**Date and time:**

Abstract: This talk is aimed at a general mathematical audience. We begin by reviewing some mathematical history around some important ideas of von Neumann, and how they fit into the stream of mathematics. We then review some general techniques for dealing with spaces and algebras of Hilbert space operators, that is, with the ‘quantum analogue’ of functions and function spaces and algebras. There will be an emphasis on the theories of operator spaces, positivity, etc. We will also mention two new theories we have been developing: 1). A generalization of the theory of operator algebras called Jordan operator algebras, joint work with Z. Wang and M. Neal. 2). A generalization of the theory of operator positivity, and its use in operator algebras, to more general algebras (eg. Banach algebras, nonselfadjoint operator algebras, etc)