Workshop in Analysis and Probability
July 5th - August 5th 2011
Schedule of Talks
As talks are scheduled, the information will be posted here. Please check back often.
Unless otherwise noted, all talks will be presented in Milner Hall, Room 317.
Refreshments will be provided 15 minutes before each talk.
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July 8 4pm-5pm |
Detelin Dosev, Weizmann Institute of Science
Commutators on some classical Banach spaces In this talk we will discuss some recent development on the problem of classifying the commutators on a class of classical Banach spaces. We will make an overview of some known results and will discuss the problem of classifying the commutators on \(C(K)\). |
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July 11 4pm-5pm |
Ken Dykema, Texas A&M University
Some natural questions in \(II_1\)-factors Two questions that have been answered in complex matrix algebras and in \(B(H)\) are: 1. Which elements are single commutators? 2. Which self-adjoints are real parts of quasi-nilpotents? The analogous questions in \(II_1\)-factors are open. We'll discuss some constructions that lead to partial answers. (Joint work with Anna Skripka.) |
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July 12 4pm-5pm |
Michael Anshelevich, Texas A&M University
Generators of some non-commutative stochastic processes The following results are well known. A Levy process is a Markov process. Its transition operators form a semigroup of contractions. The generator of this semigroup can be written down explicitly using Fourier analysis. I will discuss the same questions in the framework of free Levy processes (process with freely independent, stationary increments). Some of the new difficulties in this case are (1) there is no free Fourier transform, and (2) the transition operators no longer form a semigroup. Nevertheless, most of the results from the first paragraph carry over, with the generators written down explicitly as singular integral operators. All the free probability results needed will be quoted but not proved, so only functional analysis is necessary as background. |
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July 14 4pm-5pm |
Tanmoy Paul, Indian Statistical Institute
Ball remotality in Banach spaces For a closed and bounded set \(C\) in a Banach space and \(x_0\in X\) define \(\phi_{C}(x_0)=\sup_{z\in C}\|z-x_0\|\). Due to a result by K.S. Lau we have that if \(C\) is weakly compact there is a 'fat' set(dense \(G_\delta\)) from where there exists farthest points in \(C\). The condition weakly compact can be weakened under various conditions on the Banach space. It is easy to observe that the converse of Lau's result is not true in general. Instead of any closed bounded set we will concentrate our discussion on closed ball of a subspace and observe there are considerable amount of Banach spaces which provide subspaces, whose unit ball is not weakly compact, having this property. A special attention will be paid for function algebras of \(C(K)\) where \(K\) is a compact Hausdorff space. Various type of stability results can be derived in this regard. If time permitting I will discuss some very natural open problems on farthest points. |
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July 15 4pm-5pm |
Ciprian Foias, Texas A&M University
A sample of open problems concerning invariant probability measures of the Navier-Stokes equations The invariant probability measures of the Navier-Stokes equations play a central role in a rigorous approach connecting those equations to "the empirical theory of fully developed turbulence (initiated by Kolmogorov about 70 years ago)". Recent results show that the validity of empirical theory depends on the existence of some very special invariant measures and solutions of the Navier-Stokes equations. Our presentation of the facts above will be given in a "Functional Analysis" frame, so that the understanding of some of the standing open problems (and their motivation) does not require familiarity with the large body of knowledge which is not explicit in the functional framework. |
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July 25 11am-12pm |
Bentuo Zheng, University of Memphis
Commutators on \(Z_p=( \sum\ell_2 )_{\ell_p}\) , \( 1 < p < \infty \) Let \(T\) be a bounded linear operator on \(Z_p\). Then \(T\) is a commutator if and only if for all non zero \(\lambda\in \mathbb{C}\), the operator \(T-\lambda I\) is not \(Z_p\)-strictly singular. Recall that and operator \(T\) on \(Z_p\) is called \(Z_p\) strictly singular if the restriction of \(T\) on any subspace of \(Z_p\) that is isomorphic to \(Z_p\) is not an isomorphism. |
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July 25 4pm-5pm |
Franciszek Szafraniec,
Dilations as the reproducing kernel property I intend to present my personal look at dilation and extension theorems which are known under the acronym KSGNS. The invincible conclusion is that appearing in the title. As a reference source I recommend my paper Murphy's "Positive definite kernels and Hilbert \(C^*\)-modules" reorganized, Banach Center Publications, 89(2010), 275-295. |
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July 26 11am-12pm |
David Larson, Texas A&M University
Operator-Valued Measures, Dilations, and the Theory of Frames We investigate some natural associations between the theory of frames, the theory of operator-valued measures on sigma-algebras of sets, and the theory of normal linear maps on von Neumann algebras. In this connection frame theory itself is identified with the special case in which the domain algebra for the maps is abelian. Some key proofs have extensions to the non-abelian setting. |
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July 26 4pm-5pm |
Lixin Cheng, Xiamen University
Stability of nonsurjective \(\varepsilon\)-isometries of Banach spaces PDF file with abstract. |
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July 27 11am-12pm |
Narutaka Ozawa, Kyoto University
Survey on Weak Amenability Weakly amenability (aka Cowling-Haagerup property) is a generalization of amenability and is a convenient tool in the study of group algebras. Besides amenable groups, the class of weakly amenable groups contains many interesting examples such as free groups. I will give a survey on weak amenability. |
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July 27 4pm-5pm |
Timur Oikhberg, University of California at Irvine
Automatic continuity of orthogonality or disjointness preserving maps We investigate automatic continuity of linear maps \(T\) from \(E\) to \(X\), in the following setting: \(E\) and \(X\) are Banach spaces, and \(E\) is equipped with some kind of order (it may be either a Banach lattice, or a non-commutative \(L_p\) space). The images of disjoint (relative to the natural order of \(E\) ) elements of \(E\) satisfy certain inequality, resembling disjointness. Here is a sample of our results. (i) If \(E\) is an order continuous Banach lattice, \(T\) is bijective, and \( \|Tx + Ty\| \geq \max\{\|Tx\|, \|Ty\|\} \) whenever \(x\) and \(y\) are disjoint, then \(T\) is continuous. (ii) If \(E = L_p(\tau) \) where \( \tau \) is a faithful normal semi-finite trace on a \(II_1\) factor, and \(T : X \to Y\) is such that \(\|Tx + Ty\|^p = \|Tx||^p + \|Ty\|^p\) whenever \(x\) and \(y\) are orthogonal (that is, they have mutually orthogonal left and right support projections), then \(T\) is continuous. Part of this work was carried out in collaboration with A.M.Peralta and D.Puglisi. |
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July 28 11am-12pm |
Dan Freeman, University of Texas at Austin
Shrinking and boundedly complete Schauder frames for Banach spaces A Schauder frame for a Banach space X is a sequence \((x_i,f_i)\subseteq X\times X^*\) such that \(\sum f_i(x)x_i=x\) for all \(x\in X\). Frames can be thought of in some respect as redundant bases, and thus it is natural to consider what theorems and properties for bases can be generalized to frames. We will discuss in particular how the properties of shrinking and boundedly complete work for frames. This talk will cover joint work with K. Beanland and R. Liu. |
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July 28 4pm-5pm |
Aleksei Lissitin and Indrek Zolk, University of Tartu
Some approximation properties of Banach spaces Indrek Zolk will speak about a result (generalizing a theorem from 1988 due to Godefroy and Saphar) where under certain geometric constraints (covering a large class of Banach spaces), the commuting bounded compact approximation property (AP) on a Banach space itself can be "lifted" to its dual. He will also briefly investigate a subspace of \(c_0\) (due to Johnson and Schechtman from 1996) that fails the metric AP but has the commuting 6-bounded AP (slightly improving a result by Godefroy from 2001). Aleksei Lissitsin will speak about convex approximation properties (APs defined by a convex set of operators) which encompass the classical AP as well as its bounded versions, APs defined by operator ideals, APs of pairs of Banach spaces, and the positive AP of Banach lattices. Generalizations of several important results on the classical AP will be presented, including Grothendieck's criterion of the AP via the approximability of (weakly) compact operators and Johnson's lifting theorem of the metric AP from a Banach space to its dual space. Both communications are based on a joint work with Prof. Eve Oja. |
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July 29 11am-12pm |
Sergey Ajiev, University of New South Wales
Hölder classification of infinite-dimensional spheres, Tsar'kov's phenomenon and applications The uniform classification of infinite-dimensional spheres, developed in relation with the solution of the distortion problem is more balanced than the continuous, isometric, Lipschitz or uniform classifications of infinite-dimensional Banach spaces. It allows to transfer a group structure, group actions and other metric-related constructions from one space onto another. We show that the uniformly continuous homeomorphisms can be "upgraded" to the Hölder ones in the classical setting and establish the explicit and, occasionally, sharp exponents of the Hölder regularity for pairs of concrete spaces, including various Besov, Lizorkin-Triebel, Sobolev, sequence, Schatten-von Neumann and other Banach spaces (including lattices and more general non-commutative spaces). These results appear to have close ties with the presence of a remarkable phenomenon from the infinite-dimensional approximation theory discovered by Tsar'kov for the uniform mappings between pairs of Lebesgue spaces and the problems of extension and interpolation of the mappings between the pairs of the spaces under consideration. Among the applications of the main results and tools are multiple examples of spaces that do not allow any \(C^*\)-algebra structure but can be endowed with a homogeneous Hölder group structure, automatic stability principle in PDE and certain quantitative results related to the analysis of vector-valued functions and measures. |
Last modified: 20 May 2014, Alejandro Chavez-Dominguez