Banach March 2019

banach@mathdept.okstate.edu

2 participants
16 discussions

Abstract of a paper by Rafael Chiclana, Miguel Martin
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “The Bishop--Phelps--Bollobás property for Lipschitz maps” by Rafael Chiclana<https://arxiv.org/search/math?searchtype=author&query=Chiclana%2C+R>, Miguel Martin<https://arxiv.org/search/math?searchtype=author&query=Martin%2C+M>. Abstract: In this paper, we introduce and study a Lipschitz version of the Bishop-Phelps-Bollobás property (Lip-BPB property). This property deals with the possibility to make a uniformly simultaneous approximation of a Lipschitz map $F$ and a pair of points at which $F$ almost attains its norm by a Lipschitz map $G$ and a pair of points such that $G$ strongly attains its norm at the new pair of points. We first show that if $M$ is a finite pointed metric space and $Y$ is a finite-dimensional Banach space, then the pair $(M,Y)$ has the Lip-BPB property, and that both finiteness are needed. Next, we show that if $M$ is a uniformly Gromov concave pointed metric space (i.e.\ the molecules of $M$ form a set of uniformly strongly exposed points), then $(M,Y)$ has the Lip-BPB property for every Banach space $Y$. We further prove that this is the case of finite concave metric spaces, ultrametric spaces, and Hölder metric spaces. The extension of the Lip-BPB property from $(M,\mathbb{R})$ to some Banach spaces $Y$, the relationship with absolute sums, and some results only valid for compact Lipschitz maps, are also discussed. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1901.02956

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Abstract of a paper by Ivan Yaroslavtsev
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “On strongly orthogonal martingales in UMD Banach spaces” by Ivan Yaroslavtsev<https://arxiv.org/search/math?searchtype=author&query=Yaroslavtsev%2C+I>. Abstract: In the present paper we introduce the notion of strongly orthogonal martingales. Moreover, we show that for any UMD Banach space $X$ and for any $X$-valued strongly orthogonal martingales $M$ and $N$ such that $N$ is weakly differentially subordinate to $M$ one has that for any $1<p<\infty$ \[ \mathbb E \|N_t\|^p \leq \chi_{p, X}^p \mathbb E \|M_t\|^p,\;\;\; t\geq 0, \] with the sharp constant $\chi_{p, X}$ being the norm of a decoupling-type martingale transform and being within the range \[ \max\Bigl\{\sqrt{\beta_{p, X}}, \sqrt{\hbar_{p,X}}\Bigr\} \leq \max\{\beta_{p, X}^{\gamma,+}, \beta_{p, X}^{\gamma, -}\} \leq \chi_{p, X} \leq \min\{\beta_{p, X}, \hbar_{p,X}\}, \] where $\beta_{p, X}$ is the UMD$_p$ constant of $X$, $\hbar_{p, X}$ is the norm of the Hilbert transform on $L^p(\mathbb R; X)$, and $\beta_{p, X}^{\gamma,+}$ and $ \beta_{p, X}^{\gamma, -}$ are the Gaussian decoupling constants. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1812.08049

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Abstract of a paper by March T. Boedihardjo
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Bounded representations on $l^p$” by March T. Boedihardjo<https://arxiv.org/search/math?searchtype=author&query=Boedihardjo%2C+M+T>. Abstract: We show that (i) every bounded unital representation of an amenable group $G$ on $l^{p}$, $1<p<\infty$, is a direct summand of a representation that is approximately similar to the left regular representation of $G$ on $l^{p}$ and that (ii) if $\rho$ is a unital representation of a unital $C^{*}$-algebra $\mathcal{A}$ on $l^{p}$, $1<p<\infty$, $p\neq 2$, then $\rho$ satisfies a compactness property and $\mathcal{A}/\text{ker }\rho$ is residually finite dimensional. As a consequence, a separable unital $C^{*}$-algebra $\mathcal{A}$ is isomorphic to a subalgebra of $B(l^{p})$, $1<p<\infty$, $p\neq 2$, if and only if $\mathcal{A}$ is residually finite dimensional. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1812.11165

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Abstract of a paper by Ioakeim Ampatzoglou
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “On the non-embedding of $\ell_1$ in the James Tree Space” by Ioakeim Ampatzoglou<https://arxiv.org/search/math?searchtype=author&query=Ampatzoglou%2C+I>. Abstract: James Tree Space ($\mathcal{JT}$), introduced by R. James, is the first Banach space constructed having non-separable conjugate and not containing $\ell^1$. James actually proved that every infinite dimensional subspace of $\mathcal{JT}$ contains a Hilbert space, which implies the $\ell^1$ non-embedding. In this expository article, we present a direct proof of the $\ell^1$ non-embedding, using Rosenthal's $\ell^1$- Theorem and some measure theoretic arguments, namely Riesz's Representation Theorem. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1812.07825

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Abstract of a paper by Karol Leśnik, Jakub Tomaszewski
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Pointwise multipliers of Musielak--Orlicz spaces and factorization” by Karol Leśnik<https://arxiv.org/search/math?searchtype=author&query=Le%C5%9Bnik%2C+K>, Jakub Tomaszewski<https://arxiv.org/search/math?searchtype=author&query=Tomaszewski%2C+J>. Abstract: We prove that the space of pointwise multipliers between two distinct Musielak--Orlicz spaces is another Musielak-Orlicz space and the function defining it is given by an appropriately generalized Legendre transform. In particular, we obtain characterization of pointwise multipliers between Nakano spaces. We also discuss factorization problem for Musielak-Orlicz spaces and exhibit some differences between Orlicz and Musielak-Orlicz cases. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1812.05887

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Abstract of a paper by Trond Arnold Abrahamsen, Rainis Haller, Vegard Lima, Katriin Pirk
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Delta- and Daugavet-points in Banach spaces” by Trond Arnold Abrahamsen<https://arxiv.org/search/math?searchtype=author&query=Abrahamsen%2C+T+A>, Rainis Haller<https://arxiv.org/search/math?searchtype=author&query=Haller%2C+R>, Vegard Lima<https://arxiv.org/search/math?searchtype=author&query=Lima%2C+V>, Katriin Pirk<https://arxiv.org/search/math?searchtype=author&query=Pirk%2C+K>. Abstract: A $\Delta$-point $x$ of a Banach space is a norm one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance $2$ from $x$. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, $x$ is a Daugavet-point. A Banach space $X$ has the Daugavet property if and only if every norm one element is a Daugavet-point. We show that $\Delta$- and Daugavet-points are the same in $L_1$-spaces, $L_1$-preduals, as well as in a big class of Müntz spaces. We also provide an example of a Banach space where all points on the unit sphere are $\Delta$-points, but where none of them are Daugavet-points. We also study the property that the unit ball is the closed convex hull of its $\Delta$-points. This gives rise to a new diameter two property that we call the convex diametral diameter two property. We show that all $C(K)$ spaces, $K$ infinite compact Hausdorff, as well as all Müntz spaces have this property. Moreover, we show that this property is stable under absolute sums. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1812.02450

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Banach March 2019