In this paper, a new definition of majorization for $C^{*}$-algebras is introduced. Sherman’s inequality is extended to self-adjoint operators and positive linear maps by applying the method of premajorization used for comparing two tuples of objects. A general result in a matrix setting is established. Special cases of the main theorem are studied. In particular, a HLPK-type inequality is derived.

We present two versions of the well-known Banach contraction principle: one in the context of extended metric spaces for which the distance mapping is allowed to be infinite, the other in the context of metric spaces endowed with a compatible binary relation. We also point out that these two results and the Banach contraction principle are actually equivalent.

This article deals with the approximation properties of the bivariate operators which are the combination of Bernstein–Chlodowsky operators and the Szász operators involving Appell polynomials. We investigate the degree of approximation of the operators with the help of the complete modulus of continuity and the partial moduli of continuity. In the last section of the paper, we introduce the generalized Boolean sum (GBS) of these bivariate Chlodowsky–Szasz–Appell-type operators and examine the order of approximation in the Bögel space of continuous functions by means of the mixed modulus of smoothness.

We demonstrate that any surjective isometry $T:\mathcal{A}\to \mathcal{B}$ not assumed to be linear between unital, completely regular subspaces of complex-valued, continuous functions on compact Hausdorff spaces is of the form $$T\left(f\right)=T\left(0\right)+Re[\mu \cdot (f\circ \tau \left)\right]+iIm[\nu \cdot (f\circ \rho \left)\right],$$ where $\mu$ and $\nu$ are continuous and unimodular, there exists a clopen set $K$ with $\nu =\mu$ on $K$ and $\nu =-\mu$ on $K^c$, and $\tau$ and $\rho$ are homeomorphisms.

Let $\mathcal{M}$ be a semifinite von Neumann algebra, and let $\mathcal{A}$ be a tracial subalgebra of $\mathcal{M}$. We show that $\mathcal{A}$ is a subdiagonal algebra of $\mathcal{M}$ if and only if it has the unique normal state extension property and is a $\tau$-maximal tracial subalgebra, which is also equivalent to $\mathcal{A}$ having the unique normal state extension property and satisfying $L_2$-density.

The aim of this article is to study the Fredholm completion problem of two-by-two partial operator matrices in which the lower-left entry is unspecified and others are specified. By using the methods of operator matrix representation and operator equation, we obtain necessity and sufficiency conditions for the partial operator matrices to have a Fredholm completion with the property that the lower-right entry of its Fredholm inverses is specified.

In the present paper, we consider cubic stochastic operators, and prove that the surjectivity of such operators is equivalent to their orthogonal-preserving property. In the last section we provide a full description of orthogonal-preserving (respectively, surjective) cubic stochastic operators on the $2$-dimensional simplex.

We define and study the weak Haagerup property for $C^{\ast }$-algebras in this article. A $C^{\ast }$-algebra with the Haagerup property always has the weak Haagerup property. We prove that a discrete group has the weak Haagerup property if and only if its reduced group $C^{\ast }$-algebra also has that property. Moreover, we consider the permanence of the weak Haagerup property under a few canonical constructions of $C^{\ast }$-algebras.

Let $X,Y$ be Banach spaces. We define $${\alpha }_Y\left(X\right)=sup\hspace{0.17em}\left\{\right|T^{-1}{|}^{-1}:T:Y\to X\text{is an isomorphism with}\left|T\right|\le 1\}.$$ If there is no isomorphism from $Y$ to $X$, we set ${\alpha }_Y\left(X\right)=0$, and

$${\gamma }_Y\left(X\right)=sup\hspace{0.17em}\left\{\delta \right(T):T:X\to Y\text{is a surjective operator with}|T|\le 1\},$$ where $\delta \left(T\right)=sup\hspace{0.17em}\{\delta >0:\delta B_Y\subseteq T{B}_X\}$. If there is no surjective operator from $X$ onto $Y$, we set ${\gamma }_Y\left(X\right)=0$. We prove that for a separable space $X$, ${\alpha }_{l_1}\left(X^{\ast }\right)={\gamma }_{c_0}\left(X\right)$ and ${\alpha }_{L_1}\left(X^{\ast }\right)={\gamma }_{C(\Delta )}\left(X\right)={\gamma }_{C[0,1]}\left(X\right)$.

The set of supporting vectors of a continuous linear operator, that is, the normalized vectors at which the operator attains its norm, is decomposed into its convex components. In the complex case, the set of supporting vectors of a nonzero functional is proved to be path-connected. We also introduce the concept of generalized supporting vectors for a sequence of operators as the normalized vectors that maximize the summation of the squared norm of those operators. We determine the set of generalized supporting vectors for the particular case of a finite sequence of real matrices. Finally, we unveil the relation between the supporting vectors of a real matrix $A$ and the Tikhonov regularization ${min\hspace{0.17em}}_{x\in {\mathbb{R}}^n}\Vert Ax-b\Vert +\alpha \Vert x\Vert$ reaching the conclusion that, by an appropriate choice of $b$ and $\alpha$, the supporting vectors of $A$ can be obtained via solving the Tikhonov regularization ${min\hspace{0.17em}}_{x\in {\mathbb{R}}^n}\Vert Ax-b\Vert +\alpha \Vert x\Vert$.

We introduce a family $\mathcal{T}$ consisting of invertible matrices with exactly one nonzero entry in each row and each column. The elements of $\mathcal{T}$ are possibly mutually noncommuting, and they need not be normal or self-adjoint. We consider an operator-valued unilateral weighted shift $W$ with a uniformly bounded sequence of weights belonging to $\mathcal{T}$, and we describe its minimal reducing subspaces.

Let $K$ be a hypergroup. The purpose of this article is to study the question of involutions on algebras $M(K{)}^{\ast \ast }$, $L(K{)}^{\ast \ast }$, and $L_c(K{)}^{\ast \ast }$. We show that the natural involution of $M\left(K\right)$ has the canonical extension to $M(K{)}^{\ast \ast }$ if and only if the natural involution of $L\left(K\right)$ has the canonical extension to $L(K{)}^{\ast \ast }$. Also, we give necessary and sufficient conditions for $M(K{)}^{\ast \ast }$ and $L(K{)}^{\ast \ast }$ to admit an involution extending the natural involution of $M\left(K\right)$ when $K$ is left amenable. Finally, we find the necessary and sufficient conditions for $L_c(K{)}^{\ast \ast }$ to admit an involution.