IUMJ

Title: Hyperbolic geometry on the unit ball of $B(\mathcal{H})^{n}$ and dilation theory

Authors: Gelu Popescu

Issue: Volume 57 (2008), Issue 6, 2891-2930

Abstract:

In this paper we continue our investigation concerning the hyperbolic geometry on the noncommutative ball $[B(\mathcal{H})^n]_{1}^{-}$ of all $n$-tuples $(X_1, \ldots, X_n) \in B(\mathcal{H})^{n}$ with \[ \|X_{1} X_{1}^{*} + \cdots + X_{n} X_{n}^{*} \|^{1/2} \leq 1, \] where $B(\mathcal{H})$ is the algebra of all bounded linear operators on a Hilbert space $\mathcal{H}$, and its implications to noncommutative function theory. The central object is an intertwining operator $L_{B,A}$ of the minimal isometric dilations of $A$, $B \in [B(\mathcal{H})^{n}]_{1}^{-}$, which establishes a strong connection between noncommutative hyperbolic geometry on $[B(\mathcal{H})^{n}]_{1}^{-}$ and multivariable dilation theory. The goal of this paper is to study the operator $L_{B,A}$ and its connections to the hyperbolic metric $\delta$ on the Harnack parts $\Delta$ of $[B(\mathcal{H})^{n}]_{1}^{-}$. In particular, we show that \[ \delta(A,B) = \ln \max \left\{ \| L_{A,B} \|, \| L_{A,B}^{-1}\| \right\} \] for any $A$, $B \in \Delta$, and express $\|L_{B,A}\|$ in terms of the reconstruction operators $R_{A}$ and $R_{B}$. We study the geometric structure of the operator $L_{B,A}$ and obtain new characterizations for the Harnack domination (resp. equivalence) in $[B(\mathcal{H})^{n}]_{1}^{-}$. Finally, given a contractive free holomorphic function $F := (F_1, \ldots, F_m)$ with coefficients in $B(\mathcal{E})$ and $z$, $\xi \in \mathbb{B}_{n}$, the open unit ball of $\mathbb{C}^n$, we prove that $F(z)$ is Harnack equivalent to $F(\xi)$ and $$ \|L_{F(z), F(\xi)} \| \leq \| L_{z,\xi} | = \left(\frac{1 + \|\varphi_{z}(\xi)\|_{2}}{1 - \|\varphi_{z}(\xi)\|_{2}} \right)^{1/2},$$ where $\varphi_{z}$ is the involutive automorphism of $\mathbb{B}_n$ which takes $0$ into $z$. This result implies a Schwartz-Pick lemma for operator-valued multipliers of the Drury-Arveson space, with respect to the hyperbolic metric.