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The present article describes the precise structure of the L p -spaces of projective limit measures by introducing a category-theoretical perspective. This analysis is applied to measures on vector spaces and in particular to Gaussian measures on nuclear topological vector spaces. A simple application to constructive quantum field theory (QFT) is given through the Osterwalder–Schrader axioms. [ABSTRACT FROM AUTHOR]

We present some more foundations for a theory of real structure in operator spaces and algebras, in particular concerning the real case of the theory of injectivity, and the injective, ternary, and C ∗ -envelope. We consider the interaction between these topics and the complexification. We also generalize many of these results to the setting of operator spaces and systems acted upon by a group. [ABSTRACT FROM AUTHOR]

Let G be a generalized matrix Banach algebra associated with a Morita context (A , B , M , N , Φ , Ψ). In this paper, we investigate biprojectivity and biflatness of G, under certain conditions. Indeed, we obtain a biprojective (resp. biflat) generalized matrix Banach algebra such that M ≠ 0 and N ≠ 0. Moreover, we give a sufficient condition for biprojectivity and biflatness of G, in the case where A = N = 0. Using this fact, we obtain some results about homological properties of bi-amalgamated Banach algebras. As an application, we answer some recent open questions. [ABSTRACT FROM AUTHOR]

In this paper, the concept of left ϕ-approximate Connes amenability for a dual Banach algebra 풜 is studied, where ϕ is a weak * -continuous multiplicative linear functional on 풜 . Some characterizations for module extension dual Banach algebras and matrix algebras are given. Moreover, the notion of approximate left ϕ-Connes biprojectivity for dual Banach algebras is introduced and some concrete examples regarding this new notion are indicated. [ABSTRACT FROM AUTHOR]

We prove a change of base theorem for operator space modules over C*-algebras, analogous to the change of rings for algebraic modules. We demonstrate how this can be used to show that the category of (right) matrix normed modules and completely bounded module maps has enough injectives. [ABSTRACT FROM AUTHOR]

Combettes and Pesquet (SIAM J Optim 25:1221–1248, 2015) investigated the almost sure weak convergence of block-coordinate fixed point algorithms and discussed their applications to nonlinear analysis and optimization. This algorithmic framework features random sweeping rules to select arbitrarily the blocks of variables that are activated over the course of the iterations and it allows for stochastic errors in the evaluation of the operators. The present paper establishes results on the mean-square and linear convergence of the iterates. Applications to monotone operator splitting and proximal optimization algorithms are presented. [ABSTRACT FROM AUTHOR]

In this paper we continue the investigations of the relationship between T 0 -quasi-metric spaces and partially ordered metric spaces. Among other things, we establish an equivalence of categories between so-called maximal T 0 -quasi-metric spaces and partially ordered metric spaces produced by T 0 -quasi-metrics. In the linear context we give geometric interpretations of the obtained results. In particular, we also derive a representation theorem for injective asymmetrically normed spaces. [ABSTRACT FROM AUTHOR]

An explicit description of the algebraic and vector lattice operations on the Isbell-convex hull of an asymmetrically normed real vector space is provided. Connections between the concepts of Isbell-convexity, injectivity and Dedekind-completeness of asymmetrically normed real vector spaces are also considered. [ABSTRACT FROM AUTHOR]

We introduce bounded cohomology for (pairs of) groupoids and develop homological algebra to deal with it. We generalise results of Ivanov, Frigerio and Pagliantini to this setting and show that (under topological conditions) the bounded cohomology of a pair of topological spaces is isometrically isomorphic to the bounded cohomology of the pair of fundamental groupoids. Furthermore, we prove that bounded cohomology relative to an amenable groupoid is isometrically isomorphic to the bounded cohomology of the ambient groupoid. [ABSTRACT FROM AUTHOR]