Related papers: Property (T) for Banach algebras

We prove a number of results linking properties of actions by compact groups (both quantum and classical) on Banach spaces, such as uniform continuity, spectrum finiteness and extensibility of the actions across several constructions.…

Given a compact space $X$ and a commutative Banach algebra $A$, the character spaces of $A$-valued function algebras on $X$ are investigated. The class of natural $A$-valued function algebras, those whose characters can be described by…

A Banach space $E$ has the Grothendieck property if every (linear bounded) operator from $E$ into $c_0$ is weakly compact. It is proved that, for an integer $k>1$, every $k$-homogeneous polynomial from $E$ into $c_0$ is weakly compact if…

We prove a commutative Gelfand--Naimark type theorem, by showing that the set $C_s(X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space $X$ (provided with the supremum…

We study representations of Banach algebras on reflexive Banach spaces. Algebras which admit such representations which are bounded below seem to be a good generalisation of Arens regular Banach algebras; this class includes dual Banach…

We prove that every sofic approximation of a property (T) group is approximately isomorphic to one having geometric property (T), and more generally, a box space of graphs which has boundary geometric property (T) is approximately…

A Banach space $X$ is Grothendieck if the weak and the weak$^*$ convergence of sequences in the dual space $X^*$ coincide. The space $\ell^\infty$ is a classical example of a Grothendieck space due to Grothendieck. We introduce a…

In this paper we study conditions on a Banach space X that ensure that the Banach algebra K(X) of compact operators is amenable. We give a symmetrized approximation property of X which is proved to be such a condition. This property is…

For appropriate parameters $k,p,q$, we introduce and systematically study the class of $(k,p,q)$-differential subalgebras. This is a vast class of Banach $^*$-algebras defined by their relation with their $C^*$-envelopes. Some examples are…

We show that, for a countable discrete group $\Gamma$, property $(\mathrm{T}_{L^p})$ of Bader, Furman, Gelander and Monod is equivalent to the property that, whenever an $L^p$-representation of $\Gamma$ admits a net of almost invariant unit…

In the present paper we continue the study of the structure of a Banach algebra B(A, T_g) generated by a certain Banach algebra $A$ of operators acting in a Banach space $D$ and a group {T_g}_{g \in G} of isometries of D such that T_g A…

Let $A$ and $B$ be unital semi-simple commutative Banach algebras. In this paper we study two-variable polynomials $p$ which satisfy the following property: a map $T$ from $A$ onto $B$ such that the equality \[ \sigma (p(Tf,Tg))=\sigma…

We show that a group with Kazhdan's property $(T)$ has property $(T_{B})$ for $B$ the Haagerup non-commutative $L_{p}(\mathcal{M})$-space associated with a von Neumann algebra $\mathcal{M}$, $1

We consider certain strengthenings of property (T) relative to Banach spaces that are satisfied by high rank Lie groups. Let X be a Banach space for which, for all k, the Banach--Mazur distance to a Hilbert space of all k-dimensional…

For locally compact groups amenability and Kazhdan's property (T) are mutually exclusive in the sense that a group having both properties is compact. This is no longer true for more general Polish groups. However, a weaker result still…

We define a Banach algebra A to be dual if $A = (A_\ast)^\ast$ for a closed submodule $A_\ast$ of $A^\ast$. The class of dual Banach algebras includes all $W^\ast$-algebras, but also all algebras M(G) for locally compact groups G, all…

Let $X$ be a Banach space and $\mathcal A$ be the Banach algebra $B(X)$ of bounded (i.e. continuous) linear transformations (to be called operators) on $X$ to itself. Let $\mathcal E$ be the set of idempotents in $\mathcal A$ and $\mathcal…

We study the closed algebra B_I(G) generated by the idempotents in the Fourier-Stieltjes algebra of a locally compact group G. We show that it is a regular Banach algebra with computable spectrum G^I, which we call the idempotent…

A Banach space $X$ has Pe{\l}czy\' nski's property (V) if for every Banach space $Y$ every unconditionally converging operator $T\colon X\to Y$ is weakly compact. In 1962, Aleksander Pe{\l}czy\' nski showed that $C(K)$ spaces for a compact…

It is shown that for a locally compact group $G$, if $L^{1}(G)^{**}$ is approximately weakly amenable, then $M(G)$ is approximately weakly amenable. Then, new notions of approximate weak amenability and approximate cyclic amenability for…