In this paper, we study the noncommutative balls
where is the joint operator radius for -tuples of bounded linear operators on a Hilbert space. In particular, is the operator norm, is the joint numerical radius, and is the joint spectral radius. We introduce a Harnack type equivalence relation on , and use it to define a hyperbolic distance on the Harnack parts (equivalence classes) of . We prove that the open ball
is the Harnack part containing 0 and obtain a concrete formula for the hyperbolic distance, in terms of the reconstruction operator associated with the right creation operators on the full Fock space with generators. Moreover, we show that the -topology and the usual operator norm topology coincide on . While the open ball is not a complete metric space with respect to the operator norm topology, we prove that it is a complete metric space with respect to the hyperbolic metric . In the particular case when and , the hyperbolic metric coincides with the Poincaré–Bergman distance on the open unit ball of . We introduce a Carathéodory type metric on , the set of all -tuples of operators with joint spectral radius strictly less then 1, by setting
where the supremum is taken over all noncommutative polynomials with matrix-valued coefficients , with and for all . We obtain a concrete formula for in terms of the free pluriharmonic kernel on the noncommutative ball . We also prove that the metric is complete on and its topology coincides with the operator norm topology. We provide mapping theorems, von Neumann inequalities, and Schwarz type lemmas for free holomorphic functions on noncommutative balls, with respect to the hyperbolic metric , the Carathéodory metric , and the joint operator radius .
Cite this article
Gelu Popescu, Hyperbolic geometry on noncommutative balls. Doc. Math. 14 (2009), pp. 595–651DOI 10.4171/DM/283