Classifying Hilbert bundles. II |
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Authors: | Maurice J Dupré |
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Institution: | Department of Mathematics, Tulane University, New Orleans, Louisiana 70118 USA |
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Abstract: | A Hilbert bundle (p, B, X) is a type of fibre space p: B → X such that each fibre p?1(x) is a Hilbert space. However, p?1(x) may vary in dimension as x varies in X, even when X is connected. We give two “homotopy” type classification theorems for Hilbert bundles having primarily finite dimensional fibres. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle over (p, B, X) such that the dimension of p?1(x) is m for x in A and n otherwise. As a special case, we show that if X is a compact metric space, C+X the upper cone of the suspension SX, then the isomorphism classes of (m, n)-bundles over (SX, C+X) are in one-to-one correspondence with the members of X, Vm(n)] where Vm(n) is the Stiefel manifold. The results are all applicable to the classification of separable, continuous trace C1-algebras, with specific results given to illustrate. |
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