Related papers: Quasi-Locality and Property A

Given a Banach space X and a subspace Y, the pair (X,Y) is said to have the approximation property (AP) provided there is a net of finite rank bounded linear operators on X all of which leave the subspace Y invariant such that the net…

Let $\mathcal{G}$ be a locally compact \'{e}tale groupoid and $\mathscr{L}(L^2(\mathcal{G}))$ be the $C^*$-algebra of adjointable operators on the Hilbert $C^*$-module $L^2(\mathcal{G})$. In this paper, we discover a notion called…

The Banach space $L^p(X,\mu)$, for $X$ a compact Hausdorff measure space, is considered as a special kind of quasi *-algebra (called CQ*-algebra) over the C*-algebra $C(X)$ of continuous functions on $X$. It is shown that, for $p \geq 2$,…

Given any quasi-countable, in particular any countable inverse semigroup $S$, we introduce a way to equip $S$ with a proper and right subinvariant extended metric. This generalizes the notion of proper, right invariant metrics for discrete…

In this paper, we characterize when the $\ell^p$ uniform Roe algebra of a metric space with bounded geometry is (stably) finite and when it is properly infinite in standard form for $p\in [1,\infty)$. Moreover, we show that the $\ell^p$…

An algebra A of operators on a Banach space X is called strictly semi-transitive if for all non-zero x,y in X there exists an operator S in A such that Sx=y or Sy=x. We show that if A is norm-closed and strictly semi-transitive, then every…

We work on $\ell_p$ uniform Roe algebras associated to metric spaces, and on their mutual embedding. We generalize results of I. Farah and the authors to mutual embeddings of uniform Roe algebras of operators on $\ell_p$ spaces.…

We characterize real Banach spaces $Y$ such that the pair $(\ell_\infty ^n, Y)$ has the Bishop-Phelps-Bollob\'as property for operators. To this purpose it is essential the use of an appropriate basis of the domain space $\R^n$. As a…

Let $X$ be a Banach space and $T$ be a bounded linear operator acting in $l_p(\mathbb Z^c,X)$, $1\le p\le\infty$. The operator $T$ is called \emph{locally nuclear} if it can be represented in the form \begin{equation*}…

We introduce a weakened notion of norm attainment for bounded linear operators between Banach spaces which we call \emph{quasi norm attaining operators}. An operator $T\colon X \longrightarrow Y$ between the Banach spaces $X$ and $Y$ is…

The rigidity problem for uniform Roe algebras was recently positively solved. Before its solution was found, there were positive solutions under the assumption of certain technical geometric conditions. In this paper, we introduce weaker…

Let $X$ be a Banach space. We prove that, for a large class of Banach or quasi-Banach spaces $E$ of $X$-valued sequences, the sets $E-\bigcup _{q\in\Gamma}\ell_{q}(X)$, where $\Gamma$ is any subset of $(0,\infty]$, and $E-c_{0}(X)$ contain…

Let $1\leq p\leq q\leq\infty.$ Being motivated by the classical notions of limited, $p$-limited and coarse $p$-limited subsets of a Banach space, we introduce and study $(p,q)$-limited subsets and their equicontinuous versions and coarse…

We construct various examples of non-trivial closed ideals of the compact-by-approximable algebra $\mathfrak{A}_X =:\mathcal K(X)/\mathcal A(X)$ on Banach spaces $X$ failing the approximation property. The examples include the following:…

We show that if a rearrangement invariant Banach function space $E$ on the positive semi-axis satisfies a non-trivial lower $q-$ estimate with constant $1$ then the corresponding space $E(\nm)$ of $\tau-$measurable operators, affiliated…

We completely characterize the boundedness of the area operators from the Bergman spaces $A^p_\alpha(\mathbb{B}_ n)$ to the Lebesgue spaces $L^q(\mathbb{S}_ n)$ for all $0<p,q<\infty$. For the case $n=1$, some partial results were…

We study embeddings of uniform Roe algebras which have "large range" in their codomain and the relation of those with coarse quotients between metric spaces. Among other results, we show that if $Y$ has property A and there is an embedding…

Let $A$ be an unbounded operator on a Banach space $X$. It is sometimes useful to improve the operator $A$ by extending it to an operator $B$ on a larger Banach space $Y$ with smaller spectrum. It would be preferable to do this with some…

We give sufficient conditions on an asymptotic $\ell_p$ (for $1 < p < \infty$) Banach space which ensure the space admits an operator which is not a compact perturbation of a multiple of the identity. These conditions imply the existence of…

We introduce notions of finite presentation and co-exactness which serve as qualitative and quantitative analogues of finite-dimensionality for operator modules over completely contractive Banach algebras. With these notions we begin the…