A new approach to Hilbert \(C^*\)-modules
There is a canonical operator space structure on a \(C^*\)-module. Just as with Hilbert spaces, we can ignore the `inner product' on a \(C^*\)-module when it is convenient, and work only with the metric structure. The inner product is determined by the norm, and may be easily recovered. It is shown that many of the basic constructions with \(C^*\)-modules are familiar operator space constructions. Many apparently new results about \(C^*\)-modules are obtained, some of them unobtainable without the use of operator spaces. We give two new characterizations of \(C^*\)-modules. We also give an Eilenberg-Watts theorem, and a fundamental Morita theorem characterizing strong Morita equivalence of \(C^*\)-algebras in terms of isomorphism of \(C^*\)-module categories. Such theorems had previously been obtained by M. Rieffel for \(W^*\)-algebras. We show that the norm of an element \(u\) of the interior tensor product \(Y\otimes_\theta Z\) of two \(C^*\)-modules equals the Haagerup tensor norm, and also equals the completely bounded (but not the bounded) norm of \(u\) regarded as an operator \(\overline Y\to Z\). Here \(\overline Y\) is the conjugate \(C^*\)-module. These realizations permit one to study categorical \(C^*\)-module notions in a way which is much closer to the traditional `rings and modules' framework. For example, we obtain the basic HOM-tensor relations.
strong Morita equivalence of \(C^*\)-algebras
isomorphism of \(C^*\)-module categories
scientific article; zbMATH DE number 1031531