William Arveson

Who was William Arveson?

William Arveson was a mathematician specializing in operator algebras who worked as a professor of Mathematics at the University of California, Berkeley. Arveson obtained his Ph. D. from UCLA in 1964.

Of particular note is Arveson's work on completely positive maps. One of his earlier results in this area is an extension theorem for completely positive maps with values in the algebra of all bounded operators on a Hilbert space. This theorem led naturally to the question of injectivity of von-Neumann algebras in general, which culminated in work by Alain Connes relating injectivity to hyperfiniteness.

One of the major features of Arveson's work was the use of algebras of operators to elucidate single operator theory. In a series of papers in the 60's and 70's, Arveson introduced noncommutative analogues of several concepts from classical harmonic analysis including the Shilov and Choquet boundaries and used them very successfully in single operator theory.