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Let V⊆B(H, K) be a Hilbert C*-module over a C*- algebra A⊆B(H), and X, Y∈V. In this paper we study a problem of finding A∈B(H) such that |X+YB|- |X+YA| is a positive element in A for all B∈A. We show that such an operator exists if and only if the range of Y*X is contained in the range of Y*Y, and in this case it can be chosen to belong to A″. We also consider Hilbert C*-modules in which for every X and Y there is (a unique) A with the above property.
C*-algebra ; Hilbert C*-module ; Best approximation ; Closed range operator
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