Guy Salomon

The Fock Space in the Slice Hyperholomorphic Setting

In this paper we introduce and study some basic properties of the Fock space (also known as Segal–Bargmann space) in the slice hyperholomorphic setting. We discuss both the case of slice regular functions over quaternions and the case of slice monogenic functions with values in a Clifford algebra. In the specific setting of quaternions, we also introduce the full Fock space. This paper can be seen as the beginning of the study of infinitedimensional analysis in the quaternionic setting.

The Isomorphism Problem for Complete Pick Algebras: A Survey

Complete Pick algebras – these are, roughly, the multiplier algebras in which Pick’s interpolation theorem holds true – have been the focus of much research in the last twenty years or so. All (irreducible) complete Pick algebras may be realized concretely as the algebras obtained by restricting multipliers on Drury–Arveson space to a subvariety of the unit ball; to be precise: every irreducible complete Pick algebra has the form \(\mathcal{M}_v\;=\;\left\{f|_v\;:\;f\in\mathcal{M}_d\right\}\), where \(\mathcal{M}_d\) denotes the multiplier algebra of the Drury–Arveson space \(H_d^2\), and V is the joint zero set of some functions in \(\mathcal{M}_d\). In recent years several works were devoted to the classification of complete Pick algebras in terms of the complex geometry of the varieties with which they are associated. The purpose of this survey is to give an account of this research in a comprehensive and unified way. We describe the array of tools and methods that were developed for this program, and take the opportunity to clarify, improve, and correct some parts of the literature.

NEW TOPOLOGICAL ℂ-ALGEBRAS WITH APPLICATIONS IN LINEAR SYSTEMS THEORY

Abstract. Motivated by the Schwartz space of tempered distri-butions S ′ and the Kondratiev space of stochastic distributionsS −1 we define a wide family of nuclear spaces which are increas-ing unions of (duals of) Hilbert spaces H ′p ,p∈ N, with decreasingnorms k · k p . The elements of these spaces are functions on a freecommutative monoid. We characterize those rings in this familywhich satisfy an inequality of the form kf⋄gk p ≤ A(p−q)kfk q kgk p for all p≥ q+ d, where ⋄ denotes the convolution in the monoid,A(p−q) is a strictly positive number and dis a fixed natural num-ber (in this case we obtain commutative topological rings). Suchan inequality holds in S −1 , but not in S ′ . We give an example ofsuch a ring which contains S ′ . We characterize invertible elementsin these rings and present applications to linear system theory.1991 Mathematics Subject Classification. Primary: 46A11, 13J99. Secondary:93E03, 60H40.Key words and phrases. nuclear spaces, topological rings, Wick product, convo-lution, White noise space, V˚age inequality, Schwartz space of tempered distribu-tions, Kondratiev spaces, linear systems on commutative rings.D. Alpay thanks the Earl Katz family for endowing the chair which supportedhis research.

Full Cuntz-Krieger dilations via non-commutative boundaries: DILATION VIA NON-COMMUTATIVE BOUNDARIES

We apply Arvesons non-commutative boundary theory to dilate every Toeplitz-Cuntz-Krieger family of a directed graph

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

G

On free stochastic processes and their derivatives

to a full Cuntz-Krieger family for

Non-commutative stochastic distributions and applications to linear systems theory

G

TOPOLOGICAL CONVOLUTION ALGEBRAS

. We do this by describing all representations of the Toeplitz algebra

On Algebras Which are Inductive Limits of Banach Spaces

\mathcal{T}(G)

Hyperrigid subsets of Cuntz--Krieger algebras and the property of rigidity at zero.

that have unique extension when restricted to the tensor algebra