Past Functional Analysis Seminar

4 November 2014
17:00
Mahya Ghandehari
Abstract

The Fourier algebra of a locally compact group was first defined by Eymard in 1964. Eymard showed that this algebra is in fact the space of all coefficient functions of the left regular representation equipped with pointwise operations. The Fourier algebra is a semi-simple commutative Banach algebra, and thus it admits no non-zero continuous derivation into itself. In this talk we study weak amenability, which is a weaker form of differentiability, for Fourier algebras. A commutative Banach algebra is called weakly amenable if it admits no non-zero continuous derivations into its dual space. The notion of weak amenability was first defined and studied for certain important examples by Bade, Curtis and Dales. 

 

In 1994, Johnson constructed a non-zero continuous derivation from the Fourier algebra of the rotation group in 3 dimensions into its dual. Subsequently, using the structure theory of Lie groups and Lie algebras, this result was extended to any non-Abelian, compact, connected group. Using techniques of non-commutative harmonic analysis, we prove that semi-simple connected Lie groups and 1-connected non-Abelian nilpotent Lie groups are not weak amenable by reducing the problem to two special cases: the $ax+b$ group and the 3-dimensional Heisenberg group. These are the first examples of classes of locally compact groups with non-weak amenable Fourier algebras which do not contain closed copies of the rotation group in 3 dimensions.

  • Functional Analysis Seminar
27 May 2014
17:00
to
18:15
Steve Hofmann
Abstract
A classical theorem of F. and M. Riesz states that for a simply connected domain in the complex plane with a rectifiable boundary, harmonic measure and arc length measure on the boundary are mutually absolutely continuous. On the other hand, an example of C. Bishop and P. Jones shows that the latter conclusion may fail, in the absence of some sort of connectivity hypothesis. In this work, we nonetheless establish versions of the F. and M. Riesz theorem, in higher dimensions, in which other properties of har¬monic functions substitute for the absolute continuity of harmonic measure. These substitute properties are natural “proxies” for har¬monic measure estimates, in the sense that, in more topologically be¬nign settings, they are actually equivalent to a scale-invariant quanti¬tative version of absolute continuity.
  • Functional Analysis Seminar
13 May 2014
17:00
to
18:15
Aaron Tikuisis
Abstract
Inspired largely by the fact that commutative C*-algebras correspond to (locally compact Hausdorff) topological spaces, C*-algebras are often viewed as noncommutative topological spaces. In particular, this perspective has inspired notions of noncommutative dimension: numerical isomorphism invariants for C*-algebras, whose value at C(X) is equal to the dimension of X. This talk will focus on certain recent notions of dimension, especially decomposition rank as defined by Kirchberg and Winter. A particularly interesting part of the dimension theory of C*-algebras is occurrences of dimension reduction, where the act of tensoring certain canonical C*-algebras (e.g. UHF algebras, Cuntz' algebras O_2 and O_infinity) can have the effect of (drastically) lowering the dimension. This is in sharp contrast to the commutative case, where taking a tensor product always increases the dimension. I will discuss some results of this nature, in particular comparing the dimension of C(X,A) to the dimension of X, for various C*-algebras A. I will explain a relationship between dimension reduction in C(X,A) and the well-known topological fact that S^n is not a retract of D^{n+1}.
  • Functional Analysis Seminar

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