Noncommutative hyperbolic geometry on the unit ball of $B(H{)}^n$

In this paper we introduce a hyperbolic distance $\delta$ on the noncommutative open ball $[B(H{)}^n{]}_1$, where $B(H)$ is the algebra of all bounded linear operators on a Hilbert space $H$, which is a noncommutative extension of the Poincare-Bergman metric on the open unit ball of $C^n$. We prove that $\delta$ is invariant under the action of the group $Aut([B(H{)}^n{]}_1)$ of all free holomorphic automorphisms of $[B(\text{cH}{)}^n{]}_1$, and show that the $\delta$-topology and the usual operator norm topology coincide on $[B(H{)}^n{]}_1$. Moreover, we prove that $[B(H{)}^n{]}_1$ is a complete metric space with respect to the hyperbolic metric and obtained an explicit formula for $\delta$ in terms of the reconstruction operator. A Schwarz-Pick lemma for bounded free holomorphic functions on $[B(H{)}^n{]}_1$, with respect to the hyperbolic metric, is also obtained.