nLab
Jordan-Lie-Banach algebra

Idea

A Jordan-Lie-Banach algebra (or JLB-algebra for short) is a topological algebra that behaves like a Poisson algebra, only that the commutative product is not required to form an associative algebra, but just a Jordan algebra. Hence a JLB-algebra is a nonassociative Poisson algebra with topology.

JLB-algebras are the outcome of quantization of Poisson algebras. Often that outcome is regarded to be a non-commutative but associative C-star-algebra. But any such induces a JLB-algebra by letting the Jordan product be the symmetrized product and the Lie bracket the commutator. There is a condition relating the associator of the JLB-algebra to the Lie bracket, that characterizes those JLB-algebras that come from non-commutative associative algebras, and in the usual definition of JLB-algebra this condition is required. In that case JLB-algebras are effectively the same as $C^{*}$-algebras, the only difference being that the single assocative product is explcitly regarded as inducing the two products of a non-associative Poisson algebra.

References

A careful definition is in section 1.1 of

• Hans Halvorson, Maximal Beable Subalgebras of Quantum-Mechanical Observables (pdf)

A brief remark is on p. 80 of

• Sean Bates, Alan Weinstein, Lectures on the geometry of quantization, pdf