William Arveson

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.

In a series of papers in the 1960s and 1970s, 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.

[2] In a highly cited paper,[papers 2] Arveson made a systematic study of commutative subspace lattices, which yield a large class of nonselfadjoint operator algebras and proved among other results, the theorem that a transitive algebra containing a maximal abelian von Neumann subalgebra in B(H) must be trivial.

In the late 80's and 90's, Arveson played a leading role in developing the theory of one-parameter semigroups of *-endomorphisms on von Neumann algebras - also known as E-semigroups.

Among his achievements, he introduced product systems and proved that they are complete invariants of E-semigroups up to cocycle conjugacy.