Projective modules over function algebras

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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.