Boundary structure of bounded symmetric domains.

*(English)*Zbl 0981.32012Let \(D\) be a bounded symmetric domain with a base point \(o\) realized as the open unit ball in a complex Banach space \(E\). Given \(a\in D\) denote by \(s_a\) the symmetry of \(D\) which has \(a\) as the isolated fixed point. The transformation of the form \(s_as_o\) is called a transvection. It is known that each \(a\in D\) corresponds to the unique transvection \(g_a\) satisfying \(g_a(o)=a\).

The main theorem states: For any point \(c\in\partial D\), the limits \(s_c:=\lim_{a\to c}s_a\) and \(g_c:=\lim_{a\to c}g_a\) exist as holomorphic mappings \(D\to E\) and the convergence is locally uniform on \(D\). Moreover, the mappings \(D\times\overline D\to\overline D\) defined by \((z,a)\to s_a(z)\) and \((z,a)\to g_a(z)\) extend to continuous mappings \(\overline D\times\overline D\setminus (\partial D\times\partial D)\to\overline D\).

The existence of these limits depends on the realization of \(D\). The proof uses the fine structure of JB\({}^*\)-triples (it is known that a complex Banach space is a JB\({}^*\)-triple if and only if its unit ball is a bounded symmetric domain [see W. Kaup, Math. Z. 183, 503-529 (1983; Zbl 0519.32024)], particularly the Gelfand-Naimark theorem for JB*-triples [Y. Friedman and B. Russo, Duke Math. J. 53, 139-148 (1986; Zbl 0637.46049)]; it is added in proof that the result can be obtained without using this theorem according to some result of O. Loos). The paper also contains the description of those boundary components of \(D\) which pass through tripotents of \(E\).

The main theorem states: For any point \(c\in\partial D\), the limits \(s_c:=\lim_{a\to c}s_a\) and \(g_c:=\lim_{a\to c}g_a\) exist as holomorphic mappings \(D\to E\) and the convergence is locally uniform on \(D\). Moreover, the mappings \(D\times\overline D\to\overline D\) defined by \((z,a)\to s_a(z)\) and \((z,a)\to g_a(z)\) extend to continuous mappings \(\overline D\times\overline D\setminus (\partial D\times\partial D)\to\overline D\).

The existence of these limits depends on the realization of \(D\). The proof uses the fine structure of JB\({}^*\)-triples (it is known that a complex Banach space is a JB\({}^*\)-triple if and only if its unit ball is a bounded symmetric domain [see W. Kaup, Math. Z. 183, 503-529 (1983; Zbl 0519.32024)], particularly the Gelfand-Naimark theorem for JB*-triples [Y. Friedman and B. Russo, Duke Math. J. 53, 139-148 (1986; Zbl 0637.46049)]; it is added in proof that the result can be obtained without using this theorem according to some result of O. Loos). The paper also contains the description of those boundary components of \(D\) which pass through tripotents of \(E\).

Reviewer: V.Gichev (Omsk)

##### MSC:

32H40 | Boundary regularity of mappings in several complex variables |

32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |

58C10 | Holomorphic maps on manifolds |

17C65 | Jordan structures on Banach spaces and algebras |

17C27 | Idempotents, Peirce decompositions |