Mappings Preserving Submodules of Hilbert C*-Modules |
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Authors: | Magajna Bojan |
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Affiliation: | Department of Mathematics, University of Ljubljana Jadranska 19, Ljubljana 1000, Slovenia. E-mail: Bojan.Magajna{at}fmf.uni-lj.si |
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Abstract: | A Hilbert module over a C*-algebra B is a right B-module X,equipped with an inner product ·, · which is linearover B in the second factor, such that X is a Banach space withthe norm ||x||:=||x, x||1/2. (We refer to [8] for the basictheory of Hilbert modules; the basic example for us will beX=B with the inner product x, y=x*y.) We denote by B(X) thealgebra of all bounded linear operators on X, and we denoteby L(X) the C*-algebra of all adjointable operators. (In thebasic example X=B, L(X) is just the multiplier algebra of B.)Let A be a C*-subalgebra of L(X), so that X is an A-B-bimodule.We always assume that A is nondegenerate in the sense that [AX]=X,where [AX] denotes the closed linear span of AX. Denote by AX the algebra of all mappings on X of the form (1.1) where m is an integer and aiA, biB for all i. Mappings of form(1.1) will be called elementary, and this paper is concernedwith the question of which mappings on X can be approximatedby elementary mappings in the point norm topology. |
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