ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 65,   2   (1996)
pp.   161-169

ADJOINTABILITY OF OPERATORS ON HILBERT $C^*$-MODULES
V. M. MANUILOV

Abstract.  Can a functional $f\in H^*_A = \Hom_A(H_A;A)$ on the non-self-dual Hilbert module $H_A$ over a $C^*$-algebra $A$ be represented as an operator of some inner product by an element of the module $H_A$, this inner product being equivalent to the given one? We discuss this question and prove that for some classes of $C^*$-algebras the closure with respect to the given norm of unification of such functionals for all equivalent inner products coincides with the dual module $H^*_A$. We discuss the notion of compactness of operators in relation to representability of functionals. We also show how an operator on $H_A$ in some situations (e.g. if it is Fredholm) can be made adjointable by change of the inner product to an equivalent one.

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