Forthcoming events in this series
Maximal left ideals of operators acting on a Banach space
Abstract
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.
The heat equation in curved stripes
Abstract
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 Dauns-Hofmann Theorem and tensor products of C*-algebras
Abstract
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.
The construction of quantum dynamical semigroups by way of non-commutative Markov processes
Abstract
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.
The Dirichlet-to-Neumann operator on rough domains
Abstract
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).
Analytical aspects of isospectral drums
Abstract
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).
On the representation of tight functionals as integrals
Abstract
We consider a vector lattice L of bounded real continuous functions on a topological space X that separates the points of X and contains the constant functions. A notion of tightness for linear functionals is defined,
and by an elementary argument we prove with the aid of the classical Riesz representation theorem that every tight continuous linear functional on L can be represented by integration with respect to a Radon measure. This result leads incidentally to an simple proof of Prokhorov’s existence theorem for
the limit of a projective system of Radon measures.
Carleson embeddings and integration operators of Volterra type on Fock spaces
Abstract
We consider spaces of entire functions that are $p$-integrable
with respect to a radial weight. Such spaces are usually called
Fock spaces, and a classical example is provided by the Gaussian
weight. It turns out that a function belongs to some Fock
space if and only if its derivative belongs to a Fock space
with a (possibly) different weight. Furthermore, we characterize
the Borel measures $\mu$ for which a Fock space is continuously
embedded in $L^q(\mu0)$ with $q>0$. We then illustrate the
applicability of these results to the study of properties such as
boundedness, compactness, Schatten class membership and the invariant
subspaces of integration operators of Volterra type acting on Fock spaces.
(joint work with Jose Angel Pelaez)
A Laman theorem for non-Euclidean bar-joint frameworks.
Abstract
Laman's theorem characterises the minimally infinitesimally rigid frameworks in Euclidean space $\mathbb{R}^2$ in terms of $(2,3)$-tight graphs. An interesting problem is to determine whether analogous results hold for bar-joint frameworks in other normed linear spaces. In this talk we will show how this can be done for frameworks in the non-Euclidean spaces $(\mathbb{R}^2,\|\cdot\|_q)$ with $1\leq q\leq \infty$, $q\not=2$. This is joint work with Stephen Power.
Fine scales of decay rates of operator semigroups
Abstract
A very efficient way to obtain rates of energy decay for damped wave equations is to use operator semigroups to pass from resolvent estimates to energy estimates. This is known to give the optimal results in cases when the resolvent estimates have simple forms such as being exactly polynomial ($|s|^\alpha$). After reviewing that theory this talk will discuss cases when the resolvent estimates are slightly different, for example $|s|^\alpha/ \log |s|$ or $|s|^\alpha \log |s|$.
Huygens' Principle for Hyperbolic Equations and $L^p$ Estimates for Riesz Transforms on Manifolds via First-Order Systems
Abstract
We prove that strongly continuous groups generated by first-order systems $D$ on Riemannian manifolds have finite propagation speed. The new direct proof for self-adjoint systems also provides a new approach to the weak Huygens' principle for second-order hyperbolic equations. The techniques are also combined with the resolvent approach to sectorial operators to obtain $L^2$ off-diagonal estimates for functions of $D$, which are the starting point for obtaining $L^p$ estimates for Riesz transforms on manifolds where the heat semigroup does not satisfy pointwise Guassian bounds. The two approaches are then combined via a Calder\'{o}n reproducing formula that allows for the analysing function to interact with $D$ through the holomorphic functional calculus whilst the synthesising function interacts with $D$ through the Fourier transform. This is joint work with P.~Auscher and A.~McIntosh.
A geometric interpretation of heat kernels associated with convolution semi-groups
Spectral problems for semigroups and asymptotically disappearing solutions
Abstract
We study symmetric systems with dissipative boundary conditions. The solutions of the mixed problems for such systems are given by a contraction semigroup $V(t)f = e^{tG_b}f,\: t \geq 0$. The solutions $u = e^{tG_b}f$ with eigenfunctions $f$ of the generator $G_b$ with eigenvalues $\lambda,\: \Re \lambda
Joint numerical radius
Abstract
Let $T_1,\dots,T_n$ be bounded linear operators on a complex Hilbert space
$H$. We study the question whether it is possible to find a unit vector
$x\in H$ such that $|\langle T_jx, x\rangle|$ is large for all $j$. Thus
we are looking for a generalization
of the well-known fact for $n = 1$ that the numerical radius $w(T)$ of a
single operator T satisfies $w(T)\ge \|T\|/2$.