Manifolds of algebraic elements in JB*-triples

Given a complex Hilbert space H, we study the differential geometry of the manifold A of normal algebraic elements in the algebra Z=L(H) of bounded linear operators on H. We represent A as a disjoint union of connected subsets M of Z. Using the algebraic structure of Z, a torsionfree affine connection $\nabla$ (that is invariant under the group G=Aut (Z) of automorphisms of Z) is defined on each of these connected components and the geodesics are computed. In case M consists of elements that have a fixed finite rank r, G-invariant Riemann and Kähler structures are defined on M, which in this way becomes a totally geodesic symmetric holomorphic manifold. Similar results are established for the manifold of algebraic elements in an abstract JB*-triple.

Keywords: JB*-triples, Grassmann manifolds, Riemann manifolds.

AMS 2000 Subject Classification: 48G20, 72H51.

algelem.dvi (80K)
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Jose M. Isidro < jmisidro@zmat.usc.es >