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.

Definitions

JC algebra

A JC algebra is a real subspace of the space of self-adjoint operators on a real or complex Hilbert space, closed under the operator Jordan product ab = ½(ab + ba) and closed in the operator norm.

Jordan Banach algebra

A Jordan Banach algebra is a real Jordan algebra with a norm making it a Banach space and satisfying || ab || ≤ ||a||⋅||b||.

JB algebra

A JB algebra is a Jordan Banach algebra satisfying

\displaystyle{\|a^2\|=\|a\|^2,\,\,\, \|a^2\| \le \|a^2 + b^2\|.}

JB* algebras

A JB* algebra or Jordan C* algebra is a complex Jordan algebra with an involution aa* and a norm making it a Banach space and satisfying

JW algebras

A JW algebra is a Jordan subalgebra of the Jordan algebra of self-adjoint operators on a complex Hilbert space that is closed in the weak operator topology.

JBW algebras

A JBW algebra is a JB algebra that, as a real Banach space, is the dual of a Banach space called its predual.[1] There is an equivalent more technical definition in terms of the continuity properties of the linear functionals in the predual, called normal functionals. This is usually taken as the definition and the abstract characterization as a dual Banach space derived as a consequence.[2]

Properties of JB algebras

Properties of JB* algebras

The definition of JB* algebras was suggested in 1976 by Irving Kaplansky at a lecture in Edinburgh. The real part of a JB* algebra is always a JB algebra. Wright (1977) proved that conversely the complexification of every JB algebra is a JB* algebra. JB* algebras have been used extensively as a framework for studying bounded symmetric domains in infinite dimensions. This generalizes the theory in finite dimensions developed by Max Koecher using the complexification of a Euclidean Jordan algebra. [6]

Properties of JBW algebras

Elementary properties

Comparison of projections

Let M be a JBW factor. The inner automorphisms of M are those generated by the period two automorphisms Q(1 – 2p) where p is a projection. Two projections are equivalent if there is an inner automorphism carrying one onto the other. Given two projections in a factor, one of them is always equivalent to a subprojection of the other. If each is equivalent to a subprojection of the other, they are equivalent.

A JBW factor can be classified into three mutually exclusive types as follows:

Tomita–Takesaki theory permits a further classification of the type III case into types IIIλ (0 ≤ λ ≤ 1) with the additional invariant of an ergodic flow on a Lebesgue space (the "flow of weights") when λ = 0.[10]

Classification of JBW factors of Type I

Classification of JBW factors of Types II and III

The JBW factors not of Type I2 and I3 are all JW factors, i.e. can be realized as Jordan algebras of self-adjoint operators on a Hilbert space closed in the weak operator topology. Every JBW factor not of Type I2 or Type I3 is isomorphic to the self-adjoint part of the fixed point algebra of a period 2 *-antiautomorphism of a von Neumann algebra. In particular each JBW factor is either isomorphic to the self-adjoint part of a von Neumann factor of the same type or to the self-adjoint part of the fixed point algebra of a period 2 *-antiautomorphism of a von Neumann factor of the same type. [12] For hyperfinite factors, the class of von Neumann factors completely classified by Connes and Haagerup, the period 2 *-antiautomorphisms have been classified up to conjugacy in the automorphism group of the factor.[13]

See also

Notes

References