Operator valued frames on C*-modules

TL;DR

This paper extends the concept of operator valued frames to sigma-unital C*-algebras on Hilbert C*-modules, establishing their properties, classifications, and relations to projections in the multiplier algebra.

Contribution

It introduces operator valued frames on Hilbert C*-modules for sigma-unital C*-algebras, generalizing previous definitions and providing a detailed classification framework.

Findings

Reformulation of frames in terms of strict topology series

Characterization of frame transform and projection as limits in the multiplier algebra

Classification of frames via Murray-von Neumann equivalence classes

Abstract

Frames on Hilbert C*-modules have been defined for unital C*-algebras by Frank and Larson and operator valued frames on a Hilbert space have been studied in arXiv.0707.3272v1.[math.FA]. Goal of the present paper is to introduce operator valued frames on a Hilbert C*-module for a sigma-unital C*-algebra. Theorem 1.4 reformulates the definition given by Frank and Larson in terms of a series of rank-one operators converging in the strict topology. Theorem 2.2. shows that the frame transform and the frame projection of an operator valued frame are limits in the strict topology of a series of elements in the multiplier algebra and hence belong to it. Theorem 3.3 shows that two operator valued frames are right similar if and only if they share the same frame projection. Theorem 3.4 establishes a one to one correspondence between Murray-von Neumann equivalence classes of projections in the multiplier algebra and right similarity equivalence classes of operator valued frames and provides a parametrization of all Parseval operator-valued frames on a given Hilbert C*-module. Left similarity is then defined and Proposition 3.9 establishes when two left unitarily equivalent frames are also right unitarily equivalent.