Angular derivatives on bounded symmetric domains

Abstract

Publication informationIsrael

A noncommutative version of the Julia-Wolff-Carathéodory theorem

“…Numerous generalizations, to self‐maps of balls or polydisks in , analytic functions with values in spaces of linear operators, analytic self‐maps on domains in spaces of operators or in more general Banach spaces, etc. (see, for example, ; the list is not exhaustive) are known. This paper gives a version of this theorem for noncommutative self‐maps of the noncommutative upper half‐plane of a von Neumann algebra .…”

Bounded pluriharmonic functions on symmetric irreducible Siegel domains

“…We shall mainly follow [23], even though we shall slightly depart from its notation (cf. [27]). We shall consider only finite-dimensional Jordan triple systems and finite-dimensional Jordan algebras.…”

“…Consequently, every positive Hermitian Jordan triple system is induced by a (finitedimensional) 'JB * -triple'. We may therefore apply the results of, e.g., [27]. Definition 3.4.…”

“…Finally, det on Z 1 (e) is the usual (complex) determinant. Thus, π is a 'projection device' (according to [1]) or a 'projective device' (according to [27]). We first need a simple lemma.…”

“…In Section 2, we recall some basic facts on pluriharmonic measures on the Šilov boundary bD of the bounded symmetric domain D. In Section 3, we recall the formalism of Jordan triple systems, since it will be necessary to perform the computations of Sections 5 and 6. In Section 4, we recall a notion of restricted limits introduced in [27]; even though we shall not make use of the results of [27] on (restricted) angular limits and derivatives, we shall need some technical estimates which follow from the construction of a family of canonical 'projective' (or 'projection') devices, which may be of independented interest. Sections 5 and 6 constitute the core of the paper.…”