OA/OT Seminar Ical Atom

In this talk we will discuss two central problems in multivariable operator theory: (1) joint invariant subspaces of the tuple of shift operators on the Hardy space over the unit polydisc, and (2) a commutant lifting theorem of the tuple of shift operator on certain analytic reproducing kernel Hilbert spaces over the unit ball. If time permits, we will also talk about Nevanlinna-Pick interpolation theorem and a relevant factorization result for multipliers on reproducing kernel Hilbert spaces over the unit ball.

In this talk, we will start going over Takesaki’s annals paper that proves that every separable C*-algebra A can be represented as continuous “noncommutative” functions with values in B(H) (H separable) on the space of representations of A on H. Furthermore, the universal enveloping von Neumann algebra of A is identified with all the bounded “noncommutative” functions on the same space

An index theory for elliptic operators on a closed manifold was developed by Atiyah and Singer. For a family of such operators parametrized by points of a compact space X, they computed the K^0(X)-valued analytical index in purely topological terms. An analog of this theory for self-adjoint elliptic operators on closed manifolds was developed by Atiyah, Patodi, and Singer; the analytical index of a family in this case takes values in the K^1 group of a base space.

If a manifold has non-empty boundary, then boundary conditions come into play, and situation becomes much more complicated. The integer-valued index of a single boundary value problem was computed by Atiyah and Bott. This result was recently generalized to K^0(X)-valued family index by Melo, Schrohe, and Schick. The self-adjoint case, however, remained open.

In the talk I shall present a family index theorem for self-adjoint elliptic operators on a surface with boundary. I compute the K^1(X)-valued analytical index in terms of the topological data of the family over the boundary. The talk is based on my preprint arXiv:1809.04353.

Attached

TBA

In this talk we give an introduction to (real) infinite dimensional moment problems, i.e. for measures supported on real infinite dimensional spaces. We will focus on the following problem: when can a linear functional on a unital commutative real algebra A be represented as an integral w.r.t. a Radon measure on the real character space X(A) equipped with the Borel σ-algebra generated by the weak topology? Our main idea is to construct X(A) as a projective limit of the character spaces of all finitely generated subalgebras of A, to be able to exploit the classical finite dimensional moment theory in the infinite dimensional case. We thus obtain existence results for representing measures defined on the cylinder σ-algebra on X(A), carried by the projective limit construction. If in addition the well-known Prokhorov (ε-K) condition is fulfilled, then we can solve our problem by extending such representing measures from the cylinder to the Borel σ-algebra on X(A). These results allow us to establish e.g. infinite dimensional analogues of the classical Riesz-Haviland.

Our work was motivated by the paper [Ghasemi-Kuhlmann-Marshall: Moment problem in infinitely many variables, Israel Journal of Mathematics, Volume 212, 989-1012 (2016) ] where the case when A is the algebra of real polynomials in infinitely many variables is considered. Our projective limit technique provides alternative proofs to the results of [GKM2016].

(Joint work with Maria Infusino, Tobias Kuna and Patrick Michalski)

C*-algebras have been intensely studied in recent years, especially through the lens of classification via K-theoretic invariants. Prominent advances include results for Cuntz-Krieger algebras of directed graphs. One such result of Cuntz and Krieger shows that the K-theory groups of such algebras essentially coincide with Bowen-Franks groups of the subshift of finite type associated to the graph.

On the other hand, classifying non-self-adjoint operator algebras is an effort initiated by Arveson in his late 60s paper on algebras arising from one-sided measure preserving dynamics. This was later taken up by Davidson and Katsoulis in the topological scenario, where they classified non-self-adjoint operator algebras arising from multidimensional one-sided dynamical systems on compact Hausdroff spaces.

In this talk we will connect, through examples, these traditionally unrelated classification schemes. We survey some pertinent results from the literature and uncover a striking hierarchy of classification for irreversible and reversible operator algebras.

The theory of non-commutative (nc) rational functions which are regular at 0 is well known and studied, in terms of their minimal realizations: any such function admits a unique minimal realization centred at 0 and the domain of the function coincides with the invertibility set of the (resolvent of the) realization. In addition, a nc power series around the origin will be the power series expansion of a nc rational function if and only if a given Hankel matrix built from the coefficients of the given power series has a finite rank (Fliess-Kronecker).

In this talk, we present generalizations of these ideas to the case where the centre is non-scalar. In particular, we prove the existence and uniqueness of a minimal Fornasini-Marchesini realization for every nc rational function, centred at an arbitrary matrix point in its domain of regularity, and show that using this realization, one can evaluate the function on all of its domain (of matrices of all sizes).

Unlike the case of a scalar centre, the coefficients of the realization can not be chosen arbitrarily. We present necessary and sufficient conditions (called the linearized lost-abbey conditions) on the coefficients of a minimal realization centred at a matrix point, such that there exists a nc rational function which admits the realization.

This is a joint work with Victor Vinnikov.