Preprint: "Wavelet representations and Fock space on positive matrices" by P.E.T. Jorgensen and D.W. Kribs

[Brian Treadway (treadway@math.uiowa.edu)]

Title: Wavelet representations and Fock space on positive matrices

Authors: P.E.T. Jorgensen, D.W. Kribs

We show that every biorthogonal wavelet determines a representation by operators on Hilbert space satisfying simple identities, which captures the established relationship between orthogonal wavelets and Cuntz-algebra representations in that special case. Each of these representations is shown to have tractable finite-dimensional co-invariant doubly-cyclic subspaces. Further, motivated by these representations, we introduce a general Fock-space Hilbert space construction which yields creation operators containing the Cuntz-Toeplitz isometries as a special case.

In this paper, we wish to establish a connection between biorthogonal wavelets on the one hand [I. Daubechies, Ten Lectures on Wavelets, in CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics, Philadelphia, 1992] and representation theory for operators on Hilbert space on the other [O. Bratteli and P. Jorgensen, Endomorphisms of B(H) II, Journal of Functional Analysis 145 (1997), 323-373 http://www.math.uiowa.edu/ftp/jorgen/endomorphisms_B_H_2.ps.gz], [K. Davidson, D. Kribs, and M. Shpigel, Isometric dilations of non-commuting finite rank n-tuples, Canadian Journal of Mathematics 53 (2001), 506-545]. This is accomplished by showing that each of these wavelets yields a collection of operators acting on Hilbert space which satisfy simple identities, and which contain the Cuntz relations [J. Cuntz, Simple C*-algebras generated by isometries, Communications in Mathematical Physics 57 (1977), 173-185] as a special case. In fact, this new relationship collapses to the now well-known connection between orthogonal wavelets and representations of the Cuntz C*-algebra in that special case [O. Bratteli and P. Jorgensen, Wavelet filters and infinite-dimensional unitary groups, in Proceedings of the International Conference on Wavelet Analysis and Applications (Guangzhou, China, 1999) (D. Deng et al., eds.), International Press, Boston, 2001, pp. 35-64 http://arxiv.org/abs/math.FA/0001171]. Our second goal is to develop a framework for studying this new class of representations. Toward this end, we introduce a general Fock space Hilbert space construction which reduces to unrestricted Fock space in the familiar cases. Indeed, the natural creation operators we get can be thought of as an analogue of the Cuntz-Toeplitz creation operators to this more general setting. We regard this construction and the creation operators determined by it as interesting objects of study in their own right. Finally, our hope is that this paper will lead to further study of the relationships and objects introduced here.

Preprint at:
http://arxiv.org/abs/math.CA/0204034