Past Functional Analysis Seminar

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.

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.

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}.

We consider the layer potentials associated with operators $L=-\mathrm{div} A \nabla$ acting in the upper half-space $\mathbb{R}^{n+1}_+$, $n\geq 2$, where the coefficient matrix $A$ is complex, elliptic, bounded, measurable, and $t$-independent.  A ``Calder\'{o}n--Zygmund" theory is developed for the boundedness of the layer potentials under the assumption that solutions of the equation $Lu=0$ satisfy interior De Giorgi--Nash--Moser type estimates. In particular, we prove that $L^{^2}$ estimates for the layer potentials imply sharp $L^p$ and endpoint space estimates. The method of layer potentials is then used to obtain solvability of boundary value problems. This is joint work with Steve Hofmann and Marius Mitrea.

Contractions of Lie groups have been used by physicists to understand how classical physics is the limit ``as the speed of light tends to infinity" of relativistic physics. It turns out that a contraction can be understood as an approximate homomorphism between two Lie algebras or Lie groups, and we can use this to transfer harmonic analysis from a group to its ``limit", finding relationships which generalise the traditional results that the Fourier transform on $\R$ is the limit of Fourier series on $\TT$. We can transfer $L^p$ estimates, solutions of differential operators, etc. The interesting limiting relationship between the representation theory of the groups involved can be understood geometrically via the Kirillov orbit method.

We prove that the double layer potential operator and the gradient of the single layer potential operator are L_2 bounded for general second order divergence form systems. As compared to earlier results, our proof shows that the bounds for the layer potentials are independent of well posedness for the Dirichlet problem and of De Giorgi-Nash local estimates. The layer potential operators are shown to depend holomorphically on the coefficient matrix A\in L_\infty, showing uniqueness of the extension of the operators beyond singular integrals. More precisely, we use functional calculus of differential operators with non-smooth coefficients to represent the layer potential operators as bounded Hilbert space operators. In the presence of Moser local bounds, in particular for real scalar equations and systems that are small perturbations of real scalar equations, these operators are shown to be the usual singular integrals. Our proof gives a new construction of fundamental solutions to divergence form systems, valid also in dimension 2.

An important aspect in the study of Kato's perturbation theorem for substochastic semi-

groups is the study of the honesty of the perturbed semigroup, i.e. the consistency between

the semigroup and the modelled system. In the study of Laplacians on graphs, there is a

corresponding notion of stochastic completeness. This talk will demonstrate how the two

notions coincide.

We address the following two questions regarding the maximal left ideals of the Banach algebras B(E) of bounded operators acting on an infinite-dimensional Banach space E:

i) Does B(E) always contain a maximal left ideal which is not finitely generated?

ii) Is every finitely-generated maximal left ideal of B(E) necessarily of the form {T\in B(E): Tx = 0}? for some non-zero vector x in E?

Since the two-sided ideal F(E) of finite-rank operators is not contained in any of the maximal left ideals mentioned above, a positive answer to the second question would imply a positive answer to the first.

Our main results are:

Question i) has a positive answer for most (possibly all) infinite-dimensional Banach spaces;

Question ii) has a positive answer if and only if no finitely-generated, maximal left ideal of B(E) contains F(E); the answer to Question ii) is positive for many, but not all, Banach spaces. We also make some remarks on a more general conjecture that a unital Banach algebra is finite-dimensional whenever all its maximal left ideals are finitely generated; this is true for C*-algebras.

This is based on a recent paper with H.G. Dales, T. Kochanek, P. Koszmider and N.J. Laustsen (Studia Mathematica, 2013) and work in progress with N.J. Laustsen.

We study the inuence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the flat case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the flat case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation.

The problem of representing a (non-commutative) C*-algebra $A$ as the

algebra of sections of a bundle of C*-algebras over a suitable base

space may be viewed as that of finding a non-commutative Gelfand-Naimark

theorem. The space $\mathrm{Prim}(A)$ of primitive ideals of $A$, with

its hull-kernel topology, arises as a natural candidate for the base

space. One major obstacle is that $\mathrm{Prim}(A)$ is rarely

sufficiently well-behaved as a topological space for this purpose. A

theorem of Dauns and Hofmann shows that any C*-algebra $A$ may be

represented as the section algebra of a C*-bundle over the complete

regularisation of $\mathrm{Prim}(A)$, which is identified in a natural

way with a space of ideals known as the Glimm ideals of $A$, denoted

$\mathrm{Glimm}(A)$.

In the case of the minimal tensor product $A \otimes B$ of two

C*-algebras $A$ and $B$, we show how $\mathrm{Glimm}(A \otimes B)$ may

be constructed in terms of $\mathrm{Glimm}(A)$ and $\mathrm{Glimm} (B)$.

As a consequence, we describe the associated C*-bundle representation of

$A \otimes B$ over this space, and discuss properties of this bundle

where exactness of $A$ plays a decisive role.

Although generators of strongly continuous semigroups of contractions

on Banach spaces are characterised by the Hille-Yosida theorem, in

practice it can be difficult to verify that this theorem's hypotheses

are satisfied. In this talk, it will be shown how to construct certain

quantum Markov semigroups (strongly continuous semigroups of

contractions on C* algebras) by realising them as expectation

semigroups of non-commutative Markov processes; the extra structure

possessed by such processes is sufficient to avoid the need to use

Hille and Yosida's result.

We consider a bounded connected open set

$\Omega \subset {\rm R}^d$ whose boundary $\Gamma$ has a finite

$(d-1)$-dimensional Hausdorff measure. Then we define the

Dirichlet-to-Neumann operator $D_0$ on $L_2(\Gamma)$ by form

methods. The operator $-D_0$ is self-adjoint and generates a

contractive $C_0$-semigroup $S = (S_t)_{t > 0}$ on

$L_2(\Gamma)$. We show that the asymptotic behaviour of

$S_t$ as $t \to \infty$ is related to properties of the

trace of functions in $H^1(\Omega)$ which $\Omega$ may or

may not have. We also show that they are related to the

essential spectrum of the Dirichlet-to-Neumann operator.

The talk is based on a joint work with W. Arendt (Ulm).

Almost 50 years ago, Kac posed the now-famous question `Can one hear the

shape of a drum?', that is, if two planar domains are isospectral with respect to the Dirichlet (or Neumann) Laplacian, must they necessarily be congruent?

This question was answered in the negative about 20 years ago with the

construction of pairs of polygonal domains with special group-theoretically

motivated symmetries, which are simultaneously Dirichlet and Neumann

isospectral.

We wish to revisit these examples from an analytical perspective, recasting the

arguments in terms of elliptic forms and intertwining operators. This allows us to prove in particular that the isospectrality property holds for a far more general class of elliptic operators than the Laplacian, as it depends purely on what the intertwining operator does to the form domains. We can also show that the same type of intertwining operator cannot intertwine the Robin Laplacian on such domains.

This is joint work with Wolfgang Arendt (Ulm) and Tom ter Elst (Auckland).