Noncommutative hyperbolic geometry on the unit ball of B(H)n ✩

In this paper we introduce a hyperbolic (Poincaré–Bergman type) distance δ on the noncommutative open ball B(H)n 1 := (X1, . . . , Xn) ∈ B(H)n: X1X∗1 +···+XnX∗n 1/2 < 1 , where B(H) is the algebra of all bounded linear operators on a Hilbert space H. It is proved that δ is invariant under the action of the free holomorphic automorphism group of [B(H)n]1, i.e., δ Ψ(X),Ψ(Y) = δ(X,Y), X,Y ∈ B(H)n 1, for all Ψ ∈ Aut([B(H)n]1). Moreover, we show that the δ-topology and the usual operator norm topology coincide on [B(H)n]1. While the open ball [B(H)n]1 is not a complete metric space with respect to the operator norm topology, we prove that [B(H)n]1 is a complete metric space with respect to the hyperbolic metric δ. We obtain an explicit formula for δ in terms of the reconstruction operator RX := X∗1 ⊗R1 +···+X∗n ⊗Rn, X:= (X1, . . . , Xn) ∈ B(H)n 1, associated with the right creation operators R1, . . . , Rn on the full Fock space with n generators. In the particular case when H= C, we show that the hyperbolic distance δ coincides with the Poincaré–Bergmandistance on the open unit ball Bn := z = (z1, . . . , zn) ∈ Cn: z 2 < 1 . We obtain a Schwarz–Pick lemma for free holomorphic functions on [B(H)n]1 with respect to the hyperbolic metric, i.e., if F := (F1, . . . , Fm) is a contractive ( F ∞ 1) free holomorphic function, then δ F(X),F(Y) δ(X,Y), X,Y ∈ B(H)n 1. As consequences, we show that the Carathéodory and the Kobayashi distances, with respect to δ, coincide with δ on [B(H)n]1. The results of this paper are presented in the more general context of Harnack parts of the closed ball [B(H)n]−1 , which are noncommutative analogues of the Gleason parts of the Gelfand spectrum of a function algebra. © 2009 Elsevier Inc. All rights reserved.