I will give an overview of semigroup C*-algebras, which are C*-algebras generated by left regular representations of semigroups. The main focus will be on examples from number theory and group theory.

# Past Functional Analysis Seminar

Congruence monoids in the ring of integers are given by certain unions of arithmetic progressions. To each congruence monoid, there is a canonical way to associate a semigroup C*-algebra. I will explain this construction and then discuss joint work with Xin Li on K-theoretic invariants. I will also indicate how all of this generalizes to congruence monoids in the ring of integers of an arbitrary algebraic number field.

We consider Schroedinger operators on the whole Euclidean space or on the half-space, subject to real Robin boundary conditions. I will present the construction of a non-real potential that decays at infinity so that the corresponding Schroedinger operator has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum. This proves that the Lieb-Thirring inequalities, crucial in quantum mechanics for the proof of stability of matter, do no longer hold in the non-selfadjoint case.

I will give an overview of the localization technique: a powerful dimension-reduction tool for proving geometric and functional inequalities. Having its roots in a pioneering work of Payne-Weinberger in the 60ies about sharp Poincare’-Wirtinger inequality on Convex Bodies in Rn, recently such a technique found new applications for a range of sharp geometric and functional inequalities in spaces with Ricci curvature bounded below.

The well known Katznelson-Tzafriri theorem states that a power-bounded operator $T$ on a Banach space $X$ satisfies $\|T^n(I-T)\| \to 0$ as $n \to \infty$ if and only if the spectrum of $T$ touches the complex unit circle nowhere except possibly at the point $\{1\}$. As it turns out, the rate at which $\|T^n(I-T)\|$ goes to zero is largely determined by estimates on the resolvent of $T$ on the unit circle minus $\{1\}$ and not only is this interesting from a purely spectral and operator theoretic perspective, the applications of such quantified decay rates are myriad, ranging from the mean ergodic theorem to so-called alternating projections, from probability theory to continuous-in-time evolution equations. In this talk, we will trace the story of these so-called quantified Katznelson-Tzafriri theorems through previously known results up to the present, ending with a new result proved just a few weeks ago that largely completes the adventure.

## Further Information:

ABSTRACT: Given a metric measure space $(X,d,\mu)$, its doubling constant is given by

$$

C_\mu=\sup_{x\in X, r>0} \frac{\mu(B(x,2r))}{\mu(B(x,r))},

$$

where $B(x,r)$ denotes the open ball of radius $r$ centered at $x$. Clearly, $C_\mu\geq1$, and in the case $X$ reduces to a singleton $C_\mu=1$. One might think that for a metric space with more than one point, the constant $C_\mu$ could be very close to one. However, we will show that in general $C_\mu\geq2$. The talk is based on a joint work with Javier Soria (Barcelona).