Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1910.05888

Twisted Fourier(-Stieltjes) spaces and amenability

Hun Hee Lee·Xiao Xiong
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 results of Nebbia, Fendler, Bożejko, Losert and Ruan characterizing amenability of the underlying locally compact group through comparison of the twisted Fourier-Stieltjes space with the associated multiplier spaces.

Explore related subjects

math.OAmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hun Hee Lee, Xiao Xiong. 2025-08-12. Twisted Fourier(-Stieltjes) spaces and amenability. https://arxiv.org/abs/1910.05888

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Maximal Ergodic Theorems for Operators with Finite Peripheral Spectrum

Let $\mathcal M$ be a semifinite von Neumann algebra and $T : \mathcal{M} \to \mathcal{M}$ be a positive $L_\infty-L_1$ contraction in the sense of Junge-Xu, of which the numerical range, when viewed as an operator on $L_2(\mathcal M),$ is contained in a closed polygon with vertices on the unit circle. In this article, we prove that there exists a positive constant $C_p(T)$ such that \begin{equation}\label{abstract1stin} \Big\|\sup_{n \ge 0}\!^{+} T^n x \Big\|_p \le C_p(T)\, \|x\|_p \end{equation} for all \( x \in L_p(\mathcal{M}) \), $1<p<\infty$ extending some noncommutative maximal ergodic inequalities proved by Junge-Xu \cite{junge-Xu} and later generalized by Bekjan \cite{Bekjan2008}. In the commutative setting, similar inequalities as in \eqref{abstract1stin} hold for arbitrary $L_\infty-L_1$ contractions with the same condition in the numerical range, yielding a vast generalization of a classical maximal ergodic theorem of Stein \cite{Stein-ergodic-theorem} proved in 1960s. Moreover, we establish a noncommutative weak-type maximal inequality for convolution powers which was proved by Calderón and Bellow \cite{Bellow-Calderon} in the classical setting, complementing our strong type noncommutative maximal ergodic inequalities. Our method relies on several new polynomial identities, suitable square function estimates tailored to fit our setting and generalization of Stein's method of embedding maximal function into analytic family of operators. However, we show that even in the classical setting, the variational inequality extending \eqref{abstract1stin} holds for arbitrary operators described above, precisely when the spectrum meets the unit circle only at $1.$

math.OA↗
arXiv2026

The noncommutative topological factor theorem for rank-one product lattices

We prove a noncommutative topological factor theorem for irreducible lattices in products of real rank-one simple Lie groups. The intermediate C*-subalgebras between the reduced group C*-algebra and the boundary crossed product are exactly the crossed products arising from coordinate subproducts of the Furstenberg boundary. This follows from a more general theorem for product boundary actions, which also yields tree and mixed local-field versions. We finally show that, for lattices in connected semisimple real Lie groups with finite center and no compact factors, the scalar-expectation case of the corresponding classification is equivalent to ordinary ITAP.

math.OA↗
arXiv2026

Representation stability for compact and discrete quantum groups

We study approximate representations of locally compact quantum groups and prove stability results in the sense of Ulam in this context. Our main result is that compact and amenable discrete quantum groups are representation stable. We also show that an analogous stability result holds for unitary compressions of general amenable locally compact quantum groups without the assumption of compactness or discreteness.

math.OA↗