A projection $P_0$ on a complex Banach space is generalized $3$- circular if its linear combination with two projections $P_1$ and $P_2$ having coefficients ${\lambda }_1$ and ${\lambda }_2$, respectively, is a surjective isometry, where ${\lambda }_1$ and ${\lambda }_2$ are distinct unit modulus complex numbers different from $1$ and $P_0\oplus P_1\oplus P_2=I$. Such projections are always contractive. In this paper, we prove structure theorems for generalized $3$-circular projections acting on the spaces of all $n\times n$ symmetric and skew-symmetric matrices over $\mathbb{C}$ when these spaces are equipped with unitary congruence invariant norms.

A linear mapping $\delta$ from an algebra $\mathcal{A}$ into a left $\mathcal{A}$-module $\mathcal{M}$ is called a Jordan left derivation if $\delta \left(A^2\right)=2A\delta \left(A\right)$ for every $A\in \mathcal{A}$. We prove that if an algebra $\mathcal{A}$ and a left $\mathcal{A}$-module $\mathcal{M}$ satisfy one of the following conditions—(1) $\mathcal{A}$ is a $C^{\ast }$-algebra and $\mathcal{M}$ is a Banach left $\mathcal{A}$-module; (2) $\mathcal{A}=Alg\mathcal{L}$ with $\cap \{L_{-}:L\in {\mathcal{J}}_{\mathcal{L}}\}=\left(0\right)$ and $\mathcal{M}=B\left(X\right)$; and (3) $\mathcal{A}$ is a commutative subspace lattice algebra of a von Neumann algebra $\mathcal{B}$ and $\mathcal{M}=B\left(\mathcal{H}\right)$—then every Jordan left derivation from $\mathcal{A}$ into $\mathcal{M}$ is zero. $\delta$ is called left derivable at $G\in \mathcal{A}$ if $\delta \left(AB\right)=A\delta \left(B\right)+B\delta \left(A\right)$ for each $A,B\in \mathcal{A}$ with $AB=G$. We show that if $\mathcal{A}$ is a factor von Neumann algebra, $G$ is a left separating point of $\mathcal{A}$ or a nonzero self-adjoint element in $\mathcal{A}$, and $\delta$ is left derivable at $G$, then $\delta \equiv 0$.

We obtain two results stating the uniform openness of bilinear operators and multilinear functionals. The first result deals with Banach spaces $L^p:=L_{\mathbb{K}}^p$ (over $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$) and pointwise multiplication from $L^p\times L^q$ to $L^r$ (where $1/p+1/q=1/r$). The second result is concerned with the nontrivial $n$-linear functionals from the product $X_1\times \cdots \times X_n$ of normed spaces (over $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$) to the field $\mathbb{K}$.

Let $\mathcal{A}$ be a Rickart $*$-ring and let ${\le }^{*},{\le }^{♯},{\le }^{\oplus }$, and ${\le }_{\oplus }$ denote the star, the sharp, the core, and the dual core partial orders in $\mathcal{A}$, respectively. The sets of all $b\in \mathcal{A}$ such that $a\le b$, along with the sets of all $b\in \mathcal{A}$ such that $b\le a$, are characterized, where $a\in \mathcal{A}$ is given and where $\le$ is one of the partial orders: ${\le }^{*}$, or ${\le }^{♯}$, or ${\le }^{\oplus }$, or ${\le }_{\oplus }$. Such sets of elements that are above or below a given element under the minus partial order ${\le }^{-}$ in a Rickart ring $\mathcal{A}$ are also studied. Some recent results of Cvetković-Ilić et al. on partial orders in $\mathcal{B}\left(H\right)$, the algebra of all bounded linear operators on a Hilbert space $H$, are thus generalized.

Let $A$ and $B$ be Banach algebras, let $\theta :A\to B$ be a continuous Banach algebra homomorphism, and let $I$ be a closed ideal in $B$. Then the $l^1$-direct sum of $A$ and $I$ with a special product becomes a Banach algebra, denoted by $A{\bowtie }^{\theta }I$, which we call the generalized semidirect product of $A$ and $I$. In this article, among other things, we first characterize derivations on $A{\bowtie }^{\theta }I$ and then we compute the first cohomology group of $A{\bowtie }^{\theta }I$ when the first cohomology groups of $A$ with coefficients in $A$ and $I$ are trivial. As an application we characterize the first cohomology group of second duals of dual Banach algebras. Then we provide a nice characterization of the first cohomology group of $A{\bowtie }^{\mathrm{id}}A$. Furthermore, we calculate the first cohomology group of some concrete Banach algebras related to locally compact groups.

We define Hardy spaces of functions taking values on a Banach space $\mathcal{X}$ over nonsmooth domains. The types of functions we consider are harmonic functions on a starlike Lipschitz domain and solutions to the heat equation on a time-varying domain. Our purpose is twofold: (a) to characterize the Radon–Nikodym property of the Banach space $\mathcal{X}$ in terms of the existence of nontangential limits of $\mathcal{X}$-valued functions $u$ in the corresponding Hardy space with index $p\ge 1$, (b) to identify the function of the boundary values of $u$ in the Hardy space with index $p>1$ with an element in the space ${\mathcal{V}}_{\mathcal{X}}^p$ of measures of $p$-bounded variation in the absence of the Radon–Nikodym property of $\mathcal{X}$. This extends similar results already known on the unit disk of $\mathbb{C}$ and the semispace ${\mathbb{R}}^n\times (0,\infty )$.

We study new classes of linear preservers between ${\mathrm{C}}^{\ast }$-algebras and between ${\mathrm{JB}}^{\ast }$-triples. Let $E$ and $F$ be ${\mathrm{JB}}^{\ast }$-triples with ${\partial }_e\left(E_1\right)\ne \varnothing$. We prove that every linear map $T:E\to F$ strongly preserving Brown–Pedersen quasi-invertible elements is a triple homomorphism. Among the consequences, we establish that, given two unital ${\mathrm{C}}^{\ast }$-algebras $A$ and $B$, for each linear map $T$ strongly preserving Brown–Pedersen quasi-invertible elements, there exists a Jordan ${}^{\ast }$-homomorphism $S:A\to B$ satisfying $T\left(x\right)=T\left(1\right)S\left(x\right)$ for every $x\in A$. We also study the connections between linear maps strongly preserving Brown–Pedersen quasi-invertibility and other clases of linear preservers between ${\mathrm{C}}^{\ast }$-algebras like Bergmann-zero pairs preservers, Brown–Pedersen quasi-invertibility preservers, and extreme points preservers.

We introduce the Hardy type space for Musielak–Orlicz spaces. It includes several existing Hardy type spaces such as the Hardy–Orlicz spaces and the Hardy spaces with variable exponents. Furthermore, we develop an atomic decomposition such that the size condition just relies on the norms of Musielak–Orlicz spaces. This gives us a nature extension of the molecular decompositions to the Hardy type space for Musielak–Orlicz spaces.

The notion of Szegö-type properties of positive Borel measures is well known and widely exploited. In this paper, we consider a class of orthogonal decompositions of isometries on Hilbert spaces which correspond to Szegö-type properties of their elementary measures. Our decompositions are closely connected with some special families of invariant subspaces. It is shown that this connection holds for the decomposition constructed in the paper. We illustrate our results with several examples. We also give a short proof of Mlak’s theorem on the elementary measures of completely nonunitary contractions.

Let $A$ be a commutative unital Banach algebra and let $X$ be a compact space. We study the class of $A$-valued function algebras on $X$ as subalgebras of $C(X,A)$ with certain properties. We introduce the notion of $A$-characters of an $A$-valued function algebra $\mathcal{A}$ as homomorphisms from $\mathcal{A}$ into $A$ that basically have the same properties as evaluation homomorphisms ${\mathcal{E}}_x:f\mapsto f\left(x\right)$, with $x\in X$. We show that, under certain conditions, there is a one-to-one correspondence between the set of $A$-characters of $\mathcal{A}$ and the set of characters of the function algebra $\mathfrak{A}=\mathcal{A}\cap C\left(X\right)$ of all scalar-valued functions in $\mathcal{A}$. For the so-called natural $A$-valued function algebras, such as $C(X,A)$ and $Lip(X,A)$, we show that ${\mathcal{E}}_x$ ($x\in X$) are the only $A$-characters. Vector-valued characters are utilized to identify vector-valued spectra.

We prove that for a given Banach space $X$, the subset of norm-attaining Lipschitz functionals in ${Lip}_0\left(X\right)$ is weakly dense but not strongly dense. Then we introduce a weaker concept of directional norm attainment and demonstrate that for a uniformly convex $X$ the set of directionally norm-attaining Lipschitz functionals is strongly dense in ${Lip}_0\left(X\right)$ and, moreover, that an analogue of the Bishop–Phelps–Bollobás theorem is valid.

A finite solution to a system of equations will generate a single function normalized tight frame wavelet (Parseval’s frame wavelet) with compact support associated with a $2\times 2$ expansive integral matrix whose determinant is either $2$ or $-2$ in $L^2\left({\mathbb{R}}^2\right)$.