Noncommutative hyperbolic geometry on the unit ball of [formula omitted

Abstract

In this paper we introduce a hyperbolic (Poincaré–Bergman type) distance δ on the noncommutative open ball [ B ( H ) n ] 1 : = { ( X 1 , … , X n ) ∈ B ( H ) n : ‖ X 1 X 1 ∗ + ⋯ + X n X 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 R X : = X 1 ∗ ⊗ R 1 + ⋯ + X n ∗ ⊗ R n , X : = ( X 1 , … , X n ) ∈ [ B ( H ) n ] 1 , associated with the right creation operators R 1 , … , R n 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–Bergman distance on the open unit ball B n : = { z = ( z 1 , … , z n ) ∈ C n : ‖ 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 : = ( F 1 , … , F m ) 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.