symmetric monoidal (∞,1)-category of spectra
Types of quantum field thories
A Jordan–Lie–Banach algebra (or -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 -algebra is a nonassociative Poisson algebra with topology.
JLB-algebras are the outcome of quantization of Poisson algebras. In traditional strict deformation quantization 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 product the commutator (times ). 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 -algebras, the only difference being that the single assocative product is explcitly regarded as inducing the two products of a non-associative Poisson algebra. For more on this separation of the Lie-algebra and the Jordan algebra aspect of quantization see at order-theoretic structure in quantum mechanics.
A JLB-algebra (over the real numbers) consists of a Banach space equipped with two short bilinear operators and , respectively called the Jordan product and the Lie product, satisfying the following identities:
This definition is adapted from Section 1.1 of Halvorson, 1999. Halvorson does not include the statement that the Lie multiplication is short, and it includes a nonnegative real constant factor on the right-hand side of the associator identity (second version). However, Halvorson claims to construct an equivalence between real -algebras and complex -algebras, and this construction produces a short Lie product that satisfies .
While we're at it let's define a -algebra to be a -algebra whose underlying Banach space is equipped with a predual.
The Jordan product and Lie product are respectively the real-symmetrized and imaginary-antisymmetrized parts of an associative operation on the complexification of , defining a complex -algebra; and every -algebra likewise defines a JLB-algebra consisting of its Hermitian elements.
Specifically, starting with a JLB-algebra , we write formally as , on which we define the following operations:
If the Jordan product of the JLB-algebra has an identity , then so does the -algebra:
Conversely, starting with a -algebra , we form the subspace , on which we define the following operations (under each of which is closed):
If the -algebra has an identity, then this is also an identity for the Jordan product (so is also defined by restriction).
This all defines a functor each way between the groupoids of -algebras and -algebras, which in fact (I hope!) form an adjoint equivalence. Since we have a notion of morphism (not just isomorphism) of -algebras, we can transport this along the equivalence to get a notion of morphism of -algebras (which I would expect to be a short linear map that preserves both products) and thus a category equivalent to .
Then real -algebras are equivalent to complex -algebras.
A definition is in section 1.1 of
A brief remark is on p. 80 of