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.
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.