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On admissible topologies in JB*-triples. (English) Zbl 1327.58007

Summary: Given a complex JB*-triple \(X\), we define and study admissible topologies on \(X\), i.e., locally convex topologies {\(\tau\)} on \(X\) coarser than the norm topology, invariant under the group of surjective linear isometries of \(X\), and such that the triple product is jointly \(\tau\)-\(\tau\)-continuous on bounded subsets of \(X\). As a consequence of the joint \(\tau\)-\(\tau\)-continuity of the triple product, all holomorphic automorphisms \(g\in\mathrm{Aut}(b)\) of the open unit ball \(B\subset X\) are homeomorphisms of \((B,\tau|_B)\) and the natural action \((a,z)\mapsto g_a(z)\) is jointly \(\tau\)-\(\tau\)-continuous on \(B\times B\).

MSC:

58B12 Questions of holomorphy and infinite-dimensional manifolds
46G20 Infinite-dimensional holomorphy
57T15 Homology and cohomology of homogeneous spaces of Lie groups
17C65 Jordan structures on Banach spaces and algebras
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
14M17 Homogeneous spaces and generalizations
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