# Past Functional Analysis Seminar

An old result of Dixmier, Day and others states that every continuous bounded representation of an amenable group on Hilbert space is similar to a unitary representation. In similar vein, one can ask if amenable subalgebras of $B(H)$ are always similar to self-adjoint subalgebras. This problem was open for many years, but it was recently shown by Farah and Ozawa that in general the answer is negative; their approach goes via showing that the Dixmier--Day result is false when $B(H)$ is replaced by the Calkin algebra. In this talk, I will give some of the background, and then outline a simplified and more explicit version of their construction; this is taken from joint work with Farah and Ozawa (2014) . It turns out that the key mechanism behind these negative results is the large supply of projections in $\ell_\infty / c_0$, rather than the complicated structure of $B(H)$.

This talk will be by videolink from Warsaw. The starting-time will be a little after 17:00 due to a TCC lecture and time needed to establish video connections.

Abstract: The Haagerup approximation property for finite von Neumann algebras (i.e.von Neumann algebras with a tracial faithful normal state) has been studied for more than 30 years. The original motivation to study this property came from the case of group von Neumann algebras of discrete groups, where it corresponds to the geometric Haagerup property of the underlying group. Last few years brought a lot of interest in the Haagerup property for discrete and general locally compact quantum groups. If the discrete quantum group in question is not unimodular, the associated (quantum) group von Neumann algebra cannot be finite, so we need a broader framework for the operator algebraic property. In this talk, I will present recent developments regarding the Haagerup approximation property for arbitrary von Neumann algebras and will also discuss some questions relating it to the issues related to the classical Schoenberg correspondence. (Mainly based on joint work with Martijn Caspers.)

We present several quantified versions of Ingham’s Tauberian theorem in

which the rate of decay is determined by the behaviour of a certain boundary

function near its singularities. The proofs of these results are modified

versions of Ingham’s own proof and, in particular, involve no estimates of

contour integrals. The general results are then applied in the setting of C_0-

semigroups, giving both new proofs of previously known results and, in one

important case, a sharper result than was previously available.

Stochastic homogenization shows that solutions to an elliptic problem with rapidly oscillating, ergodic random coefficients can be effectively described by an elliptic problem with homogeneous, deterministic coefficients. The definition of the latter is based on the construction of a "corrector" and invokes an elliptic operator that acts on the probability space of admissible coefficient fields. While qualitative homogenization is well understood and classical, quantitative results (e.g. estimates on the homogenization error and approximations to the homogenized coefficients) have only been obtained recently. In the talk we discuss an optimal estimate on the associated semigroup (usually called the "random walk in the random environment") and show that it decays with an algebraic rate. The result relies on a link between a Spectral Gap of a Glauber dynamics on the space of coefficient fields (a notion that we borrow from statistical mechanics) and heat kernel estimates. As applications we obtain moment bounds on the corrector and an optimal convergence rate for the approximation of the homogenized coefficients via periodic representative volume elements.

We begin by reviewing different stability types for abstract differential equations and strongly continuous semigroups on Hilbert spaces. We concentrate on exponential stability, polynomial stability, and strong stability with a finite number of singularities on the imaginary axis. We illustrate each stability type with examples from partial differential equations and control theory. In the second part of the talk we study the preservation of strong and polynomial stabilities of a semigroup under bounded perturbations of its generator. As the main results we present conditions for preservation of these two stability types under finite rank and trace class perturbations. In particular, the conditions require that certain graph norms of the perturbing operators are sufficiently small. In the final part of the talk we consider robust output tracking for linear systems, and explain how this control problem motivates the study of preservation of polynomial stability of semigroups. In particular, the solution of this problem requires determining which uncertainties in the parameters of the controlled system preserve the stability of the closed-loop system consisting of the system and the dynamic controller. We show that if the reference signal to be tracked is a nonsmooth periodic function, it is impossible to stabilize the closed-loop system exponentially, but polynomial stability is achievable under suitable assumptions. Subsequently, the uncertainties in the parameters of the system can be represented as a bounded perturbation to the system operator of the polynomially stable closed-loop system.