Inductive limit structure of automorphisms

我正在研究无理旋转C*-代数A的傅里叶变换自同构f的George A.Elliott的归纳极限问题(它依赖于无理参数theta)。(如果U和V是满足酉海森堡对易关系的A的正则生成元，则f(U)=V，f(V)=U*.)在我的多次尝试中，它被证明是一个困难的问题。然而，我能够解决这个问题的AF方面(在我最后一次NSERC拨款的任期内)。也就是说，当参数theta是属于稠密G-Delta集的无理数时，A在傅立叶变换下的不动点子代数是AF代数(参见我所附的Crelle‘s Journal论文)。后来，在与Wolfgang Lueck和N.Christopher Phillips的合作下，我们已经能够得到所有无理数的这个AF结果[15](这部分使用了我在[11]中的早期工作)。(这个研究计划的AF方面在我早在1999年的NSERC研究提案中被提到，现在已经解决了。)(更具体地说，关于由两个圆代数和两个矩阵代数组成的C*-子代数)。这为这个问题的肯定答案提供了很好的证据。-还有另外两个A的正则自同构还没有被很好地理解，即3阶和6阶(我称之为“三次”和“十六进制”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)和我一起承担研究这些。由于NSERC授予他两个NSERC USRA，我们能够完成关于这一方向的两篇论文(参见[2]和[14])，其中一篇最近已被接受发表在算子理论杂志上。我们希望最终也能解决并建立这些自同构的感应极限结构。尽管Lueck-Phillips-Walters[15]的方法可能会证明这一点，但这些自同构的AF问题仍然悬而未决。

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.