Matricial Function Theory and Weighted Shifts

Abstract

Let 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 T+(E), whose restrictions to M are normal, are parametrized by certain discs or balls D(E, σ) indexed by the normal * -representations σ of M . Each disc has analytic structure, and each element F ∈ T+(E) gives rise to an operator-valued function Fσ on D(E, σ) 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 maps are developed. Our perspective has been inspired by work of Vladimir Müller (J Oper Theory 20(1):3-20, 1988) 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 (Mem Am Math Soc 200(941):vi+91, 2009), who investigated similar questions.

Interpolation and Commutant Lifting with Weights

“…Suppose k ≥ 1. By the discussion preceding Theorem 7.6 in [9], Z ′(k) ⊗I H = U σ * k (Z (k) ⊗I H )U σ k ; moreover, by Lemma 7.2 of [9],…”

“…) * is completely positive, linear, and ultraweakly continuous by Lemma 4.4 of [9]; in addition, Φ z < 1. Therefore, in the Banach algebra of completely bounded maps on σ(M) ′ , ∞ j=0 Φ j z converges in norm.…”

“…The Weighted Hardy Algebra. We now turn to the weighted setting of [9]. Throughout, the sequence X = {X k } ∞ k=0 will denote an admissible sequence; this means that X k ∈ ϕ k (M) c for each k ∈ N 0 , X k ≥ 0 for each k ∈ N 0 , X 0 = 0, X 1 is invertible, and finally that lim…”

“…by Theorem 4.5 of [9], where the summation on the left-hand side of the equation is with respect to the norm topology on σ(M) ′ , and the summation on the right-hand side is with respect to the ultraweak topology on σ(M) ′ . Let z ∈ I(σ E • ϕ, σ).…”

Weighted Cuntz-Krieger Algebras

“…In [10] the second author, with P. Muhly, introduced and studied algebras of weighted shifts on the Fock space associated with a correspondence. This is a far reaching generalization of the classical weighted shift (on ℓ 2 ).…”