Inductive limit structure of automorphisms

我正在研究George A. Elliott的归纳极限问题关于无理性旋转C*代数A的傅里叶变换自同构f（它依赖于无理性参数）(如果U和V是A满足酉Heisenberg交换关系的正则生成子，则f（U) = V, f(V) = U*。）在我多次尝试解决它的过程中，它被证明是一个难题。然而，我能够解决问题的AF方面（在我最后一次获得NSERC资助的任期内）。也就是说，A在傅里叶变换下的不动点子代数是AF代数，当参数是无理数，属于密集的g集合（参见我所附的Crelle's Journal论文）。后来，在与Wolfgang Lueck和N. Christopher Phillips的联合工作中，我们已经能够获得所有无理数b[15]的AF结果（部分使用了我在[11]的早期工作）。（我在1999年的NSERC研究计划中提到了这个研究项目的AF方面，现在已经解决了。）（更具体地说，C*-子代数是两个圆代数和两个矩阵代数的直接和）。这为这个问题的肯定答案提供了很好的证据。- A还有另外两个正则自同构，即3阶和6阶的自同构（我称之为“三次”和“六次”变换）。我已经招募了我的一个非常聪明的本科生（朱利安·巴克）来和我一起研究这些。感谢NSERC授予他两个NSERC USRA，我们能够在这个方向上完成两篇论文（见[2]和[14]），其中一篇最近被《算子理论杂志》接受发表。我们希望最终也能解决并建立这些自同构的归纳极限结构。尽管Lueck-Phillips-Walters b[15]的方法可能会证明这一点，但这些的AF问题仍然是开放的。

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.