Noncommutative hyperbolic geometry on the unit ball ofB(H)n

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.,). 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 ✩ Research supported in part by an NSF grant.

Noncommutative Christoffel-Darboux kernels

“…A particularly nice example of a convex set is the unit ball; in the noncommutative case, it is the row ball. Popescu has extensively studied nc function theory on the row ball (see, for example, [28,29,30,31,32]). Davidson and Pitts [18,16,17] have studied the free semigroup algebra and, in doing so, studied nc bounded analytic functions on the row ball (see also [35,36]).…”