multipliers associated with the firstSegment
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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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We study the dual relationship between quantum group convolution maps $L^1(mathbb{G})rightarrow L^{infty}(mathbb{G})$ and completely bounded multipliers of $widehat{mathbb{G}}$. For a large class of locally compact quantum groups $mathbb{G}$ we completely isomorphically identify the mapping ideal of row Hilbert space factorizable convolution maps with $M_{cb}(L^1(widehat{mathbb{G}}))$, yielding a quantum Gilbert representation for completely bounded multipliers. We also identify the mapping ideals of completely integral and completely nuclear convolution maps, the latter case coinciding with $ell^1(widehat{bmathbb{G}})$, where $bmathbb{G}$ is the quantum Bohr compactification of $mathbb{G}$. For quantum groups whose dual has bounded degree, we show that the completely compact convolution maps coincide with $C(bmathbb{G})$. Our techniques comprise a mixture of operator space theory and abstract harmonic analysis, including Fubini tensor products, the non-commutative Grothendieck inequality, quantum Eberlein compactifications, and a suitable notion of quasi-SIN quantum group, which we introduce and exhibit examples from the bicrossed product construction. Our main results are new even in the setting of group von Neumann algebras $VN(G)$ for quasi-SIN locally compact groups $G$.
In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompsons group $mathcal{F}$. We introduce the notion of unique factorization semigroup which contains Thompsons semigroup $mathcal{S}$ and the free semigroup $mathcal{F}_n$ on $n$ generators ($geq2$). Let $mathfrak{B}(mathcal{S})$ and $mathfrak{B}(mathcal{F}_n)$ be the Banach algebras generated by the left regular representations of $mathcal{S}$ and $mathcal{F}_n$, respectively. It is proved that all derivations on $mathfrak{B}(mathcal{S})$ and $mathfrak{B}(mathcal{F}_n)$ are automatically continuous, and every derivation on $mathfrak{B}(mathcal{S})$ is induced by a bounded linear operator in $mathcal{L}(mathcal{S})$, the weak closed Banach algebra consisting of all bounded left convolution operators on $l^2(mathcal{S})$. Moreover, we show that the first continuous Hochschild cohomology group of $mathfrak{B}(mathcal{S})$ with coefficients in $mathcal{L}(mathcal{S})$ vanishes. These conclusions provide positive indications for the left amenability of Thompsons semigroup.
The noncommutative Fourier transform of the irrational rotation C*-algebra is shown to have a K-inductive structure (at least for a large concrete class of irrational parameters, containing dense $G_delta$s). This is a structure for automorphisms that is analogous to Huaxin Lins notion of tracially AF for C*-algebras, except that it requires more structure from the complementary projection.
We present several operat
The Fourier(-Stieltjes) algebras on locally compact groups are important commutative Banach algebras in abstract harmonic analysis. In this paper we introduce a generalization of the above two algebras via twisting with respect to 2-cocycles on the group. We also define and investigate basic properties of the associated multiplier spaces with respect to a pair of 2-cocycles. We finally prove a twisted version of the result of Bo.{z}ejko/Losert/Ruan characterizing amenability of the underlying locally compact group through the comparison of the twisted Fourier-Stieltjes space with the associated multiplier spaces.
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