Noncommutative hyperbolic geometry on the unit ball of Bn(H)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:‖X1X1∗+⋯+XnXn∗‖1/2<1}, where B(H)B(H) is the algebra of all bounded linear operators on a Hilbert space HH. It is proved that δ   is invariant under the action of the free holomorphic automorphism group of [Bn(H)]1[B(H)n]1, i.e.,δ(Ψ(X),Ψ(Y))=δ(X,Y),X,Y∈[B(H)n]1, for all Ψ∈Aut([Bn(H)]1)Ψ∈Aut([B(H)n]1). Moreover, we show that the δ  -topology and the usual operator norm topology coincide on [Bn(H)]1[B(H)n]1. While the open ball [Bn(H)]1[B(H)n]1 is not a complete metric space with respect to the operator norm topology, we prove that [Bn(H)]1[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 operatorRX:=X1∗⊗R1+⋯+Xn∗⊗Rn,X:=(X1,…,Xn)∈[B(H)n]1, associated with the right creation operators R1,…,RnR1,…,Rn on the full Fock space with n   generators. In the particular case when H=CH=C, we show that the hyperbolic distance δ coincides with the Poincaré–Bergman distance on the open unit ballBn:={z=(z1,…,zn)∈Cn:‖z‖2<1}. We obtain a Schwarz–Pick lemma for free holomorphic functions on [Bn(H)]1[B(H)n]1 with respect to the hyperbolic metric, i.e., if F:=(F1,…,Fm)F:=(F1,…,Fm) is a contractive (‖F‖∞⩽1‖F‖∞⩽1) free holomorphic function, thenδ(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 [Bn(H)]1[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.