Frames for Hilbert C*-modules

There is growing evidence that Hilbert C*-module theory and the theory of wavelets and Gabor (i.e. Weyl-Heisenberg) frames are tightly related to each other in many aspects. Both the research fields can benefit from achievements of the other field. The goal of the talk given at the mini-workshop was to give an introduction to the theory of module frames and to Hilbert C*modules showing key analogies, and how to overcome the existing obstacles of Hilbert C*-module theory in comparison to Hilbert space theory. The theory of module frames of countably generated Hilbert C*-modules over unital C*algebras was discovered and investigated studying an approach to Hilbert space frame theory by Deguang Han and David R. Larson [7]. Surprisingly, almost all of the concepts and results can be reobtained in the Hilbert C*-module setting. This has been worked out in joint work with D. R. Larson in [4, 5, 6]. Complementary results have been obtained by T. Kajiwara, C. Pinzari and Y. Watatani in [8] using other techniques and motivations. Frames have been also used by D. Bakić and B. Guljaš in [1] calling them quasi-bases. Meanwhile, the case of Hilbert C*modules over non-unital C*-algebras has been investigated by I. Raeburn and S. J. Thompson [14], as well as by D. Bakić and B. Guljaš discovering standard frames even for this class of countably generated Hilbert C*-modules in a well-defined larger multiplier module. However, many problems still have to be solved. How to link core C*-theory to wavelet theory was first observed by M. A. Rieffel in 1997, cf. [15]. His approach has been worked out by J. A. Packer and M. A. Rieffel [12, 13], and by P. J. Wood [16, 17] in great detail. As major results a framework in terms of Hilbert C*modules has been obtained sharing most of the basic structures with generalized multi-resolution analysis for key classes of wavelet and Gabor frames. The Gabor case has been investigated by P. G. Casazza, M. A. Coco and M. C. Lammers [2, 3], and by F. Luef [11] obtaining an adapted to the Gabor situation variant of the Hilbert C*-module approach. In particular, the results by J. A. Packer and M. A. Rieffel in [13] indicate that the described operator algebraic approach to the wavelet theory in L2(R2) is capable to give new deep insights into classical wavelet theory.