Free holomorphic functions on the unit ball of B(H)n

In this article we prove that quasi-multiplicative (with respect to the usual length function) mappings on the permutation group $\SSn$ (or, more generally, on arbitrary amenable Coxeter groups), determined by self-adjoint contractions fulfilling the braid or Yang-Baxter relations, are completely positive. We point out the connection of this result with the construction of a Fock representation of the deformed commutation relations $d_id_j^*-\sum_{r,s} t_{js}^{ir} d_r^*d_s=\delta_{ij}\id$, where the matrix $t_{js}^{ir}$ is given by a self-adjoint contraction fulfilling the braid relation. Such deformed commutation relations give examples for operator spaces as considered by Effros, Ruan and Pisier. The corresponding von Neumann algebras, generated by $G_i=d_i+d_i^*$, are typically not injective.

https://arxiv.org/pdf/funct-an/9408002.pdf

Completely Positive Maps on Coxeter Groups, Deformed Commutation Relations, and Operator Spaces

We develop a dilation theory on noncommutative varieties determined by row contractions T: = [T 1 ,..., T n ] subject to constraints such as p(T 1 ,...,T n ) = 0, p ∈ P, where T is a set of noncommutative polynomials. The model n-tuple is the universal row contraction [B 1 ,...,B n ] satisfying the same constraints as T, which turns out to be, in a certain sense, the maximal constrained piece of the n-tuple [S 1 ,..., S n ] of left creation operators on the full Fock space on n generators. The theory is based on a class of constrained Poisson kernels associated with T and representations of the C*-algebra generated by B 1 ,...,B n and the identity. Under natural conditions on the constraints we have uniqueness for the minimal dilation. A characteristic function Θ T is associated with any (constrained) row contraction T and it is proved that I - Θ T Θ* T = K T K* T , where K T is the (constrained) Poisson kernel of T. Consequently, for pure constrained row contractions, we show that the characteristic function is a complete unitary invariant and provide a model. We show that the curvature invariant and Euler characteristic asssociated with a Hilbert module generated by an arbitrary (resp. commuting) row contraction T can be expressed only in terms of the (resp. constrained) characteristic function of T. We provide a commutant lifting theorem for pure constrained row contractions and obtain a Nevanlinna-Pick interpolation result in our setting.

Operator theory on noncommutative varieties

We develop a theory of holomorphic functions in several noncommuting (free) variables and thus provide a framework for the study of arbitrary n-tuples of operators. The main topics are the following: Free holomorphic functions and Hausdorff derivations; Cauchy, Liouville, and Schwartz type results for free holomorphic functions; Algebras of free holomorphic functions; Free analytic functional calculus and noncommutative Cauchy transforms; Weierstrass and Montel theorems for free holomorphic functions; Free pluriharmonic functions and noncommutative Poisson transforms; Hardy spaces of free holomorphic functions.

Free holomorphic functions on the unit ball of B(H)^n. II

In this volume the author studies noncommutative domains $\mathcal{D}_f\subset B(\mathcal{H})^n$ generated by positive regular free holomorphic functions $f$ on $B(\mathcal{H})^n$, where $B(\mathcal{H})$ is the algebra of all bounded linear operators on a Hilbert space $\mathcal{H}$. Table of Contents: Introduction; Operator algebras associated with noncommutative domains; Free holomorphic functions on noncommutative domains; Model theory and unitary invariants on noncommutative domains; Commutant lifting and applications; Bibliography. (MEMO/205/964)

Operator Theory on Noncommutative Domains

We introduce a notion called `maximal commuting piece' for tuples of Hilbert space operators. Given a commuting tuple of operators forming a row contraction there are two commonly used dilations in multivariable operator theory. Firstly there is the minimal isometric dilation consisting of isometries with orthogonal ranges and hence it is a noncommuting tuple. There is also a commuting dilation related with a standard commuting tuple on Boson Fock space. We show that this commuting dilation is the maximal commuting piece of the minimal isometric dilation. We use this result to classify all representations of Cuntz algebra O_n coming from dilations of commuting tuples.

/pdf/standard-noncommuting-and-commuting-dilations-of-commuting-4jmlp598sy.pdf

Standard noncommuting and commuting dilations of commuting tuples

Motivated by a result on weak Markov dilations, we define a notion of characteristic function for ergodic and coisometric row contractions with a one-dimensional invariant subspace for the adjoints. This extends a definition given by G. Popescu. We prove that our characteristic function is a complete unitary invariant for such tuples and show how it can be computed.

Characteristic Functions for Ergodic Tuples

We introduce characteristic functions for certain contractive liftings of row contractions. These are multi-analytic operators which classify the liftings up to unitary equivalence and provide a kind of functional model. The most important cases are subisometric and coisometric liftings. We also identify the most general setting which we call reduced liftings. We derive properties of these new characteristic functions and discuss the relation to Popescu's definition for completely non-coisometric row contractions. Finally we apply our theory to completely positive maps and prove a one-to-one correspondence between the fixed point sets of completely positive maps related to each other by a subisometric lifting.

Characteristic Functions of Liftings

Given a contractive tuple of Hilbert space operators satisfying certainA-relations we show that there exists a unique minimal dilation to generators of Cuntz-Krieger algebras or its extension by compact operators. This Cuntz-Krieger dilation can be obtained from the classical minimal isometric dilation as a certain maximalA-relation piece. We define a maximal piece more generally for a finite set of polynomials inn noncommuting variables. We classify all representations of Cuntz-Krieger algebrasO
 A obtained from dilations of commuting tuples satisfyingA-relations. The universal properties of the minimal Cuntz-Krieger dilation and the WOT-closed algebra generated by it is studied in terms of invariant subspaces.

/pdf/minimal-cuntz-krieger-dilations-and-representations-of-cuntz-2j8x4gxz1p.pdf

Minimal Cuntz-Krieger dilations and representations of Cuntz-Krieger algebras

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Papers

Standard noncommuting and commuting dilations of commuting tuples

Characteristic Functions for Ergodic Tuples

Characteristic Functions of Liftings

Characteristic Functions of Liftings

Minimal Cuntz-Krieger dilations and representations of Cuntz-Krieger algebras