Grant: Discovery Grants Program - Individual

I am working on George A. Elliott's inductive limit problem for the Fourier transform automorphism f of the irrational rotation C*-algebra A (which depends on an irrational parameter theta). (If U and V are the canonical generators of A satisfying the unitary Heisenberg commutation relation, then f(U) = V and f(V) = U*.)  It proved to be a difficult problem during my many attempts to solve it. However, I was able to solve the AF aspect of the problem (during the tenure of my last NSERC grant). Namely, that the fixed point subalgebra of A under the Fourier transform is an AF algebra when the parameter theta is irrational number belonging to a dense G-delta set (see my enclosed Crelle's Journal paper).  Later on, in joint work with Wolfgang Lueck and N. Christopher Phillips, we have been able to obtain this AF result for all irrationals [15] (which partly used my earlier work in [11]).  (This AF aspect of the research program was mentioned in my last NSERC research proposal back in 1999, and it is now solve.) (more specifically on C*-subalgebras that are direct sums of two circle algebras and two matrix algebras). This gives good evidence for an affirmative answer to the problem. - There are two other canonical automorphism of A that are not well understood, namely of orders 3 and 6 (which I call the "cubic" and "hexic" transforms).  I have recruited one of my very bright undergraduate students (Julian Buck) to join me to undertake to study these. Thanks to NSERC's awarding him two NSERC USRA's, we were able to complete two papers on this direction (see [2] and [14]), one of which has recently been accepted for publication in the Journal of Operator Theory.  We hope eventually to address and establish the inductive limit structure of these automorphisms also.  The AF problem for these is still open, although it is possible that the methods of Lueck-Phillips-Walters [15] may work to show this.