# The Radii Problems for Holomorphic Mappings in J^{*}-algebras

## Abstract

The so-called *J* ^{*}-algebras, introduced by L.A. Harris, are closed subspaces of the space *L*(*H*) of bounded linear operators over a Hilbert space *H* which preserves a kind of Jordan triple product structure. The open unit ball of any *J* ^{*}-algebra is a natural generalization of the open unit disk in the complex plane. In particular, any *C* ^{*}-algebra can be realized as a *J* ^{*}-algebra.

In turn, the problems related to the stability of solutions to automorphic differential equations under perturbations can be often transformed to the geometric function theory study on *J* ^{*}-algebras.

In this direction we solve some radii problems for holomorphically accretive mappings acting in the unit ball of a unital *J* ^{*}-algebra.

We also characterize a class of biholomorphic mappings having a symmetrically- spiral structure.

## Keywords

Holomorphically dissipative mapping spirallike mapping J^{*}-algebra leaf-composed domain

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