A survey of quantale theory
by
Pedro Resende
Instituto Superior Técnico

Quantales are complete lattices equipped with an associative multiplication that distributes over arbitrary joins in each variable. They are meant to provide a "quantization" of the notion of locale, or pointfree space, by means of which one should be able to define a category of "spaces" dual to the category of not necessarily commutative C*-algebras, generalizing the duality that in the commutative case arises from the Gelfand-Naimark representation theorem. While so far this goal has been achieved only partially, this talk aims at describing some of the main developments in quantale theory, from the point of view both of its purely algebraic aspects and of how it is, or may become, related to other areas, such as operator algebras, noncommutative geometry, and computer science.

Date received: February 28, 2003