Matricial Function Theory and Weighted Shifts

TL;DR

This paper extends the theory of tensor algebras of $W^{*}$-correspondences by incorporating operator-valued weights, revealing new connections with completely positive maps and providing a framework for modeling operator algebras via weighted shifts.

Contribution

It introduces a weighted version of tensor algebra representations, develops new proofs, and links the theory to weighted shift operators and completely positive maps.

Findings

Weighted tensor algebras are parametrized similarly to unweighted cases.

Operator-valued weights lead to new analytic and algebraic structures.

Connections with weighted shift operators and completely positive maps are established.

Abstract

Let $\mathcal{T}_{+}(E)$ be the tensor algebra of a $W^{*}$-correspondence $E$ over a $W^{*}$-algebra $M$. In earlier work, we showed that the completely contractive representations of $\mathcal{T}_{+}(E)$, whose restrictions to $M$ are normal, are parametrized by certain discs or balls $\overline{D(E,\sigma)}$ indexed by the normal $*$-representations $\sigma$ of $M$. Each disc has analytic structure, and each element $F\in \mathcal{T}_{+}(E) $ gives rise to an operator-valued function $\widehat{F}_{\sigma}$ on $\overline{D(E,\sigma)}$ that is continuous and analytic on the interior. In this paper, we explore the effect of adding operator-valued weights to the theory. While the statements of many of the results in the weighted theory are anticipated by those in the unweighted setting, substantially different proofs are required. Interesting new connections with the theory of completely positive are developed. Our perspective has been inspired by work of Vladimir M\"{u}ller in which he studied operators that can be modeled by parts of weighted shifts. Our results may be interpreted as providing a description of operator algebras that can be modeled by weighted tensor algebras. Our results also extend work of Gelu Popescu, who investigated similar questions.