Noncommutative Hyperbolic Geometry on the Unit Ball of B(h)

In this paper we introduce a hyperbolic (Poincaré-Bergman type) distance δ on the noncommutative open ball [B(H)]1 := n (X1, . . . ,Xn) ∈ B(H) n : ‖X1X ∗ 1 + · · ·+XnX ∗ n‖ 1/2 < 1 o , 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)]1, i.e., δ(Ψ(X),Ψ(Y )) = δ(X, Y ), X, Y ∈ [B(H)]1, for all Ψ ∈ Aut([B(H)]1). Moreover, we show that the δ-topology and the usual operator norm topology coincide on [B(H)]1. While the open ball [B(H)]1 is not a complete metric space with respect to the operator norm topology, we prove that [B(H)]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) ]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 PoincaréBergman distance on the open unit ball Bn := {z = (z1, . . . , zn) ∈ C n : ‖z‖2 < 1}. We obtain a Schwarz-Pick lemma for free holomorphic functions on [B(H)]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)]1. As consequences, we show that the Carathéodory and the Kobayashi distances, with respect to δ, coincide with δ on [B(H)]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. Introduction Poincaré’s discovery of a conformally invariant metric on the open unit disc D := {z ∈ C : |z| < 1} of the complex plane was a cornerstone in the development of complex function theory. The hyperbolic (Poincaré) distance is defined on D by δP (z, w) := tanh −1 ∣