Frames in Hilbert C*-Modules and Morita Equivalent C*-Algebras

Abstract

We show that the property of a C*-algebra that all its Hilbert modules have a frame, in the case of σ-unital C*-algebras, is preserved under Rieffel–Morita equivalence. In particular, we show that a σ-unital continuous-trace C*-algebra with trivial Dixmier–Douady class, all of whose Hilbert modules admit a frame, has discrete spectrum. We also show this for the tensor product of any commutative C*-algebra with the C*-algebra of compact operators on any Hilbert space.

Topological and frame properties of certain pathological $C^*$-algebras

“…Hence, by proving that the criterion is not satisfied for some module, we, as a consequence, show that there is no standard frame for this module (which, by the way, does not mean that there is no outer standard frame in sense of [3], see [10,Remark 3.3]). In [11], [9], [12], [13] there is also a searching for modules without frames, more precisely, for algebras A such that it is possible to say surely is there an A-module without frames or not.…”

Frame-Less Hilbert C*-Modules

“…Hilbert A-module with no standard frame. Moreover, if two C * -algebras A and B are Morita equivalent and A is σ -unital, then the property of A that every Hilbert A-module admits a standard frame inherits to B, cf [2,. Theorem 2.4].…”