**William B. Arveson** (22 November 1934 – 15 November 2011) was a mathematician specializing in operator algebras who worked as a professor of Mathematics at the University of California, Berkeley.^{ [1] }

Arveson obtained his Ph.D. from UCLA in 1964 with thesis advisor Henry Dye and thesis *Prediction theory and group representations*.

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.^{ [papers 1] } 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 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 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.

- Books

- Arveson, William (1976),
*An Invitation to C*-Algebras*, Graduate Texts in Mathematics, New York: Springer-Verlag, ISBN 0387901760 - Arveson, William (1984),
*Ten Lectures On Operator Algebras*, CBMS Regional Conference Series in Mathematics, New York: American Mathematical Society, ISBN 0821807056 - Arveson, William (2002),
*A Short Course on Spectral Theory*, Graduate Texts in Mathematics, New York: Springer-Verlag, ISBN 0387953000 - Arveson, William (2003),
*Noncommutative dynamics and E-semigroups*, Springer Monographs in Mathematics, New York: Springer-Verlag, ISBN 0-387-00151-4, MR 1978577

- Papers

- ↑ Arveson, William B. (1969), "Subalgebras of C*-algebras",
*Acta Mathematica*,**123**: 141–224, doi: 10.1007/bf02392388 , MR 0253059 - ↑ Arveson, William (1974), "Operator algebras and invariant subspaces",
*Annals of Mathematics*, Second Series,**100**(3): 433–532, doi:10.2307/1970956, JSTOR 1970956, MR 0365167

In mathematics, specifically in functional analysis, a **C ^{∗}-algebra** is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra

In mathematics, a **von Neumann algebra** or **W*-algebra** is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.

**Noncommutative geometry** (**NCG**) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of *spaces* that are locally presented by noncommutative algebras of functions. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions.

In functional analysis, a discipline within mathematics, given a C*-algebra *A*, the **Gelfand–Naimark–Segal construction** establishes a correspondence between cyclic *-representations of *A* and certain linear functionals on *A*. The correspondence is shown by an explicit construction of the *-representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal.

In functional analysis, a branch of mathematics, an **operator algebra** is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.

In mathematics, an **amenable group** is a locally compact topological group *G* carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive invariant measure on subsets of *G*, was introduced by John von Neumann in 1929 under the German name "messbar" in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "*mean*".

**Free probability** is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of independence, and it is connected with free products. This theory was initiated by Dan Voiculescu around 1986 in order to attack the free group factors isomorphism problem, an important unsolved problem in the theory of operator algebras. Given a free group on some number of generators, we can consider the von Neumann algebra generated by the group algebra, which is a type II_{1} factor. The isomorphism problem asks whether these are isomorphic for different numbers of generators. It is not even known if any two free group factors are isomorphic. This is similar to Tarski's free group problem, which asks whether two different non-abelian finitely generated free groups have the same elementary theory.

In the field of mathematics known as functional analysis, the **invariant subspace problem** is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. Many variants of the problem have been solved, by restricting the class of bounded operators considered or by specifying a particular class of Banach spaces. The problem is still open for separable Hilbert spaces.

In mathematics, **operator theory** is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis.

In abstract algebra, a **Jordan algebra** is a nonassociative algebra over a field whose multiplication satisfies the following axioms:

- .

In the theory of von Neumann algebras, a part of the mathematical field of functional analysis, **Tomita–Takesaki theory** is a method for constructing **modular automorphisms** of von Neumann algebras from the polar decomposition of a certain involution. It is essential for the theory of type III factors, and has led to a good structure theory for these previously intractable objects.

In conformal field theory and representation theory, a **W-algebra** is an associative algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov, and the name "W-algebra" comes from the fact that Zamolodchikov used the letter W for one of the elements of one of his examples.

In mathematics, the **Chevalley–Shephard–Todd theorem** in invariant theory of finite groups states that the ring of invariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group is generated by pseudoreflections. In the case of subgroups of the complex general linear group the theorem was first proved by G. C. Shephard and J. A. Todd (1954) who gave a case-by-case proof. Claude Chevalley (1955) soon afterwards gave a uniform proof. It has been extended to finite linear groups over an arbitrary field in the non-modular case by Jean-Pierre Serre.

In mathematics, a **commutation theorem** explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace. The first such result was proved by Francis Joseph Murray and John von Neumann in the 1930s and applies to the von Neumann algebra generated by a discrete group or by the dynamical system associated with a measurable transformation preserving a probability measure. Another important application is in the theory of unitary representations of unimodular locally compact groups, where the theory has been applied to the regular representation and other closely related representations. In particular this framework led to an abstract version of the Plancherel theorem for unimodular locally compact groups due to Irving Segal and Forrest Stinespring and an abstract Plancherel theorem for spherical functions associated with a Gelfand pair due to Roger Godement. Their work was put in final form in the 1950s by Jacques Dixmier as part of the theory of **Hilbert algebras**. It was not until the late 1960s, prompted partly by results in algebraic quantum field theory and quantum statistical mechanics due to the school of Rudolf Haag, that the more general non-tracial Tomita–Takesaki theory was developed, heralding a new era in the theory of von Neumann algebras.

In mathematics, a **nuclear C*-algebra** is a C*-algebra *A* such that the injective and projective C*-cross norms on *A*⊗*B* are the same for every C*-algebra *B*. This property was first studied by Takesaki (1964) under the name "Property T", which is not related to Kazhdan's property T.

A **Schröder–Bernstein property** is any mathematical property that matches the following pattern

In mathematics, **Jordan operator algebras** are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are real numbers, the algebras are called **Jordan Banach algebras**. The theory has been extensively developed only for the subclass of **JB algebras**. The axioms for these algebras were devised by Alfsen, Schultz & Størmer (1978). Those that can be realised concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product and the operator norm are called **JC algebras**. The axioms for complex Jordan operator algebras, first suggested by Irving Kaplansky in 1976, require an involution and are called **JB* algebras** or **Jordan C* algebras**. By analogy with the abstract characterisation of von Neumann algebras as C* algebras for which the underlying Banach space is the dual of another, there is a corresponding definition of **JBW algebras**. Those that can be realised using ultraweakly closed Jordan algebras of self-adjoint operators with the operator Jordan product are called **JW algebras**. The JBW algebras with trivial center, so-called **JBW factors**, are classified in terms of von Neumann factors: apart from the exceptional 27 dimensional Albert algebra and the **spin factors**, all other JBW factors are isomorphic either to the self-adjoint part of a von Neumann factor or to its fixed point algebra under a period two *-anti-automorphism. Jordan operator algebras have been applied in quantum mechanics and in complex geometry, where Koecher's description of bounded symmetric domains using Jordan algebras has been extended to infinite dimensions.

In mathematics, **semi-simplicity** is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A **semi-simple object** is one that can be decomposed into a sum of *simple* objects, and simple objects are those that do not contain non-trivial proper sub-objects. The precise definitions of these words depends on the context.

- ↑ "William Arveson Death Notice: William Arveson's Obituary by the San Francisco Chronicle". Legacy.com. Retrieved 2012-04-26.
- ↑ Davidson, Kenneth R. (2012), "The mathematical legacy of William Arveson" (PDF),
*Journal of Operator Theory*,**68**(2): 101–127, arXiv: 1209.6294 , Bibcode:2012arXiv1209.6294D

- Markiewicz, Daniel; Jorgensen, Palle E. T. (August 2015), "William B. Arveson: A Tribute" (PDF),
*Notices of the American Mathematical Society*, Providence, RI: American Mathematical Society,**62**(7): 761–769, doi: 10.1090/noti1264 - Directory of operator algebraists
- Bill Arveson's homepage
- William Barnes Arveson at the Mathematics Genealogy Project

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