Banach August 2017

banach@mathdept.okstate.edu

1 participants
13 discussions

Joe Diestel
by Bentuo Zheng (bzheng) 20 Aug '17

20 Aug '17

Dear Colleagues, With great regret, we share the news of the passing of Professor Joe Diestel (Kent State University) on August 17, 2017. Joe's influence on the academic, and non-academic, communities in Kent was profound. He will be very much missed. Sincerely, Bentuo Zheng Associate Professor, Department of Mathematical Sciences, College of Arts and Sciences [UofM logo] The University of Memphis 359 Dunn Hall Memphis, TN 38152<http://www.memphis.edu/emailsignatures/emailsignaturemac.php#> 901.678<http://www.memphis.edu/emailsignatures/emailsignaturemac.php#>.3534 | memphis.edu<http://www.memphis.edu/> [UofM tw]<https://www.facebook.com/uofmemphis> [UofM tw] <https://twitter.com/uofmemphis> [UofM tw] <https://instagram.com/uofmemphis>

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Abstract of a paper by March Boedihardjo
by Bentuo Zheng (bzheng) 01 Aug '17

01 Aug '17

This is an announcement for the paper “Weak closure of ultrapowers of operators on $L_p$” by March Boedihardjo<https://arxiv.org/find/math/1/au:+Boedihardjo_M/0/1/0/all/0/1>. Abstract: Let $1<p<\infty$. We find the closure of ultrapowers of operators on $L_p$ in the weak operator topology when the ultrafilter is selective. As a consequence, we show that the commutant of $B(L_p)$ in its ultrapower may or may not be trivial depending on the ultrafilter assuming the existence of a selective nonprincipal ultrafilter. This extends a result of Farah, Phillips and Stepr\=ans. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.09658

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Abstract of a paper by Luis García-Lirola, Colin Petitjean, Antonin Procházka, Abraham Rueda Zoca
by Bentuo Zheng (bzheng) 01 Aug '17

01 Aug '17

This is an announcement for the paper “Extremal structure and Duality of Lipschitz free spaces” by Luis García-Lirola<https://arxiv.org/find/math/1/au:+Garcia_Lirola_L/0/1/0/all/0/1>, Colin Petitjean<https://arxiv.org/find/math/1/au:+Petitjean_C/0/1/0/all/0/1>, Antonin Procházka<https://arxiv.org/find/math/1/au:+Prochazka_A/0/1/0/all/0/1>, Abraham Rueda Zoca<https://arxiv.org/find/math/1/au:+Zoca_A/0/1/0/all/0/1>. Abstract: We analyse the relationship between different extremal notions in Lipschitz free spaces (strongly exposed, exposed, preserved extreme and extreme points). We prove in particular that every preserved extreme point of the unit ball is also a denting point. We also show in some particular cases that every extreme point is a molecule, and that a molecule is extreme whenever the two points, say $x$ and $y$, which define it satisfy that the metric segment $[x,y]$ only contains $x$ and $y$. The most notable among them is the case when the free space admits an isometric predual with some additional properties. As an application, we get some new consequences about norm-attainment in spaces of vector valued Lipschitz functions. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.09307

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Abstract of a paper by Kevin Beanland, Tomasz Kania, Niels Jakob Laustsen
by Bentuo Zheng (bzheng) 01 Aug '17

01 Aug '17

This is an announcement for the paper “The algebras of bounded operators on the Tsirelson and Baernstein spaces are not Grothendieck spaces” by Kevin Beanland<https://arxiv.org/find/math/1/au:+Beanland_K/0/1/0/all/0/1>, Tomasz Kania<https://arxiv.org/find/math/1/au:+Kania_T/0/1/0/all/0/1>, Niels Jakob Laustsen<https://arxiv.org/find/math/1/au:+Laustsen_N/0/1/0/all/0/1>. Abstract: We show that if the Banach algebra $\mathcal{B}(X)$ of bounded operators on a Banach space $X$ is a Grothendieck space, then $X$ is reflexive, and we give two new examples of reflexive Banach spaces $X$ for which $\mathcal{B}(X)$ is not a Grothendieck space, namely $X=T$ (the Tsirelson space) and $X=B_p$(the $p$th Baernstein space) for $p\in (1, \infty)$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.08399

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Abstract of a paper by Adi Tcaciuc
by Bentuo Zheng (bzheng) 01 Aug '17

01 Aug '17

This is an announcement for the paper “The almost-invariant subspace problem for Banach spaces” by Adi Tcaciuc<https://arxiv.org/find/math/1/au:+Tcaciuc_A/0/1/0/all/0/1>. Abstract: We show that for any bounded operator $T$ acting on an infinite dimensional Banach space there exists a rank one operator $F$ such that $T+F$ has invariant subspace of infinite dimension and codimension. This extends to arbitrary Banach spaces a previous result that was proved only in the reflexive case. We also show that, for any fixed $\epsilon>0$, there exists $F$ as above such that $\|F\|<\epsilon$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.07836

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Abstract of a paper by Timur Oikhberg
by Bentuo Zheng (bzheng) 01 Aug '17

01 Aug '17

This is an announcement for the paper “Injectivity and projectivity in $p$-multinormed spaces” by Timur Oikhberg<https://arxiv.org/find/math/1/au:+Oikhberg_T/0/1/0/all/0/1>. Abstract: We find large classes of injective and projective $p$-multinormed spaces. In fact, these classes are universal, in the sense that every $p$-multinormed space embeds into (is a quotient of) an injective (resp. projective) $p$-multinormed space. As a consequence, we show that any $p$-multinormed space has a canonical representation as a subspace of a quotient of a Banach lattice. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.07640

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Abstract of a paper by P.M. Berná, O. Blasco, G. Garrigós, E. Hernández. T. Oikhberg
by Bentuo Zheng (bzheng) 01 Aug '17

01 Aug '17

This is an announcement for the paper “Embeddings and Lebesgue-type inequalities for the greedy algorithm in Banach spaces” by P.M. Berná<https://arxiv.org/find/math/1/au:+Berna_P/0/1/0/all/0/1>, O. Blasco<https://arxiv.org/find/math/1/au:+Blasco_O/0/1/0/all/0/1>, G. Garrigós<https://arxiv.org/find/math/1/au:+Garrigos_G/0/1/0/all/0/1>, E. Hernández. T. Oikhberg<https://arxiv.org/find/math/1/au:+Oikhberg_E/0/1/0/all/0/1>. Abstract: We obtain Lebesgue-type inequalities for the greedy algorithm for arbitrary complete seminormalized biorthogonal systems in Banach spaces. The bounds are given only in terms of the upper democracy functions of the basis and its dual. We also show that these estimates are equivalent to embeddings between the given Banach space and certain discrete weighted Lorentz spaces. Finally, the asymptotic optimality of these inequalities is illustrated in various examples of non necessarily quasi-greedy bases. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.07513

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Abstract of a paper by March Boedihardjo
by Bentuo Zheng (bzheng) 01 Aug '17

01 Aug '17

This is an announcement for the paper “Multiplication operators on $L^p$” by March Boedihardjo<https://arxiv.org/find/math/1/au:+Boedihardjo_M/0/1/0/all/0/1>. Abstract: We show that every operator on $L_p, 1<p<\infty$ defined by multiplication by the identity function on $\mathbb{C}$ is a compact perturbation of an operator that is diagonal with respect to an unconditional basis. We also classify these operators up to similarity modulo compact operators and up to approximate similarity. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.04798

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Abstract of a paper by Arkady Kitover, Mehmet Orhon
by Bentuo Zheng (bzheng) 01 Aug '17

01 Aug '17

This is an announcement for the paper “The dual Radon - Nikodym property for finitely generated Banach $C(K)$-Modules” by Arkady Kitover<https://arxiv.org/find/math/1/au:+Kitover_A/0/1/0/all/0/1>, Mehmet Orhon<https://arxiv.org/find/math/1/au:+Orhon_M/0/1/0/all/0/1>. Abstract: We extend the well-known criterion of Lotz for the dual Radon-Nikodym property (RNP) of Banach lattices to finitely generated Banach $C(K)$-modules and Banach $C(K)$-modules of finite multiplicity. Namely, we prove that if $X$ is a Banach space from one of these classes then its Banach dual $X^*$ has the RNP iff $X$ does not contain a closed subspace isomorphic to $\ell_1$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.04655

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Abstract of a paper by Maciej Ciesielski
by Bentuo Zheng (bzheng) 01 Aug '17

01 Aug '17

This is an announcement for the paper “Lower and upper local uniform $K$-monotonicity in symmetric spaces” by Maciej Ciesielski<https://arxiv.org/find/math/1/au:+Ciesielski_M/0/1/0/all/0/1>. Abstract: Using the local approach to the global structure of a symmetric space $E$ we establish a relationship between strict $K$- monotonicity, lower (resp. upper) local uniform $K$-monotonicity, order continuity and the Kadec-Klee property for global convergence in measure. We also answer the question under which condition upper local uniform $K$-monotonicity concludes upper local uniform monotonicity. Finally, we present a correlation between $K$-order continuity and lower local uniform $K$-monotonicity in a symmetric space $E$ under some additional assumptions on $E$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.02632

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Banach August 2017