# Past Functional Analysis Seminar

We consider a system of coupled second order integro-differential evolution equations in a Hilbert space, which is partially damped through memory effects. A global existence theorem regarding the solutions to its Cauchy problem is given, only under basic conditions that the memory kernels possess positive definite primitives but without nonnegative/decreasing assumptions. Following this, we find an approach to successfully obtain the stability of the system energy and various decay rates. Moreover, the abstract results are applied to several concrete systems in the real world, including the Timoshenko type. This is a joint work with Professor Ti-Jun Xiao (Fudan University) and Professor Jin Liang (Shanghai Jiaotong University)

A subshift is characterized by a set of allowable words on $d$ symbols. In a sense it encodes the allowable operations an automaton performs. In the late 1990's Matsumoto constructed a C*-algebra associated to a subshift, deriving initially his motivation from the work of Cuntz-Krieger. These C*-algebras were then studied in depth in a series of papers. In 2009 Shalit-Solel discovered a relation of the subshift algebras with their variants of operator algebras related to homogeneous ideals. In particular a subshift corresponds to a monomial ideal under this prism.

In a recent work with Shalit we take a closer look at these cases and study them in terms of classification programmes on nonselfadjoint operator algebras and Arveson's Programme on the C*-envelope. We investigate two nonselfadjoint operator algebras from one SFT and show that they completely classify the SFT: (a) up to the same allowable words, and (b) up to local conjugacy of the quantized dynamics. In addition we discover that the C*-algebra fitting Arveson's Programme is the quotient by the generalized compacts, rather than taking unconditionally all compacts as Matsumoto does. Actually there is a nice dichotomy that depends on the structure of the monomial ideal.

Nevertheless in the process we accomplish more in different directions. This happens as our case study is carried in the intersection of C*-correspondences, subproduct systems, dynamical systems and subshifts. In this talk we will give the basic steps of our results with some comments on their proofs.

The Bergman space $A_2(\mathbb D)$ is the closed subspace of $L^2(\mathbb D)$ consisting of analytic functions, where $\mathbb D$ denotes the unit

disk. One considers the projection from $L^2(\mathbb D)$ into $A_2(\mathbb D)$, such a projection can be written as an integral operator

with a singular kernel. In this talk, we will present the recent advances on the one weight and two weight theory for the

Bergman projection, in particular we will discuss the Sarason Conjecture for the Bergman space, sharp weighted estimates for the Bergman projection and a description of a $B_{\infty}$ class that has been until now absent. This is joint work with A. Aleman and S. Pott from Lund University (Sweden).

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 $B(H)$.