Past Functional Analysis Seminar

22 October 2013
17:00
to
18:25
David McConnell
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.
  • Functional Analysis Seminar
11 June 2013
17:00
to
18:15
Alex Belton
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.
  • Functional Analysis Seminar
14 May 2013
17:00
to
18:07
Tom ter Elst
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).
  • Functional Analysis Seminar
9 May 2013
17:00
to
18:10
James Kennedy
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).
  • Functional Analysis Seminar
23 April 2013
17:00
to
18:08
David Edwards
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.
  • Functional Analysis Seminar
5 March 2013
17:00
to
18:16
Olivia Constantin
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)
  • Functional Analysis Seminar
26 February 2013
17:00
to
18:16
Derek Kitson
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.
  • Functional Analysis Seminar
12 February 2013
17:00
to
18:16
Charles Batty
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 $|s|^\alpha \log |s|$.

  • Functional Analysis Seminar

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