We generalize a result of Kamalov and we show that if $G$ is an amenable discrete group with an action $\alpha$ on a finite nuclear unital $C^{\ast }$-algebra $A$ such that the reduced crossed product $A{⋊}_{\alpha ,r}G$ has property $T$, then $G$ is finite and $A$ is finite-dimensional. As an application, an infinite discrete group $H$ is nonamenable if and only if the corresponding uniform Roe algebra $C_u^{\ast }\left(H\right)$ has property $T$.

Considering $n\times n\times n$ stochastic tensors $\left(a_{ijk}\right)$ (i.e., nonnegative hypermatrices in which every sum over one index $i$, $j$, or $k$, is 1), we study the polytope (${\Omega }_n$) of all these tensors, the convex set ($L_n$) of all tensors in ${\Omega }_n$ with some positive diagonals, and the polytope (${\Delta }_n$) generated by the permutation tensors. We show that $L_n$ is almost the same as ${\Omega }_n$ except for some boundary points. We also present an upper bound for the number of vertices of ${\Omega }_n$.

We present some results on the monotonicity of some traces involving functions of self-adjoint operators with respect to the natural ordering of their associated quadratic forms. The relation between these results and Löwner’s Theorem is discussed. We also apply these results to complete a proof of the Wegner estimate for continuum models of random Schrödinger operators as given in a 1994 paper by Combes and Hislop.

Let $\mathcal{M}$ be a von Neumann algebra with a faithful normal semifinite trace $\tau$. The noncommutative Hardy space $H^p\left(\mathcal{M}\right)$ associates with $\mathcal{A}$, which is a subdiagonal algebra of $\mathcal{M}$. We define the Hankel operator $H_t$ on $H^p\left(\mathcal{M}\right)$, and we obtain that the norm $\Vert H_t\Vert$ is equal to $d(t;\mathcal{A})$ and is also the equivalent of the $BMO\left({\mathcal{M}}^{\mathrm{sa}}\right)$ norm of $t$ for every $t\in \mathcal{M}$, where ${\mathcal{M}}^{\mathrm{sa}}$ are the self-adjoint operators in $\mathcal{M}$.

Spectral theory and functional calculus for unbounded self-adjoint operators on a Hilbert space are usually treated through von Neumann’s Cayley transform. Using ideas of Woronowicz, we redevelop this theory from the point of view of multiplier algebras and the so-called bounded transform (which establishes a bijective correspondence between closed operators and pure contractions). This also leads to a simple account of the affiliation relation between von Neumann algebras and self-adjoint operators.

In this article, we prove the weak bound for an $n$-dimensional Hardy operator on a central Morrey space. Meanwhile, we obtain the precise operator norm, and we give the weak bounds for the conjugate Hardy operator on Lebesgue space with power weights. The corresponding operator norms are also computed. As an application, we obtain an estimate for the gamma function.

Let $\mathcal{B}\left(H\right)$ be the algebra of all bounded linear operators on a complex Hilbert space $H$, and let $\mathcal{S}$ be a norm ideal in $\mathcal{B}\left(H\right)$. For $A,B\in \mathcal{B}\left(H\right)$, define the elementary operator $M_{\mathcal{S},A,B}$ on $\mathcal{S}$ by $M_{\mathcal{S},A,B}\left(X\right)=AXB$ ($X\in \mathcal{S}$). The aim of this paper is to give necessary and sufficient conditions under which the equality $V\left(M_{\mathcal{S},A,B}\right)=\overline{co}\left(W\right(A\left)W\right(B\left)\right)$ holds. Here $V\left(T\right)$ and $W\left(T\right)$ denote the algebraic numerical range and spatial numerical range of an operator $T$, respectively, and $\overline{co}(\Omega )$ denotes the closed convex hull of a subset $\Omega \subseteq \mathbb{C}$.

We introduce a certain property of commutative Banach algebras which we call property $\mathrm{O}\mathbb{B}$. We prove that every bounded disjointness-preserving linear map from a commutative Banach algebra with the aforesaid property to any semisimple, commutative Banach algebra is a weighted composition map. Further, it is shown that a variety of important Banach algebras in harmonic analysis have the property $\mathrm{O}\mathbb{B}$.

The exact value of the Schäffer-type constants are investigated under the absolute normalized norms on ${\mathbb{R}}^2$ by means of their corresponding continuous convex functions on $[0,1]$. Moreover, some sufficient conditions which imply uniform normal structure are presented. These results improve some known results.

It is well known, as a consequence of a theorem of Richard Arens, that a commutative Fréchet locally $m$-convex algebra $E$ with unit does not have dense finitely generated ideals. We shall see that this result can no longer be true if $E$ is not complete and metrizable. We observe that the same is true for the theorem of Arens; that is, this theorem can no longer be true if $E$ is not complete and metrizable. Moreover, several conditions for a unital commutative (not necessarily complete) locally $m$-convex algebra are given, for which all maximal ideals have codimension one.

For Banach spaces $X$ and $Y$, let $K(X,Y)$ denote the space of all compact operators from $X$ to $Y$ endowed with the operator norm. We give sufficient conditions for subsets of $K(X,Y)$ to be relatively compact. We also give some necessary and sufficient conditions for the Dunford–Pettis relatively compact property of some spaces of operators.

Let ${\hat{L}}_p\left(\mathrm{M}\right)$ be the space of bounded $L_p\left(\mathrm{M}\right)$-quasimartingales. We prove that, with equivalent norms, $\left({\hat{L}}_{p_0}\right(\mathrm{M}),{\hat{L}}_{p_1}(\mathrm{M}){)}_{\theta ,p}={\hat{L}}_p(\mathrm{M})$, where $1<p_0,p_1\le \infty$, $1<\theta <1$, and $\frac{1}{p}=\frac{1-\theta }{p_0}+\frac{\theta }{p_1}$. We also prove that, for $1<p<q<\infty$, $\left({\widehat{BMO}}^c\right(\mathrm{M}),{\hat{\mathrm{H}}}_p^c(\mathrm{M}){)}_{\frac{p}{q},q}={\hat{\mathrm{H}}}_q^c(\mathrm{M})$ and $\left({\widehat{BMO}}^r\right(\mathrm{M}),{\hat{\mathrm{H}}}_p^r(\mathrm{M}){)}_{\frac{p}{q},q}={\hat{\mathrm{H}}}_q^r(\mathrm{M})$, where ${\hat{\mathrm{H}}}_p\left(\mathrm{M}\right)$ and $\widehat{BMO}\left(\mathrm{M}\right)$ are, respectively, the Hardy space and the bounded mean oscillation space of noncommutative quasimartingales.

This article investigates a bijective map $\Phi$ between two von Neumann algebras, one of which has no central abelian projections, satisfying $\Phi \left(\right[[A,B{]}_{\ast },C{]}_{\ast })=\left[\right[\Phi \left(A\right),\Phi \left(B\right){]}_{\ast },\Phi \left(C\right){]}_{\ast }$ for all $A,B,C$ in the domain, where $[A,B{]}_{\ast }=AB-B{A}^{*}$ is the skew Lie product of $A$ and $B$. We show that the map $\Phi \left(I\right)\Phi$ is a sum of a linear $\ast$-isomorphism and a conjugate linear $\ast$-isomorphism, where $\Phi \left(I\right)$ is a self-adjoint central element in the range with $\Phi (I{)}^2=I$.

In this paper, upper semicontinuity and continuity of the set-valued metric generalized inverse $T^{\partial }$ in nearly dentable spaces are investigated using the methods of Banach space geometry. Moreover, it is proved that if $X$ is a nearly dentable space and if $C$ is a closed convex set of $X$, then $C$ is approximatively compact if and only if $P_C\left(x\right)$ is compact for any $x\in X$.

We give some new estimates for the essential norm of weighted composition operators on the space $\mathit{BMOA}$. As a corollary, we obtain a new characterization for the compactness of weighted composition operators on the space $\mathit{BMOA}$.