Recent zbMATH articles in MSC 28A20https://www.zbmath.org/atom/cc/28A202021-05-28T16:06:00+00:00WerkzeugEquivalence between GLT sequences and measurable functions.https://www.zbmath.org/1459.470092021-05-28T16:06:00+00:00"Barbarino, Giovanni"https://www.zbmath.org/authors/?q=ai:barbarino.giovanniAuthor's abstract: The theory of generalized locally Toeplitz (GLT) sequences of matrices has been developed in order to study the asymptotic behaviour of particular spectral distributions when the dimension of the matrices tends to infinity. Key concepts in this theory are the notions of approximating classes of sequences (a.c.s.)\ and spectral symbols that lead to defining a metric structure on the space of matrix sequences and provide a link with the measurable functions. In this paper, we prove additional results regarding theoretical aspects, such as the completeness of the matrix sequences space with respect to the metric a.c.s.\ and the identification of the space of GLT sequences with the space of measurable functions.
Reviewer: Tomasz Natkaniec (Gdańsk)The work of Lewis Bowen on the entropy theory of non-amenable group actions.https://www.zbmath.org/1459.370042021-05-28T16:06:00+00:00"Thouvenot, Jean-Paul"https://www.zbmath.org/authors/?q=ai:thouvenot.jean-paulThe aim of this paper is to present the achievements of Lewis Bowen. The main focus is on the problem of isomorphism of Bernoulli actions of countable non-amenable groups which he solved brilliantly in the remarkable papers [\textit{L. P. Bowen}, Ann. Math. (2) 171, No. 2, 1387--1400 (2010; Zbl 1201.37007); \textit{L. Bowen}, J. Am. Math. Soc. 23, No. 1, 217--245 (2010; Zbl 1201.37005)]. Here two invariants were introduced, which led to many developments. One of them is the extension of the definition of sofic measure theoretic entropy. There are also nice implications related to spectral theory.
Reviewer: Michael L. Blank (Moskva)Ideal convergent subsequences and rearrangements for divergent sequences of functions.https://www.zbmath.org/1459.400022021-05-28T16:06:00+00:00"Balcerzak, Marek"https://www.zbmath.org/authors/?q=ai:balcerzak.marek"Popławski, Michał"https://www.zbmath.org/authors/?q=ai:poplawski.michal"Wachowicz, Artur"https://www.zbmath.org/authors/?q=ai:wachowicz.arturSummary: Let $\mathfrak I$ be an ideal on $\mathbb N$ which is analytic or coanalytic. Assume that $(f_n)$ is a sequence of functions with the Baire property from a Polish space $X$ into a Polish space $Z$, which is divergent on a comeager set. We investigate the Baire category of $\mathfrak I$-convergent subsequences and rearrangements of $(f_n)$. Our result generalizes a theorem of \textit{R. R. Kallman} [Int. J. Math. Math. Sci. 22, No. 4, 709--712 (1999; Zbl 0952.28003)]. A similar theorem for subsequences is obtained if $(X,\mu)$ is a $\sigma$-finite complete measure space and a sequence $(f_n)$ of measurable functions from $X$ to $Z$ is $\mathfrak I$-divergent $\mu$-almost everywhere. Then the set of subsequences of $(f_n)$, $\mathfrak I$-divergent $\mu$-almost everywhere, is of full product measure on $\{0,1\}^{\mathbb N}$. Here we assume additionally that $\mathfrak I$ has property~$(G)$.