Abstract. In this paper we continue the study of free holomorphic functions on the noncommutative ball [B(H) n n]1: = (X1,..., Xn) ∈ B(H) n: ‖X1X ∗ 1 + · · · + XnX ∗ n‖1/2 o < 1, where B(H) is the algebra of all bounded linear operators on a Hilbert space H, and n = 1, 2,... or n = ∞. Several classical results from complex analysis have free analogues in our noncommutative setting. We prove a maximum principle, a Naimark type representation theorem, and a Vitali convergence theorem, for free holomorphic functions with operator-valued coefficients. We introduce the class of free holomorphic functions with the radial infimum property and study it in connection with factorizations and noncommutative generalizations of some classical inequalities obtained by Schwarz and Harnack. The Borel-Carathéodory theorem is extended to our noncommutative setting. Using a noncommutative generalization of Schwarz’s lemma and basic facts concerning the free holomorphic automorphisms of the noncommutative ball [B(H) n]1, we obtain an analogue of Julia’s lemma for free holomorphic functions F: [B(H) n]1 → [B(H) m]1. We also obtain Pick-Julia theorems for free holomorphic functions with operator-valued coefficients. We provide a noncommutative generalization of a classical inequality due to Lindelöf, which turns out to be sharper then the noncommutative von Neumann inequality. Finally, we introduce a pseudohyperbolic metric on [B(H) n]1 which is invariant under the action of the free holomorphic automorphism group of [B(H) n]1 and turns out to be a noncommutative extension of the pseudohyperbolic distance on Bn, the open unit ball of C n. In this setting, we obtain a Schwarz-Pick type lemma. We also provide commutative versions of these results for operator-valued multipliers of the Drury-Arveson space.

Let E be a W ∗-correspondence over a von Neumann algebra M and let H ∞ (E) be the associated Hardy algebra. If σ is a faithful normal representation of M on a Hilbert space H, then one may form the dual correspondence E σ and represent elements in H ∞ (E) as B(H)valued functions on the unit ball D(E σ) ∗. The functions that one obtains are called Schur class functions and may be characterized in terms of certain Pick-like kernels. We study these functions and relate them to system matrices and transfer functions from systems theory. We use the information gained to describe the automorphism group

Introduction Our object is to study domains which are the region of negative definiteness of an operator-valued Hermitian form defined on a space of operators and to investigate the biholomorphic linear fractional transformations between them. This is a unified setting in which to consider operator balls, operator half-planes, strictly J-contractive operators, strictly J-dissipative operators, etc., and the biholomorphic images of these domains under linear fractional transformations. Our approach is close in spirit to that of Potapov [28], Krein and Smuljan [27] and Smuljan [33]. At the same time, because we consider subspaces of operators, our circular domains include the matrix balls which E. Cartan [6] obtained as the classical bounded symmetric domains and they include the Siegel domains of genus 2 and 3 which Pyatetskii-Shapiro [29] associates with these domains as well as the infinite dimensional analogues of both types of domains given in [18] and [19]. Thus

A ternary ring ofoperators is an “off-diagonal corner ” of a C∗-algebra and the predual ofa ternary ring ofoperators (ifit exists) is ofthe form pR∗q for some von Neumann algebra R and projections p and q in R. In this paper, we prove that a subspace ofthe predual ofa ternary ring ofoperators is completely 1-complemented ifand only ifit is completely isometrically isomorphic to the predual ofsome ternary ring ofoperators. We next give an operator space characterization ofthe preduals ofseparable injective von Neumann algebras. Finally, we prove some concrete results about the finite dimensional completely 1-complemented subspaces ofa von Neumann algebra predual. 1. Introduction. Douglas proved that the 1-complemented subspaces of L1(X) are just the subspaces which are isometric to L1(Y) for some measure space Y (see [D]).

Abstract. We introduce a version of Voiculescu-Brown approximation entropy for isometric automorphisms of Banach spaces and develop within this framework the connection between dynamics and the local theory of Banach spaces as discovered by Glasner and Weiss. Our fundamental result concerning this contractive approximation entropy, or CA entropy, characterizes the occurrence of positive values both geometrically and topologically. This leads to various applications; for example, we obtain a geometric description of the topological Pinsker factor and show that a C ∗-algebra is type I if and only if every multiplier inner ∗-automorphism has zero CA entropy. We also examine the behaviour of CA entropy under various product constructions and determine its value in many examples, including isometric automorphisms of ℓp for 1 ≤ p ≤ ∞ and noncommutative tensor product shifts. 1.

Abstract. The tetrablock is shown to be inhomogeneous and its automorphism group is determined. A type of Schwarz lemma for the tetrablock is proved. The action of the automorphism group is described in terms of a certain natural foliation by complex geodesic discs. 1.

We define the domain of a linear fractional transformation in a space of operators and show that both the affine automorphisms and the compositions of symmetries act transitively on these domains. Further, we show that Liouville's theorem holds for domains of linear fractional transformations and, with an additional trace class condition, so does the Riemann removable singularities theorem. We also show that every biholomorphic mapping of the operator domain I ! Z Z is a linear isometry when the space of operators is a complex Jordan subalgebra of L(H) with the removable singularity property and that every biholomorphic mapping of the operator domain I + Z 1 Z 1 ! Z 2 Z 2 is a linear map obtained by multiplication on the left and right by J-unitary and unitary operators, respectively. 0. Introduction. This paper introduces a large class of finite and infinite dimensional symmetric affinely homogeneous domains which are not holomorphically equivalent to any bounded domain. Thes...

Abstract. Recent results of Davidson-Paulsen-Raghupathi-Singh give necessary and sufficient conditions for the existence of a solution to the Nevanlinna-Pick interpolation problem on the unit disk with the additional restriction that the interpolant should have the value of its derivative at the origin equal to zero. This concrete mild generalization of the classical problem is prototypical of a number of other generalized Nevanlinna-Pick interpolation problems which have appeared in the literature (for example, on a finitely-connected planar domain or on the polydisk). We extend the results of Davidson-Paulsen-Raghupathi-Singh to the setting where the interpolant is allowed to be matrixvalued and elaborate further on the analogy with the theory of Nevanlinna-Pick interpolation on a finitely-connected planar domain.

We give a new and clever proof of the Russo–Dye theorem for JB ∗-algebras, which depends on certain recent tools due to the present author. The proof given here is quite different from the known proof by J. D. M. Wright and M. A. Youngson. The approach adapted here is motivated by the corresponding C ∗-algebra results due to L. T. Gardner, R. V. Kadison and G. K. Pedersen. Accordingly, it yields more precise information. Incidentally, we obtain an alternate proof of Russo–Dye Theorem for C ∗-algebras. A couple of further results due to Kadison and Pedersen have been extended to JB ∗-algebras as

Let X and Y be right, full, Hilbert C ∗-modules over the algebras A and B respectively and let T: X → Y be a linear surjective isometry. Then T can be extended to an isometry of the linking algebras. T then is a sum of two maps: a (bi-)module map (which is completely isometric and preserves the inner product) and a map that reverses the (bi-)module actions. If A (or B) is a factor von Neumann algebra then every isometry T: X → Y is either a (bi-)module map or reverses the (bi-)module actions. 1 Supported by Technion V.P.R. Fund–Steigman Research Fund, Technion V.P.R. Fund–Fund for the Promotion of Sponsored Reserach and the Fund for the Promotion of Research at the