William Arveson
Encyclopedia
William Arveson was a mathematician
specializing in operator algebra
s 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 operator
s 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.
In a highly cited paper, 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 masa 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 semigoups 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
specializing in operator algebra
Operator algebra
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...
s who worked as a professor of Mathematics at the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
. Arveson obtained his Ph. D. from UCLA
University of California, Los Angeles
The University of California, Los Angeles is a public research university located in the Westwood neighborhood of Los Angeles, California, USA. It was founded in 1919 as the "Southern Branch" of the University of California and is the second oldest of the ten campuses...
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
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
with values in the algebra of all bounded operator
Bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
s on a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
. This theorem led naturally to the question of injectivity of von-Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
algebras in general, which culminated in work by Alain Connes
Alain Connes
Alain Connes is a French mathematician, currently Professor at the Collège de France, IHÉS, The Ohio State University and Vanderbilt University.-Work:...
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
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
including the Shilov and Choquet boundaries and used them very successfully in single operator theory.
In a highly cited paper, 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 masa 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 semigoups 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.