Operator algebras and operator theory

这一建议涉及对Hilbert空间(算子)上的连续线性映射及其生成的代数(算子代数)的研究。我们正在寻找揭示其内部工作的算子代数的有趣的结构性质。一般说来，我们希望将解析不变量与与代数相关的某些基础对象的代数或组合不变量联系起来。我们计划在我们最近取得的一些显著成功的基础上再接再厉。希望和期待取得重大进展，为学科做出重要贡献。我们最近的工作对这个领域产生了很大的影响，我们处于有利的地位，将继续产生重大的影响。*多变量算子理论试图研究(有限的)(通常是可交换的)算子集。这样一个集合的代数已经很发达了，但分析还处于比较初级的阶段。我们已经为这类集合建立了一个强大的泛函演算，这应该会导致强大的新方法。我们对寻找不变子空间(三角型)很感兴趣。在这些算符集合的泛代数中有一个非常有趣的理想结构，我们试图改进我们前面的分析。这就引出了给定函数在范数受限的小集合上的插补问题。*非交换凸性寻求将经典凸性理论和逼近理论的思想推广到算子背景下。矩阵凸集具有与高维相关的附加结构。Arveson的一个著名猜想提出了一个与近似论有关的非常有趣的问题。我们在对易的背景下回答了这个问题，这导致了经典理论的新发展和更强的逼近结果。我们希望将这一点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。这些通常是通过Cuntz-Kreiger C*-代数来研究的。这里我们研究的是弱闭非自伴算子代数。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被称为超反身性的反身性的量化版本。这已被建立在自由半群代数上，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普遍存在。*我们研究了复球上与簇相关的代数之间的同构问题。我们在同质品种上取得了成功，但在一般情况下，存在许多障碍。我们正在寻找新的几何不变量来提供新的信息。特别是，我们试图证明，如果变元适当地接近，那么它们的乘子代数在空间上是等价的(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).