Abstract
We show that there are many sets in the boundary of a bounded symmetric domain that determine the values and norm of holomorphic functions on the domain having continuous extensions to the boundary. We provide an analogue of the Bergmann-Shilov boundary for finite rank JB*-triples.
Highlights
We show that there are many sets in the boundary of a bounded symmetric domain that determine the values and norm of holomorphic functions on the domain having continuous extensions to the boundary
Recall that an open unit ball B of a complex Banach space Z is homogeneous with respect to biholomorphic mappings if, and only if, Z carries an algebraic structure that renders it a JB∗-triple, defined below [8]
Bounded symmetric domains are the infinite dimensional analogues of the Hermitian symmetric spaces but, by Kaup’s Riemann Mapping Theorem [8], we may alternatively introduce them as those domains in a Banach space which are biholomorphically equivalent to the unit ball of a JB∗-triple
Summary
Finite rank JB∗-triples have maximal tripotents and are reflexive Banach spaces. We recall that if a JB∗-triple has finite rank, the boundary of its unit ball is the disjoint union of holomorphic boundary components defined as follows.
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