Banach April 2017

banach@mathdept.okstate.edu

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Abstract of a paper by Eusebio Gardella, Hannes Thiel
by Bentuo Zheng (bzheng) 01 Apr '17

01 Apr '17

This is an announcement for the paper “Extending representations of Banach algebras to their biduals” by Eusebio Gardella<https://arxiv.org/find/math/1/au:+Gardella_E/0/1/0/all/0/1>, Hannes Thiel<https://arxiv.org/find/math/1/au:+Thiel_H/0/1/0/all/0/1>. Abstract: We show that a representation of a Banach algebra $A$ on a Banach space $X$ can be extended to a canonical representation of $A^{**}$ on $X$ if and only if certain orbit maps $A\rightarrow X$ are weakly compact. We apply this to study when the essential space of a representation is complemented. This provides a tool to disregard the difference between degenerate and nondegenerate representations on Banach spaces. As an application we show that a $C^*$-algebra $A$ has an isometric representation on an $L^p$ -space, for $p\in [1, \infty)\{2}$, if and only if $A$ is commutative The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.00882

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Abstract of a paper by Samya Kumar Ray
by Bentuo Zheng (bzheng) 01 Apr '17

01 Apr '17

This is an announcement for the paper “On Multivariate Matsaev's Conjecture” by Samya Kumar Ray<https://arxiv.org/find/math/1/au:+Ray_S/0/1/0/all/0/1>. Abstract: We present various multivariate generalizations of the Matsaev's conjecture in different settings, namely on $L^p$-spaces, non-commutative $L^p$-spaces and semigroups. We show that the multivariate Matsaev's conjecture holds true for any commuting tuple of isometries on $L^p$-spaces. We prove a similar result for Schatten-$p$ classes. We also show that any two parameter strongly continuous semigroup of contractions on a Hilbert space satisfies the multivariate Matsaev's conjecture for semigroups. At the end, we discuss some open questions. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.00733

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Abstract of a paper by Sander C. Hille, Tomasz Szarek, Daniel T.H. Worm, Maria Ziemlanska
by Bentuo Zheng (bzheng) 01 Apr '17

01 Apr '17

This is an announcement for the paper “On a Schur-like property for spaces of measures” by Sander C. Hille<https://arxiv.org/find/math/1/au:+Hille_S/0/1/0/all/0/1>, Tomasz Szarek<https://arxiv.org/find/math/1/au:+Szarek_T/0/1/0/all/0/1>, Daniel T.H. Worm<https://arxiv.org/find/math/1/au:+Worm_D/0/1/0/all/0/1>, Maria Ziemlanska<https://arxiv.org/find/math/1/au:+Ziemlanska_M/0/1/0/all/0/1>. Abstract: A Banach space has the Schur property when every weakly convergent sequence converges in norm. We prove a Schur-like property for measures: if a sequence of finite signed Borel measures on a Polish space is such that it is bounded in total variation norm and such that for each bounded Lipschitz function the sequence of integrals of this function with respect to these measures converges, then the sequence converges in dual bounded Lipschitz norm or Fortet-Mourier norm to a measure. Moreover, we prove three consequences of this result: the first is equivalence of concepts of equicontinuity in the theory of Markov operators, the second is the derivation of weak sequential completeness of the space of signed Borel measures on Polish spaces from our main result and the third concerns conditions for the coincidence of weak and norm topologies on sets of measures that are bounded in total variation norm with additional properties. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.00677

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Banach April 2017