We characterize linear maps on von Neumann algebras which leave every unital subalgebra invariant. We use this characterization to determine linear maps which respect unitary conjugation, answering a question of M. S. Moslehian.

In this paper we extend the notion of quasi-multipliers to the dual of a Banach algebra $A$ whose second dual has a mixed identity. We consider algebras satisfying weaker condition than Arens regularity. Among others we prove that for an Arens regular Banach algebra which has a bounded approximate identity the space $QM_{r}(A^{*})$ of all bilinear and separately continuous right quasi-multipliers of $A^{*}$ is isometrically isomorphic to $A^{**}.$ We discuss the strict topology on $QM_{r}(A^{*})$ and apply our results to $C^{*}-$algebras and to the group algebra of a compact group.

In the present work, a common fixed point theorem for self maps on cone metric spaces is proved. Also two examples, which shows that our main theorem is generalized version of main theorems of [A. Branciari, Int. J. Math. Math. Sci., 29 (2002), no. 9, 531-536] and [L.G. Huang and X. Zhang, J. Math. Anal. Appl. 332 (2007), no. 2, 1468-1476] are given.

An element of a Banach algebra is EP, if it commutes with its Moore-Penrose inverse. We present a number of new characterizations of EP elements in Banach algebra.

The main objective of this paper is a study of some new generalizations of Hilbert's and Hardy-Hilbert's type inequalities. We apply our general results to homogeneous functions. Also, we obtain the best possible constants when the parameters satisfy appropriate conditions.

We show that the full group $C^*$-algebra of the free product of two nontrivial countable amenable discrete groups, where at least one of them has more than two elements, is primitive. We also show that in many cases, this $C^*$-algebra is antiliminary and has an uncountable family of pairwise inequivalent, faithful irreducible representations.

The relevance of modulation spaces for deformation quantization, Landau--Weyl quantization and noncommutative quantum mechanics became clear in recent work. We continue this line of research and demonstrate that $Q_s(\mathbb{R}^{2d})$ is a good class of symbols for Landau-Weyl quantization and propose that the modulation spaces $M^p_{v_s}(\mathbb{R}^{2d})$ are natural generalized Shubin classes for the Weyl calculus. This is motivated by the fact that the Shubin class $Q_s(\mathbb{R}^{2d})$ is the modulation space $M^2_{v_s}(\mathbb{R}^{2d})$. The main result gives estimates of the singular values of pseudodifferential operators with symbols in $M^p_{v_s}(\mathbb{R}^{2d})$ for the standard Weyl calculus and for the Landau--Weyl calculus.

Let $1< p\neq 2<\infty$, $\mathcal{E}>0$ and let $T$ be an isomorphism from $\ell_p(\ell_2)$ into $L_p[0,1]$. Then there is a subspace $Y\subset \ell_p(\ell_2)$, $(1+\mathcal{E})$-isomorphic to $\ell_p(\ell_2)$ such that $T_{|Y}$ is an $(1+\mathcal{E})$-isomorphism and $T\left(Y\right)$ is $K_p$-complemented in $L_{p}\left[0,1\right]$, with $K_p$ depending only on $p$. Moreover, $K_p\le (1+\mathcal{E})\gamma_p$ if $p>2$ and $K_p\le (1+\mathcal{e})\gamma_{p/(p-1)}$ if $1<p<2$, where $\gamma_r$ is the $L_r$ norm of a standard Gaussian variable.

By introducing the concept of limited completely continuous operators between two arbitrary Banach spaces $X$ and $Y$, we give some properties of this concept related to some well known classes of operators and specially, related to the Gelfand-Phillips property of the space $X$ or $Y$. Then some necessary and sufficient conditions for the Gelfand--Phillips property of closed subspace $M$ of some operator spaces, with respect to limited complete continuity of some operators on $M$, so-called, evaluation operators, are verified.

Approximation spaces, in their many presentations, are well known mathematical objects and many authors have studied them for long time. They were introduced by Butzer and Scherer in 1968 and, independently, by Y. Brudnyi and N. Kruglyak in 1978, and popularized by Pietsch in his seminal paper of 1981. Pietsch was interested in the parallelism that exists between the theories of approximation spaces and interpolation spaces, so that he proved embedding, reiteration and representation results for approximation spaces. In particular, embedding results are a natural part of the theory since its inception. The main goal of this paper is to prove that, for certain classes of approximation schemes $(X,\{A_n\})$ and sequence spaces $S$, if $S_1\subset S_2\subset c_0$ (with strict inclusions) then the approximation space $A(X,S_1,\{A_n\})$ is properly contained into $A(X,S_2,\{A_n\})$. We also initiate a study of strict inclusions between interpolation spaces, for Petree's real interpolation method.

A theorem of Dixmier states that each bounded linear functional $f$ on the algebra of bounded linear operators on a separable Hilbert space is a direct sum of a trace functional $g$ and a singular functional $h$, vanishing on the compact operators, such that $\Vert f \Vert= \Vert g \Vert +\Vert h \Vert$. We use elementary methods to construct, via the state space of a $C^\ast$-algebra, a Banach space of $C^\ast$ matrices that contains a closed subspace on which a version of Dixmier's theorem is proved. When the $C^\ast$-algebra is taken to be the complex numbers our approach gives elementary and transparent proofs of Dixmier's theorem and the trace formula $\rm{tr}(AB) = \rm{tr}(BA)$, without using the operator theoretical machineries used in the known proofs.

Using a recently obtained criterion of $C^*$-reflexivity for commutative $C^*$-algebras, we show that the $C^*$-algebra of continuous functions on the Higson corona is not $C^*$-reflexive. This implies that $C^*$-reflexivity doesn't pass to quotient $C^*$-algebras.

In this short note, a new approach is provided to prove that every nonzero continuous cosine function on a compact group $G$ is the normalized character of a representation of $G$ into the special unitary group $SU(2)$.

Let $X$ be a Baire space, $Y$ be a $W$-space and $Z$ be a regular topological space. We will show that every $KC$-function $f:X \times Y\to Z$ is strongly quasi-continuous at each point of $X \times Y$. In particular, when $X$ is a Baire space and $Y$ is Corson compact, every $KC$-function $f$ from $X \times Y$ to a Moore space $Z$ is jointly continuous on a dense subset of $X \times Y$. We also give a few applications of our results on continuity of group actions.

In this paper we survey the results on the Dunkl-Williams inequality in normed linear spaces. These are related to the geometry of normed linear spaces, the characterizations of inner product spaces, some inequalities regarding operators on Hilbert spaces and elements of Hilbert $C^*$-modules.