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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. 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 $\mathrm{i}/2$). 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. 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 $A$ equipped with two short bilinear operators $(-)\circ(-)$ and $(-)\bullet(-)$, 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 $r$ on the right-hand side of the associator identity (second version). However, Halvorson claims to construct an equivalence between real $JLB$-algebras and complex $C^*$-algebras, and this construction produces a short Lie product that satisfies $r = 1$.
Another consequence of this definition is that the Jordan product makes $A$ into a Jordan algebra (and hence into a JB-algebra).
While we're at it let's define a $JLBW$-algebra to be a $JLB$-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 $A$, defining a complex $C^*$-algebra; and every $C^*$-algebra likewise defines a JLB-algebra consisting of its Hermitian elements.
Specifically, starting with a JLB-algebra $A$, we write $A \oplus A$ formally as $A + \mathrm{i} A$, on which we define the following operations:
If the Jordan product of the JLB-algebra has an identity $1$, then so does the $C^*$-algebra:
Conversely, starting with a $C^*$-algebra $A$, we form the subspace $sa(A) = \{x\colon A \;|\; x^* = x\}$, on which we define the following operations (under each of which $sa(A)$ is closed):
If the $C^*$-algebra has an identity, then this is also an identity for the Jordan product (so $1$ is also defined by restriction).
This all defines a functor each way between the groupoids of $C^*$-algebras and $JLB$-algebras, which in fact (I hope!) form an adjoint equivalence. Since we have a notion of morphism (not just isomorphism) of $C^*$-algebras, we can transport this along the equivalence to get a notion of morphism of $JLB$-algebras (which I would expect to be a short linear map that preserves both products) and thus a category $JLB Alg$ equivalent to $C^* Alg$.
Then real $JLBW$-algebras are equivalent to complex $W^*$-algebras.
A definition is in section 1.1 of
A brief remark is on p. 80 of
Last revised on September 27, 2014 at 09:16:05. See the history of this page for a list of all contributions to it.