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Suppose X and Y are Banach spaces, K is a compact Hausdorff space, and $C(K, X)$ is the Banach space of all continuous X-valued functions (with the supremum norm). We will study some strongly bounded operators $T:C(K, X)\to Y$ with representing measures $m:\Sigma \to L(X,Y)$, where $L(X,Y)$ is the Banach space of all operators $T:X\to Y$ and $\Sigma$ is the $\sigma$-algebra of Borel subsets of K. The classes of operators that we will discuss are the Grothendieck, p-limited, p-compact, limited, operators with completely continuous, unconditionally converging, and p-converging adjoints, compact, and absolutely summing. We give a characterization of the limited operators (resp. operators with completely continuous, unconditionally converging, p-convergent adjoints) in terms of their representing measures. Full article

Here all rings have identities. Let R be a ring and let R-mod denote the additive category of left finitely generated R-modules. Note that if R is a noetherian ring, then R-mod is an abelian category and every R-module is a finite direct sum of indecomposable R-modules. Finite Group Modular Representation Theory concerns the study of left finitely generated $\mathcal{O}G$-modules where G is a finite group and $\mathcal{O}$ is a complete discrete valuation ring with $\mathcal{O}/J\left(\mathcal{O}\right)$ a field of prime characteristic p. Thus $\mathcal{O}G$ is a noetherian $\mathcal{O}$-algebra. The Green Theory in this area yields for each isomorphism type of finitely generated indecomposable (and hence for each isomorphism type of finitely generated simple $\mathcal{O}G$-module) a theory of vertices and sources invariants. The vertices are derived from the set of p-subgroups of G. As suggested by the above, in Basic Definition and Main Results for Rings Section, let $\mathrm{\Sigma }$ be a fixed subset of subrings of the ring R and we develop a theory of $\mathrm{\Sigma }$-vertices and sources for finitely generated R-modules. We conclude Basic Definition and Main Results for Rings Section with examples and show that our results are compatible with a ring isomorphic to R. For Idempotent Morita Equivalence and Virtual Vertex-Source Pairs of Modules of a Ring Section, let e be an idempotent of R such that $R=ReR$. Set $B=eRe$ so that B is a subring of R with identity e. Then, the functions $eR{\otimes }_R\ast :R-\mathrm{mod}\to B-\mathrm{mod}$ and $Re{\otimes }_B\ast :B-\mathrm{mod}\to R-\mathrm{mod}$ form a Morita Categorical Equivalence. We show, in this Section, that such a categorical equivalence is compatible with our vertex-source theory. In Two Applications with Idemptent Morita Equivalence Section, we show such compatibility for source algebras in Finite Group Block Theory and for naturally Morita Equivalent Algebras. Full article

Let $\mathcal{A}$ be an algebra of subsets of a set $\mathsf{\Omega }$ and $ba\left(\mathcal{A}\right)$ the Banach space of bounded finitely additive scalar-valued measures on $\mathcal{A}$ endowed with the variation norm. A subset $\mathcal{B}$ of $\mathcal{A}$ is a Nikodým set for $ba\left(\mathcal{A}\right)$ if each countable $\mathcal{B}$-pointwise bounded subset M of $ba\left(\mathcal{A}\right)$ is norm bounded. A subset $\mathcal{B}$ of $\mathcal{A}$ is a Grothendieck set for $ba\left(\mathcal{A}\right)$ if for each bounded sequence ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ in $ba\left(\mathcal{A}\right)$ the $\mathcal{B}$-pointwise convergence on $ba\left(\mathcal{A}\right)$ implies its $ba{\left(\mathcal{A}\right)}^{*}$-pointwise convergence on $ba\left(\mathcal{A}\right)$. A subset $\mathcal{B}$ of an algebra $\mathcal{A}$ is a strong-Nikodým (Grothendieck) set for $ba\left(\mathcal{A}\right)$ if in each increasing covering $\{{\mathcal{B}}_n:n\in \mathbb{N}\}$ of $\mathcal{B}$ there exists ${\mathcal{B}}_m$ which is a Nikodým (Grothendieck) set for $ba\left(\mathcal{A}\right)$. The answer of the following open question for an algebra $\mathcal{A}$ of subsets of a set $\mathsf{\Omega }$, proposed by Valdivia in 2013, has not yet been found: Is it true that if $\mathcal{A}$ is a Nikodým set for $ba\left(\mathcal{A}\right)$ then $\mathcal{A}$ is a strong Nikodým set for $ba\left(\mathcal{A}\right)$? In this paper we surveyed some results related to this Valdivia’s open question, as well as the corresponding problem for strong Grothendieck sets. The new Propositions 1 and 3 provide more simplified proofs, particularly in their application to Theorems 1 and 2, which were the main results surveyed. Moreover, the proofs of almost all other propositions are wholly or partially original. Full article

A subset $\mathcal{B}$ of an algebra $\mathcal{A}$ of subsets of a set $\Omega$ has property $\left(N\right)$ if each $\mathcal{B}$-pointwise bounded sequence of the Banach space $ba\left(\mathcal{A}\right)$ is bounded in $ba\left(\mathcal{A}\right)$, where $ba\left(\mathcal{A}\right)$ is the Banach space of real or complex bounded finitely additive measures defined on $\mathcal{A}$ endowed with the variation norm. $\mathcal{B}$ has property $\left(G\right)$ [$\left(VHS\right)$] if for each bounded sequence [if for each sequence] in $ba\left(\mathcal{A}\right)$ the $\mathcal{B}$-pointwise convergence implies its weak convergence. $\mathcal{B}$ has property $\left(sN\right)$ [$\left(sG\right)$ or $\left(sVHS\right)$] if every increasing covering $\{{\mathcal{B}}_n:n\in \mathbb{N}\}$ of $\mathcal{B}$ contains a set ${\mathcal{B}}_p$ with property $\left(N\right)$ [$\left(G\right)$ or $\left(VHS\right)$], and $\mathcal{B}$ has property $\left(wN\right)$ [$\left(wG\right)$ or $\left(wVHS\right)$] if every increasing web $\{{\mathcal{B}}_{n_1{n}_2\cdots n_m}:n_i\in \mathbb{N},1\le i\le m,m\in \mathbb{N}\}$ of $\mathcal{B}$ contains a strand $\{{\mathcal{B}}_{p_1{p}_2\cdots p_m}:m\in \mathbb{N}\}$ formed by elements ${\mathcal{B}}_{p_1{p}_2\cdots p_m}$ with property $\left(N\right)$ [$\left(G\right)$ or $\left(VHS\right)$] for every $m\in \mathbb{N}$. The classical theorems of Nikodým–Grothendieck, Valdivia, Grothendieck and Vitali–Hahn–Saks say, respectively, that every $\sigma$-algebra has properties $\left(N\right)$, $\left(sN\right)$, $\left(G\right)$ and $\left(VHS\right)$. Valdivia’s theorem was obtained through theorems of barrelled spaces. Recently, it has been proved that every $\sigma$-algebra has property $\left(wN\right)$ and several applications of this strong Nikodým type property have been provided. In this survey paper we obtain a proof of the property $\left(wN\right)$ of a $\sigma$-algebra independent of the theory of locally convex barrelled spaces which depends on elementary basic results of Measure theory and Banach space theory. Moreover we prove that a subset $\mathcal{B}$ of an algebra $\mathcal{A}$ has property $\left(wWHS\right)$ if and only if $\mathcal{B}$ has property $\left(wN\right)$ and $\mathcal{A}$ has property $\left(G\right)$. Full article

We call a subset $\mathcal{M}$ of an algebra of sets $\mathcal{A}$ a Grothendieck set for the Banach space $ba\left(\mathcal{A}\right)$ of bounded finitely additive scalar-valued measures on $\mathcal{A}$ equipped with the variation norm if each sequence ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ in $ba\left(\mathcal{A}\right)$ which is pointwise convergent on $\mathcal{M}$ is weakly convergent in $ba\left(\mathcal{A}\right)$ , i.e., if there is $\mu \in ba\left(\mathcal{A}\right)$ such that ${\mu }_n\left(A\right)\to \mu \left(A\right)$ for every $A\in \mathcal{M}$ then ${\mu }_n\to \mu$ weakly in $ba\left(\mathcal{A}\right)$ . A subset $\mathcal{M}$ of an algebra of sets $\mathcal{A}$ is called a Nikodým set for $ba\left(\mathcal{A}\right)$ if each sequence ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ in $ba\left(\mathcal{A}\right)$ which is pointwise bounded on $\mathcal{M}$ is bounded in $ba\left(\mathcal{A}\right)$ . We prove that if $\mathsf{\Sigma }$ is a $\sigma$ -algebra of subsets of a set $\mathsf{\Omega }$ which is covered by an increasing sequence $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ of subsets of $\mathsf{\Sigma }$ there exists $p\in \mathbb{N}$ such that ${\mathsf{\Sigma }}_p$ is a Grothendieck set for $ba\left(\mathcal{A}\right)$ . This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a $\sigma$ -algebra $\mathsf{\Sigma }$ is covered by an increasing sequence $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ of subsets, there is $p\in \mathbb{N}$ such that ${\mathsf{\Sigma }}_p$ is a Nikodým set for $ba\left(\mathsf{\Sigma }\right)$ . This also refines the Grothendieck result stating that for each $\sigma$ -algebra $\mathsf{\Sigma }$ the Banach space ${\ell }_{\infty }\left(\mathsf{\Sigma }\right)$ is a Grothendieck space. Some applications to classic Banach space theory are given. Full article