Positivity of a $2\times 2$ operator matrix implies $\sqrt{\Vert A\Vert \cdot \Vert C\Vert }\ge \Vert B\Vert$ for operator norm $\Vert \cdot \Vert$. This can be considered as an operator version of the Schwarz inequality. In this situation, for $A,C\ge 0$, there is a natural notion of geometric mean $A♯C$, for which $\sqrt{\Vert A\Vert \cdot \Vert C\Vert }\ge \Vert A♯C\Vert$. In this paper, we study under what conditions on $A$, $B$, and $C$ or on $B$ alone the norm inequality $\sqrt{\Vert A\Vert \cdot \Vert C\Vert }\ge \Vert B\Vert$ can be improved as $\Vert A♯C\Vert \ge \Vert B\Vert$.

For a locally compact group $G$ and a compact subgroup $H$ of $G$, the square-integrable representations of group $G$ and homogeneous space $G/H$ are described. Also, the connection between the existence of admissible wavelets for locally compact groups and their homogeneous spaces is compared. Moreover, some properties of admissible wavelets and wavelet constants for homogeneous space $G/H$ are investigated, when $G$ is unimodular.

In this note, we prove that the Birkhoff–James orthogonality, as well as the strong Birkhoff–James orthogonality, is a symmetric relation in a full Hilbert $\mathcal{A}$-module $V$ if and only if at least one of the underlying $C^{\ast }$-algebras $\mathcal{A}$ or $\mathbf{K}\left(V\right)$ is isomorphic to $\mathbb{C}$.

Enlightened by the notion of perturbation of $C^{\ast }$-algebras, we introduce, and study briefly in this article, a notion of closeness of groups. We show that if two groups are “close enough” to each other, and one of them has the property that the orders of its elements have a uniform finite upper bound, then these two groups are isomorphic (but in general they are not). We also study groups that are close to abelian groups, as well as an equivalence relation induced by closeness.

We study the extension property of isometries on the unit sphere of the ${\ell }^1$-sum of strictly normed spaces, which is a special case of Tingley’s isometric extension problem. In this paper, we will give some sufficient conditions such that such isometries can be extended to the whole space.

In this exposition-type note we present detailed proofs of certain assertions concerning several algebraic properties of the cone and cylinder algebras. These include a determination of the maximal ideals, the solution of the Bézout equation, and a computation of the stable ranks by elementary methods.

In this paper, we prove strong convergence theorems by two hybrid methods for semigroups of not necessarily continuous mappings in Hilbert spaces. Using these results, we prove strong convergence theorems for discrete semigroups generated by generalized hybrid mappings and semigroups of nonexpansive mappings in Hilbert spaces.

We present a group structure on $\mathbb{D}$ via the automorphisms which fix the point $1$. Through the induced group action, each point of $\mathbb{D}$ produces an equivalence class that turns out to be a Blaschke sequence. We show that the corresponding Blaschke products are minimal/atomic solutions of the functional equation $\psi \circ \phi =\lambda \psi$, where $\lambda$ is a unimodular constant and $\phi$ is an automorphism of the unit disk. We also characterize all Blaschke products that satisfy this equation, and we study its application in the theory of composition operators on model spaces $K_{\Theta }$.

For Figà-Talamanca–Herz algebras $A_p\left(G\right)$, $1<p<\infty$, of a locally compact group $G$ and a closed subgroup $H$ of $G$, we prove an injection theorem for local Ditkin sets.

The main aim of this survey article is to present recent developments of matrix versions of the arithmetic–geometric mean inequality. Among others, we show improvements and generalizations of the arithmetic–geometric mean inequality for unitarily invariant norms via the Hadamard product, and for singular values via the operator monotone functions.

A classical theorem of G. Köthe states that the Banach spaces $X$ with the property that all bounded linear maps $X\to Y$ into an arbitrary Banach space $Y$ can be lifted with respect to bounded linear surjections onto $Y$ are up to topological linear isomorphism precisely the spaces ${\ell }^1\left(A\right)$. We extend this result to the category of normed linear spaces and bounded linear maps. This answers a question raised by A. Ya. Helemskiĭ.

In this article we will consider locally convex topologies $\tau$ on ${\ell }_1$ which are coarser than the weak topology on the unit ball and such that the unit vector basic sequence $\left(e_n\right)$ is $\tau$-convergent. We characterize these topologies depending on the $\tau$-fixed point property for left reversible semigroups on $({\ell }_1,\Vert \cdot {\Vert }_1)$. We will apply our results to the case of different weak${}^{\ast }$ topologies on ${\ell }_1$.

We determine when contractive idempotents in the measure algebra of a locally compact group commute. We consider a dynamical version of the same result. We also look at some properties of groups of measures whose identity is a contractive idempotent.

In this note, as a particular case of a more general result, we obtain the following theorem.

Let $\Omega \subseteq {\mathbf{R}}^n$ be a nonempty bounded open set, and let $f:\overline{\Omega }\to {\mathbf{R}}^n$ be a continuous function which is $C^1$ in $\Omega$. Then, at least one of the following assertions holds:

(a) $f(\Omega )\subseteq conv\left(f\right(\partial \Omega \left)\right)$.

(b) There exists a nonempty open set $X\subseteq \Omega$, with $\overline{X}\subseteq \Omega$, satisfying the following property: for every continuous function $g:\Omega \to {\mathbf{R}}^n$ which is $C^1$ in $X$, there exists $\tilde{\lambda }\ge 0$ such that, for each $\lambda >\tilde{\lambda }$, the Jacobian determinant of the function $g+\lambda f$ vanishes at some point of $X$.

As a consequence, if $n=2$ and $h:\Omega \to \mathbf{R}$ is a nonnegative function, for each $u\in C^2(\Omega )\cap C^1\left(\overline{\Omega }\right)$ satisfying in $\Omega$ the Monge–Ampère equation

$$u_{xx}{u}_{yy}-u_{xy}^2=h,$$ one has

$$\nabla u(\Omega )\subseteq conv(\nabla u(\partial \Omega \left)\right).$$

We show that, given a compact Hausdorff space $\Omega$, there is a compact group $\mathbb{G}$ and a homeomorphic embedding of $\Omega$ into $\mathbb{G}$, such that the restriction map $\mathrm{A}\left(\mathbb{G}\right)\to C(\Omega )$ is a complete quotient map of operator spaces. In particular, this shows that there exist compact groups which contain infinite cb-Helson subsets, answering a question raised by Choi and Samei. A negative result from that paper is also improved.

Properties of spectral synthesis are exploited to show that, for a large class of commutative hypergroups and for every compact hypergroup, every closed, reflexive, left-translation-invariant subspace of $L^{\infty }\left(K\right)$ is finite-dimensional. Also, we show that, for a class of hypergroups which includes many commutative hypergroups and all $\mathcal{Z}$-hypergroups, every derivation of $L^1\left(K\right)$ into an arbitrary Banach $L^1$-bimodule is continuous.

We introduce the notion of a positive spectral measure on a $\sigma$-algebra, taking values in the positive projections on a Banach lattice. Such a measure generates a bounded positive representation of the bounded measurable functions. If $X$ is a locally compact Hausdorff space and if $\pi$ is a positive representation of $C_0\left(X\right)$ on a KB-space, then $\pi$ is the restriction to $C_0\left(X\right)$ of such a representation generated by a unique regular positive spectral measure on the Borel $\sigma$-algebra of $X$. The relation between a positive representation of $C_0\left(X\right)$ on a Banach lattice and—if it exists—a generating positive spectral measure on the Borel $\sigma$-algebra are further investigated; here and elsewhere, phenomena occur that are specific for the ordered context.

In this paper, we review aspects of the biography and mathematical work of Professor Anthony To-Ming Lau.