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,
n≥2.
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
Abstract: 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.
November 25th
December 2nd
December 9th
December
16th
Wen Huang,
University of Science and Technology of China
Stable sets and unstable sets in positive entropy systems
Abstract: 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.