Analysis   Seminar    

Unless specified, all seminars are Wednesday 4-5pm at  250 Math Building.                      

September 23rd                       Organizational Meeting    


September 30th                        Hanfeng Li,     SUNY at Buffalo   
         Hilbert C*-modules admitting no frames

                                                Abstract: It is a consequence of Kasparov's stabilization theorem that every countably generated
                                                Hilbert C*-module over a unital C*-algebra has frames. I will show that this fails in general for
                                                arbitrary Hilbert C*-modules.
October 14th                            Jon Kraus,   SUNY at Buffalo
A generalization of Hilbert C*-modules

Abstract: We will discuss a generalization (due to David Blecher) of Hilbert C*-modules where
                                                the C*-algebra is replaced by an arbitrary operator algebra (a norm closed subalgebra of the
                                                bounded operators B(H) on a Hilbert space H). The generalization is based on a characterization
                                                of Hilbert C*-modules that does not involve inner products (or adjoints). We will also discuss Hilbert
W*-modules and their generalization (where the W*-algebra is replaced by an operator algebra
                                                which contains the identity operator and is closed in the weak* topology of B(H)).

October 28th                            Quanlei Fang,    SUNY at Buffalo
Commutators and localization on the Drury-Arveson space

                                                Abstract: Let f be a multiplier for the Drury-Arveson space Hn2 of the unit ball, and let ζ1, ..., ζn
                                                denote the coordinate functions.We show that for each 1 i n, the commutator [Mf*, Mζi]
                                                belongs to the Schatten class Cp, p>2n. This leads to a localization result for multipliers. 

November 4th                          Jingbo Xia,     SUNY at Buffalo
                                                Defect operators associated with submodules of the Hardy module

                                                Abstract: Let H2(S) be the Hardy space on the unit sphere S in Cn, n2. Then H2(S) is a natural
                                                Hilbert module over the ball algebra A(B). Let Mz1 , ..., Mzn be the module operators corresponding
                                                to the multiplication by the coordinated functions. Each submodule Μ⊂H2(S) gives rise to
                                                the module operators ZM,j = Mzj|M,j= 1, ..., n, on M. In this paper we establish the following
                                                commonly believed, but never previously proven result: whenever M≠{0}, the sum of the commutators
                                                                         [ZM, 1*, ZM, 1]+...+[ZM, n*, ZM, n]
                                                does not belong to the Schatten class Cn. This is a joint work with Quanlei Fang.

November 18th                        Byung Jay Kahng,     Canisius College
Some remarks on duality in the locally compact quantum group setting

In abstract harmonic analysis, among the most important result is the Pontryagin duality,
                                                which holds at the level of locally compact abelian (LCA) groups. Also, at the LCA group level,
                                                the notion of Fourier transform is defined. For further generalization, we consider the category of
                                                quantum groups, where Pontryagin-type, self-duality holds. Our quantum groups are locally compact
                                                quantum groups, in the C*-algebra or von Neumann algebra framework.
                                                By using the notion of the multiplicative unitary operators and the generalized Fourier transform, we
can enhance our understanding of the duality picture at the quantum group level. In particular, we will
consider a case of a certain coalgebra deformation of the quantum double, and its dual counterpart.

December 16th                         Wen Huang,    University of Science and Technology of China
Stable sets and unstable sets in positive entropy systems

Stable sets and unstable sets of a dynamical system with positive entropy are investigated.
                                                It is shown that in any invertible system with positive entropy, there is a measure-theoretically “rather big”
                                                set such that for any point from the set, the intersection of the closure of the stable set and the closure
                                                of the unstable set of the point has positive entropy. Moreover, for several kinds of specific systems,
                                                the lower bound of Hausdorff dimension of these sets is estimated. Particularly the lower bound of the
                                                Hausdorff dimension of such sets appearing in a positive entropy diffeomorphism on a smooth Riemannian
                                                manifold is given in terms of the metric entropy and of Lyapunov exponent.