Jordan operator algebra

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 *-antiautomorphism. 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.

JBW

JBW Cars was a British racing car manufacturer in the late 1950s, who were a Formula One constructor from 1959 to 1961.

Brian Naylor, an amateur driver from Stockport, financed his mechanic Fred Wilkinson to build a number of sports cars that were competitive against second-level national fields but were not competitive on the international stage. As well as constructing their own sports racers, JBW also prepared and entered Cooper single-seater racing cars for Naylor to drive in Formula Two and occasional Formula One events.

Formula One

Following two seasons competing in Formula Two races with the JBW Cooper T43 and T45, Brian Naylor decided to enter Formula One proper in 1959. To achieve this Fred Wilkinson constructed a car — the JBW Type 1 — that was strongly influenced by the contemporary Cooper designs, and carried its Maserati 2.5 L (150 cu in) engine behind the driver. The first appearance of the JBW-Maserati was at the pre-season BRDC International Trophy non-Championship race at Silverstone, on 2 May 1959. Naylor arrived too late to post a qualifying time, and retired on lap 41 of the race with gearbox failure. On 18 July Naylor appeared in the JBW car at the 1959 British Grand Prix at Aintree. Although he managed to qualify ahead of such well known drivers as Tony Brooks, Jack Fairman and Ivor Bueb, again the JBW let Naylor down and the car's transmission failed on lap 18. One final outing that year resulted in a failure even to qualify, after Naylor suffered an accident during practice for the Gold Cup race at Oulton Park on 26 September.

JBW (disambiguation)

JBW may refer to:

*-algebra

In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints.

Terminology

*-ring

In mathematics, a *-ring is a ring with a map * : A → A that is an antiautomorphism and an involution.

More precisely, * is required to satisfy the following properties:

for all x, y in A.

This is also called an involutive ring, involutory ring, and ring with involution. Note that the third axiom is actually redundant, because the second and fourth axioms imply 1* is also a multiplicative identity, and identities are unique.

Algebra

Algebra (from Arabic "al-jabr" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the Middle East, by mathematicians such as al-Khwārizmī (780 – 850) and Omar Khayyam (1048–1131).

†-algebra

A †-algebra (or, more explicitly, a †-closed algebra) is the name occasionally used in physics for a finite-dimensional C*-algebra. The dagger, †, is used in the name because physicists typically use the symbol to denote a hermitian adjoint, and are often not worried about the subtleties associated with an infinite number of dimensions. (Mathematicians usually use the asterisk, *, to denote the hermitian adjoint.) †-algebras feature prominently in quantum mechanics, and especially quantum information science.

References