Operator algebras and operator theory

本文研究希尔伯特空间上的连续线性映射（算子）及其生成的代数（算子代数）。我们正在寻找揭示其内部工作原理的算子代数的有趣的结构性质。一般来说，我们希望将解析不变量与与代数相关的某些底层对象的代数不变量或组合不变量联系起来。我们计划以最近取得的一些显著成就为基础。希望和期望是取得重大进展，并为该学科做出重要贡献。我们最近的工作对该领域产生了很大的影响，我们有能力继续产生重大影响。******多变量算子理论旨在研究一组（通常是可交换的）算子。这种集合的代数已经很发达了，但是分析还处于初级阶段。我们已经为这样的集合建立了一个强大的泛函演算，这将导致强大的新方法。我们感兴趣的是寻找不变的子空间（三角形式）。对于这些算子集，在通用代数中有一个非常有趣的理想结构，我们试图完善我们之前的分析。这就引出了关于给定函数在范数上有约束的小集合上的插值问题。******非交换凸性寻求将经典凸性理论和近似理论的思想推广到算子上下文。矩阵凸集具有与高维相关的附加结构。阿韦森的一个著名猜想提出了一个与近似理论有关的非常有趣的问题。我们已经在交换条件下回答了这个问题，这导致了经典理论的新发展和更强的近似结果。我们希望把它推广到非交换的情况。******与任何有向图相关联，都有几个算子代数。这些通常通过Cuntz-Kreiger C*-代数来研究。这里我们转而研究弱闭非自伴随算子代数。这导致了有趣的结构和许多问题。我们的结果与C*-代数表示的不变量有关。我们目前正在研究一种定量版本的反身性，即超反身性。这已经在自由半群代数中成立了，这是一个只有一个顶点的图的情况，我们有充分的理由相信它将在一般情况下成立。******我们研究了复球上变异代数之间的同构问题。对于同质品种，我们已经取得了成功，但在一般情况下，还有许多障碍。我们正在寻找新的几何不变量来提供新的信息。特别是，我们试图表明，如果变种是适当接近的，那么它们的乘数代数是空间等效的（相似的）。

This proposal concerns the study of continuous linear maps on Hilbert space (operators) and the algebras that they generate (operator algebras). We are seeking interesting structural properties of operator algebras that reveal their inner workings. Generally we wish to relate analytic invariants with algebraic or combinatorial invariants of some underlying object associated to the algebra. We plan to build on some of our notable recent successes. The hope and expectation is to make significant progress, and make important contributions to the discipline. Our recent work has had a strong influence on the field, and we are well positioned to continue to have a significant impact.******Multivariable operator theory seeks to study a (finite) set of (usually commuting) operators. The algebra of such a set is well developed, but the analysis is in a more rudimentary stage. We have established a strong functional calculus for such sets, and this should lead to powerful new methods. We are interested in finding invariant subspaces (triangular forms). There is a very interesting ideal structure in the universal algebra for these sets of operators, and we seek to refine our earlier analysis. This leads to questions about interpolation of given function on some small set with constraints on the norm.******Non-commutative convexity seeks to generalize ideas from classical convexity theory and approximation theory to the operator context. A matrix convex set has additional structure associated to higher dimensions. A famous conjecture of Arveson lays out a very interesting question related to approximation theory. We have answered this question in the commutative setting, which led to new developments in the classical theory and stronger approximation results. We hope to extend this to the non-commutative case.******Associated to any directed graph, there are several operator algebras. These are often studied via the Cuntz-Kreiger C*-algebra. Here we instead study the weakly closed nonself-adjoint operator algebra. This leads to interesting structure and many questions. Our results relate back to invariants for the representations of the C*-algebra. We are currently working on a quantitative version of reflexivity known as hyper-reflexivity. This has been established for free semigroup algebras, which is the case of a graph with one vertex, and we have strong reasons to believe that it will follow in general.******We have studied the problem of isomorphism between algebras associated to varieties on the complex ball. We have been successful for homogeneous varieties, but in the general case, there are many obstacles. We are seeking new geometric invariants that will provide new information. In particular, we are trying to show that if the varieties are suitably close, then their multiplier algebras are spatially equivalent (similar).