Projective modules over function algebras

函数论中最基本的代数之一是在闭单位圆盘上连续而在开单位圆盘上全纯的复变函数的圆盘代数。Carlson等[CCFW]给出了圆盘代数上非零射影的Hilbert模的例子。在回答Halmos的一个著名问题时，Pisier[P]在$A(D)$上构造了一个Hilbert模，它是由一个不同于压缩算子的算子$T$生成的。这个项目的主要目的之一是给出圆盘代数上的Hilbert模是射影的准则。在一条明显不同的发展路线上，Cuntz和Quillen[CQ]引入了拟自由代数这一类。如果非交换微分形式空间是投射的$A$-双模，则代数$A$是拟自由的。一个基本的例子是代数曲线上的坐标函数的代数。建议的PHD项目将研究函数代数上的投影模，以及与Cuntz和Quillen的联系和结果。Cuntz和Quillen的工作还没有被吸收到算子空间理论的文献中，也没有在多变量算子理论中完全实现。

One of the most basic algebras in function theory is the disc algebra $A(D)$ of complex functions that are continuous on the closed unit disc and holomorphic on the open unit disc. Carlson et al [CCFW] gave examples of Hilbert modules over the disc algebra that are nonzero and projective. Answering a well-known question of Halmos, Pisier [P] constructed a Hilbert module over $A(D)$ that is generated by an operator $T$ that is not similar to a contractive operator. One of the main aims of the project is to produce criteria for Hilbert modules over the disc algebra to be projective. In an apparently separate line of development, Cuntz and Quillen [CQ] introduced the class of quasi- free algebras. An algebra $A$ is quasi-free if the space of noncommutative differential forms is a projective $A$-bimodule. A fundamental example is the algebra of coordinate functions on an algebraic curve. The proposed PhD project will investigate projective modules over function algebras and the connection with and the results of Cuntz and Quillen. The work of Cuntz and Quillen has not yet been absorbed into the literature of operator space theory or fully realized in multivariable operator theory.