Banach May 2018

banach@mathdept.okstate.edu

1 participants
12 discussions

Informal Analysis Seminar February 24-25, 2018
by Artem Zvavitch 05 Sep '19

05 Sep '19

Dear Colleagues, The Analysis group at Kent State University is happy to announce a meeting of the Informal Analysis Seminar, which will be held at the Department of Mathematical Sciences at Kent State University, February 24-25. The seminar will feature plenary speakers Robert Connelly (Cornell University), and Peter Sternberg (Indiana University) Each speaker will deliver a four hour lecture series designed to be accessible for graduate students. Funding is available to cover the local and travel expenses of a limited number of participants. Graduate students, postdoctoral researchers, and members of underrepresented groups are particularly encouraged to apply for support. A poster session will be held for researchers to display their work. Graduate students are particularly encouraged to submit a poster. Posters can be submitted electronically in PDF format. Further information, and an online registration form, can be found online http://www.math.kent.edu/informal We encourage you to register as soon as possible, but to receive support and/or help with hotel reservation, please register before January 29, 2018. Finally, please feel free to forward this email to any colleagues or students who you think may be interested in attending. Best regards, The Kent State Analysis Group

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Abstract of a paper by Saak S. Gabriyelyan, Sidney A. Morris
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “A topological group observation on the Banach--Mazur separable quotient problem” by Saak S. Gabriyelyan<https://arxiv.org/search?searchtype=author&query=Gabriyelyan%2C+S+S>, Sidney A. Morris<https://arxiv.org/search?searchtype=author&query=Morris%2C+S+A>. Abstract: The Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient space, has remained unsolved for 85 years, but has been answered in the affirmative for special cases such as reflexive Banach spaces. It is also known that every infinite-dimensional non-normable Fr\'{e}chet space has an infinite-dimensional separable quotient space, namely $R^{\omega}$. It is proved in this paper that every infinite-dimensional Fr\'{e}chet space (including every infinite-dimensional Banach space), indeed every locally convex space which has a subspace which is an infinite-dimensional Fr\'{e}chet space, has an infinite-dimensional (in the topological sense) separable metrizable quotient group, namely $T^{\omega}$, where $T$ denotes the compact unit circle group. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.02652

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Abstract of a paper by Michiya Mori, Narutaka Ozawa
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Mankiewicz's theorem and the Mazur--Ulam property for $C^*$-algebras” by Michiya Mori<https://arxiv.org/search?searchtype=author&query=Mori%2C+M>, Narutaka Ozawa<https://arxiv.org/search?searchtype=author&query=Ozawa%2C+N>. Abstract: We prove that every unital $C^*$-algebra $A$, possibly except for the $2$ by $2$ matrix algebra, has the Mazur--Ulam property. Namely, every surjective isometry from the unit sphere $S_A$ of $A$ onto the unit sphere $S_Y$ of another normed space $Y$ extends to a real linear map. This extends the result of A. M. Peralta and F. J. Fernandez-Polo who have proved the same under the additional assumption that both $A$ and $Y$ are von Neumann algebras. In the course of the proof, we strengthen Mankiewicz's theorem and prove that every surjective isometry from a closed unit ball with enough extreme points onto an arbitrary convex subset of a normed space is necessarily affine. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.10674

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Abstract of a paper by Antonio Avilés, Gonzalo Martínez-Cervantes, José Rodríguez
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Weak$^*$-sequential properties of Johnson-Lindenstrauss spaces” by Antonio Avilés<https://arxiv.org/search?searchtype=author&query=Avil%C3%A9s%2C+A>, Gonzalo Martínez-Cervantes<https://arxiv.org/search?searchtype=author&query=Mart%C3%ADnez-Cervantes%2C…>, José Rodríguez<https://arxiv.org/search?searchtype=author&query=Rodr%C3%ADguez%2C+J>. Abstract: A Banach space $X$ is said to have Efremov's property $(\epsilon)$ if every element of the weak$^*$-closure of a convex bounded set $C\subset X^*$ is the weak$^*$-limit of a sequence in $C$. By assuming the Continuum Hypothesis, we prove that there exist maximal almost disjoint families of infinite subsets of $\mathbb{N}$ for which the corresponding Johnson-Lindenstrauss spaces enjoy (resp. fail) property $(\epsilon)$. This is related to a gap in [A. Plichko, Three sequential properties of dual Banach spaces in the weak$^*$ topology, Topology Appl. 190 (2015), 93--98] and allows to answer (consistently) questions of Plichko and Yost. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.10350

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Abstract of a paper by Vladimir P. Fonf, Sebastian Lajara, Stanimir Troyanski, Clemente Zanco
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper Norming subspaces of Banach spaces” by Vladimir P. Fonf<https://arxiv.org/search?searchtype=author&query=Fonf%2C+V+P>, Sebastian Lajara<https://arxiv.org/search?searchtype=author&query=Lajara%2C+S>, Stanimir Troyanski<https://arxiv.org/search?searchtype=author&query=Troyanski%2C+S>, Clemente Zanco<https://arxiv.org/search?searchtype=author&query=Zanco%2C+C>. Abstract: We show that, if $X$ is a closed subspace of a Banach space $E$ and $Z$ is a closed subspace of $E^*$ such that $Z$ is norming for $X$ and $X$ is total over $Z$ (as well as $X$ is norming for $Z$ and $Z$ is total over $X$), then $X$ and the pre-annihilator of $Z$ are complemented in $E$ whenever $Z$ is $w^*$-closed or $X$ is reflexive. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.09968

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Abstract of a paper by Karim Khanaki
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Remarks on Banach spaces determined by their finite dimensional subspaces” by Karim Khanaki<https://arxiv.org/search?searchtype=author&query=Khanaki%2C+K>. Abstract: A separable Banach space $X$ is said to be finitely determined if for each separable space $Y$ such that $X$ is finitely representable (f.r.) in $Y$ and $Y$ is f.r. in $X$ then $Y$ is isometric to $X$. We provide a direct proof (without model theory) of the fact that every finitely determined space $X$ (isometrically) contains every (separable) space $Y$ which is finitely representable in $X$. We also point out how a similar argument proves the Krivine-Maurey theorem on stable Banach spaces, and give the model theoretic interpretations of some results. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.08446

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Abstract of a paper by Pablo M. Berná
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Equivalence between almost-greedy bases and semi-greedy bases” by Pablo M. Berná<https://arxiv.org/search?searchtype=author&query=Bern%C3%A1%2C+P+M>. Abstract: In this paper we show that almost-greedy bases are equivalent to semi-greedy bases in the context of Schauder bases in Banach spaces. Moreover, using this result, we answer a question asked in [3]. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.05730

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Abstract of a paper by Richard J. Smith, Stanimir Troyanski
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Approximation of norms on Banach spaces” by Richard J. Smith<https://arxiv.org/search?searchtype=author&query=Smith%2C+R+J>, Stanimir Troyanski<https://arxiv.org/search?searchtype=author&query=Troyanski%2C+S>. Abstract: Relatively recently it was proved that if $\Gamma$ is an arbitrary set, then any equivalent norm on $c_0(\Gamma)$ can be approximated uniformly on bounded sets by polyhedral norms and $C^{\infty}$ smooth norms, with arbitrary precision. We extend this result to more classes of spaces having uncountable symmetric bases, such as preduals of the `discrete' Lorentz spaces $d(w, 1,\gamma)$, and certain symmetric Nakano spaces and Orlicz spaces. We also show that, given an arbitrary ordinal number $\alpha$, there exists a scattered compact space $K$ having Cantor-Bendixson height at least $\alpha$, such that every equivalent norm on $C(K)$ can be approximated as above. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.05660

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Abstract of a paper by Stephen J. Dilworth, Van Kien Nguyen
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Greedy Algorithms and Kolmogorov Widths in Banach Spaces” by Stephen J. Dilworth<https://arxiv.org/find/math/1/au:+Dilworth_S/0/1/0/all/0/1>, Van Kien Nguyen<https://arxiv.org/search?searchtype=author&query=Nguyen%2C+V+K>. Abstract: Let $X$ be a Banach space and $K$ be a compact subset in $X$. We consider a greedy algorithm for finding $n$-dimensional subspace $V_n\subset X$ which can be used to approximate the elements of $K$. We are interested in how well the space $V_n$ approximates the elements of $K$. For this purpose we compare the greedy algorithm with the Kolmogorov, width which is the best possible error one can approximate $K$ by $n$−dimensional subspaces. Various results in this direction have been given, e.g., in Binev et al. (SIAM J. Math. Anal. (2011)), DeVore et al. (Constr. Approx. (2013)) and Wojtaszczyk (J. Math. Anal. Appl. (2015)). The purpose of the present paper is to continue this line. We shall show that under some additional assumptions the results in the above-mentioned papers can be improved. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.03935

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Abstract of a paper by Richard J. Smith
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Topology, isomorphic smoothness and polyhedrality in Banach spaces” by Richard J. Smith<https://arxiv.org/search?searchtype=author&query=Smith%2C+R+J>. Abstract: In recent decades, topology has come to play an increasing role in some geometric aspects of Banach space theory. The class of so-called $w*$-locally relatively compact sets was introduced recently by Fonf, Pallares, Troyanski and the author, and were found to be a useful topological tool in the theory of isomorphic smoothness and polyhedrality in Banach spaces. We develop the topological theory of these sets and present some Banach space applications. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.02899

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Banach May 2018