Description of morphisms from a Hilbert module over a \(C^*\)-algebra into this algebra (Q5942015)

A Hilbert module over a \(C^*\)-algebra \(A\) is a left \(A\)-module which is a Hilbert space and \(\langle ax,y\rangle =\langle x,a^*y\rangle\), \(a\in A\), \(x, y\in H\). Further information on (projective) Hilbert modules may be found in \textit{A. Yu Khelemskij} [Commun. Algebra 26, 977-996 (1998; Zbl 0910.46056)]. Note that there are several different meanings of ``Hilbert module'' in the literature (some references are \textit{P. S. Muhly} and \textit{B. Solel} [Mem. Am. Math. Soc. 559 (1995; Zbl 0857.46031)]) and \textit{E. C. Lance} [Hilbert \(C^*\)-modules. A toolkit for operator algebraists, Cambridge Univ. Press (1995; Zbl 0822.46080)]. It is shown that each left Hilbert \(A\)-module \(H\) is isometrically isomorphic to a module of the form \((\sum_{p\in \Sigma}H_p\otimes K_p)\oplus H_\circ\), where the \(H_p\) are left \(A\)-modules of the form \(Ap\), the \(K_p\) are arbitrary Hilbert spaces, and the module \(H_\circ\) contains no submodule isometrically isomorphic to \(H_p\) for any \(p\in \Sigma\). Here \(\Sigma\) denotes the set of equivalence classes of all elementary projections of the algebra \(A\) in the Murray-von Neumann sense. Recall that a projection \(p\) of \(A\) is said to be elementary whenever \(pAp\) is one-dimensional. The authors completely describe arbitrary module morphisms from a left Hilbert module \(H\) over a \(C^*\)-algebra \(A\) into the naturally defined \(A\)-module \(A\).