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Volker Runde

Publications and source records attributed to Volker Runde.

At least 19 recordsLinked to original sources
arXiv2025

On operator Connes-amenability of the Fourier-Stieltjes algebra

Runde and Spronk showed in 2004 that there are non-amenable groups $G$, including $\mathbb F_2$, {whose Fourier-Stieltjes algebra, $B(G)$,} is operator Connes-amenable. This result was surprising since the measure algebra $M(G)$ is Connes-amenable if and only if $G$ is amenable, which might lead one to guess that $B(G)$ should be operator Connes-amenable if and only if $G$ is amenable. This leads to the question: for which groups $G$ is $B(G)$ operator Connes-amenable? We make progress on this problem by {exhibiting} the first examples of groups {for which $B(G)$ is not operator Connes-amenable}. More specifically, we show that $B(G)$ is not operator Connes-amenable when $G$ is a non-compact locally compact group with property (T) and finite almost periodic compactification, or when $G$ is a discrete group without the factorization property.

math.FA↗
arXiv2019

(Non-)amenability of the Fourier algebra in the cb-multiplier norm

For a locally compact group $G$, let $A(G)$ denote its Fourier algebra, $M_{cb}(A(G))$ the completely bounded multipliers of $A(G)$, and $A_{M_cb}(G)$ the closure of $A(G)$ in $M_{cb}(A(G))$. We show that, if $A_{M_cb}(G)$ is amenable, then $a(G_d)$, the almost periodic compactification of the discretization of $G$, has an abelian subgroup of finite index. As a consequence, $A_{M_cb}(G)$ cannot be amenable if $G$ contains a copy of $\free_2$, the free group in two generators, as a closed subgroup.

math.FA↗
arXiv2016

Operator ultra-amenability

Extending M.\ Daws' definition of ultra-amenable Banach algebras, we introduce the notion of operator ultra-amenability for completely contractive Banach algebras. For a locally compact group $G$, we show that the operator ultra-amenability of $A(G)$ imposes severe restrictions on $G$. In particular, it forces $G$ to be a discrete, amenable group with no infinite abelian subgroups. For various classes of such groups, this means that $G$ is finite.

math.FA↗
arXiv2015

On positive definiteness over locally compact quantum groups

The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on "square roots" of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.

math.OA↗
arXiv2014

Ergodic theory for quantum semigroups

Recent results of L. Zsido, based on his previous work with C. P. Niculescu and A. Stroh, on actions of topological semigroups on von Neumann algebras, give a Jacobs-de Leeuw-Glicksberg splitting theorem at the von Neumann algebra (rather than Hilbert space) level. We generalize this to the framework of actions of quantum semigroups, namely Hopf-von Neumann algebras. To this end, we introduce and study a notion of almost periodic vectors and operators that is suitable for our setting.

math.OA↗
arXiv2012

Why Banach algebras?

We give a brief overview of the area of Banach algebras, intended for a general mathematical audience.

math.HO↗
arXiv2012

Beurling-Figa-Talamanca-Herz algebras

For a locally compact group $G$ and $p \in (1,\infty)$, we define and study the Beurling-Figa-Talamanca-Herz algebras $A_p(G,ω)$. For $p=2$ and abelian $G$, these are precisely the Beurling algebras on the dual group $\hat{G}$. For $p =2$ and compact $G$, our approach subsumes an earlier one by H. H. Lee and E. Samei. The key to our approach is not to define Beurling algebras through weights, i.e., possibly unbounded continuous functions, but rather through their inverses, which are bounded continuous functions. We prove that a locally compact group $G$ is amenable if and only if one - and, equivalently, every - Beurling-Figa-Talamanca-Herz algebra $A_p(G,ω)$ has a bounded approximate identity.

math.FA↗
arXiv2011

Factorization of completely bounded maps through reflexive operator spaces with applications to weak almost periodicity

Let $(M,Γ)$ be a Hopf--von Neumann algebra, so that $M_\ast$ is a completely contractive Banach algebra. We investigate whether the product of two elements of $M$ that are both weakly almost periodic functionals on $M_\ast$ is again weakly almost periodic. For that purpose, we establish the following factorization result: If $M$ and $N$ are injective von Neumann algebras, and if $x, y \in M \bar{\otimes} N$ correspond to weakly compact operators from $M_\ast$ to $N$ factoring through reflexive operator spaces $X$ and $Y$, respectively, then the operator corresponding to $xy$ factors through the Haagerup tensor product $X \otimes^h Y$ provided that $X \otimes^h Y$ is reflexive. As a consequence, for instance, for any Hopf--von Neumann algebra $(M,Γ)$ with $M$ injective, the product of a weakly almost periodic element of $M$ with a completely almost periodic one is again weakly almost periodic.

math.FA↗
arXiv2011

A new and simple proof of Schauder's theorem

Schauder's theorem asserts that a bounded linear operator between Banach spaces is compact if ad only if its adjoint is. We give a new proof of this result, which is both short and completely elementary in the sense that it does not depend on anything beyond basic functional analysis, i.e., the Hahn--Banach theorem and some of its consequences; in particular, we avoid the Arzela--Ascoli theorem (and any kind of related diagonal argument).

math.FA↗
arXiv2011

Completely almost periodic functionals

Using the notion of complete compactness introduced by H. Saar, we define completely almost periodic functionals on completely contractive Banach algebras. We show that, if $(M,Γ)$ is a Hopf--von Neumann algebra with $M$ injective, then the space of completely almost periodic functionals on $M_\ast$ is a $C^\ast$-subalgebra of $M$.

math.FA↗
arXiv2010

B(l^p) is never amenable

We show that, if $E$ is a Banach space with a basis satisfying a certain condition, then the Banach algebra $\ell^\infty({\cal K}(\ell^2 \oplus E))$ is not amenable; in particular, this is true for $E = \ell^p$ with $p \in (1,\infty)$. As a consequence, $\ell^\infty({\cal K}(E))$ is not amenable for any infinite-dimensional ${\cal L}^p$-space. This, in turn, entails the non-amenability of ${\cal B}(\ell^p(E))$ for any ${\cal L}^p$-space $E$, so that, in particular, ${\cal B}(\ell^p)$ and ${\cal B}(L^p[0,1])$ are not amenable.

math.FA↗
arXiv2009

Reiter's properties (P_1) and (P_2) for locally compact quantum groups

A locally compact group $G$ is amenable if and only if it has Reiter's property $(P_p)$ for $p=1$ or, equivalently, all $p \in [1,\infty)$, i.e., there is a net $(m_α)_α$ of non-negative norm one functions in $L^p(G)$ such that $\lim_α\sup_{x \in K} \| L_{x^{-1}} m_α- m_α\|_p = 0$ for each compact subset $K \subset G$ ($L_{x^{-1}} m_α$ stands for the left translate of $m_α$ by $x^{-1}$). We extend the definitions of properties $(P_1)$ and $(P_2)$ from locally compact groups to locally compact quantum groups in the sense of J. Kustermans and S. Vaes. We show that a locally compact quantum group has $(P_1)$ if and only if it is amenable and that it has $(P_2)$ if and only if its dual quantum group is co-amenable. As a consequence, $(P_2)$ implies $(P_1)$.

math.OA↗
arXiv2009

(Non-)amenability of B(E)

In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra $B(E)$ of all bounded linear operators on a Banach space $E$ could ever be amenable if $\dim E = \infty$. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros--Haydon result that solves the "scalar plus compact problem": there is an infinite-dimensional Banach space $E$, the dual of which is $\ell^1$, such that $B(E) = K(E)+ \mathbb{C} \id_E$. Still, $B(\ell^2)$ is not amenable, and in the past decade, $ B(\ell^p)$ was found to be non-amenable for $p=1,2,\infty$ thanks to the work of C. J. Read, G. Pisier, and N. Ozawa. We survey those results, and then--based on joint work with M. Daws--outline a proof that establishes the non-amenability of $B(\ell^p)$ for all $p \in [1,\infty]$.

math.FA↗
arXiv2009

Co-representations of Hopf-von Neumann algebras on operator spaces other than column Hilbert space

Recently, M. Daws introduced a notion of co-representation of abelian Hopf--von Neumann algebras on general reflexive Banach spaces. In this note, we show that this notion cannot be extended beyond subhomogeneous Hopf--von Neumann algebras. The key is our observation that, for a von Neumann algebra $\M$ and a reflexive operator space $E$, the normal spatial tensor product $\M \bar{\tensor} \CB(E)$ is a Banach algebra if and only if $\M$ is subhomogeneous or $E$ is completely isomorphic to column Hilbert space.

math.OA↗
arXiv2008

Biflatness and biprojectivity of the Fourier algebra

We show that the biflatness - in the sense of A. Ya. Helemskii - of the Fourier algebra $A(G)$ of a locally compact group $G$ forces $G$ to either have an abelian subgroup of finite index or to be non-amenable without containing $F_2$, the free group in two generators, as a closed subgroup. An analogous dichotomy is obtained for biprojectivity.

math.FA↗
arXiv2008

Norm one idempotent cb-multipliers with applications to the Fourier algebra in the cb-multiplier norm

For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely bounded multipliers of $A(G)$, and let $A_{Mcb}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we describe the closed ideals of $A_{Mcb}(G)$ with an approximate identity bounded by 1, and we characterize those $G$ for which $A_{Mcb}(G)$ is 1-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.)

math.FA↗
arXiv2008

Column and row operator spaces over QSL_p-spaces and their use in abstract harmonic analysis

The notions of column and row operator space were extended by A. Lambert from Hilbert spaces to general Banach spaces. In this paper, we use column and row spaces over quotients of subspaces of general $L_p$-spaces to equip several Banach algebras occurring naturally in abstract harmonic analysis with canonical, yet not obvious operator space structures that turn them into completely bounded Banach algebras. We use these operator space structures to gain new insights on those algebras.

math.FA↗