# nLab Jordan-Lie-Banach algebra

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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. 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.

## Definition

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:

• Jordan commutativity: $x \circ y = y \circ x$;
• Lie anticommutativity: $x \bullet x = 0$, or equivalently (given bilinearity) $x \bullet y = -y \bullet x$;
• the Jacobi identity (Lie self-derivation): $x \bullet (y \bullet z) = (x \bullet y) \bullet z + y \bullet (x \bullet z)$, or equivalently (given anticommutativity) $x \bullet (y \bullet z) + y \bullet (z \bullet x) + z \bullet (x \bullet y) = 0$;
• Jordan derivation: $x \bullet (y \circ z) = (x \bullet y) \circ z + y \circ (x \bullet z)$;
• the associator identity: $x \circ (y \circ z) = (x \circ y) \circ z + y \bullet (x \bullet z)$, or equivalently (given anticommutativity) $(x \circ y) \circ z - x \circ (y \circ z) = (x \bullet z) \bullet y$;
• the $B$-identity: ${\|x \circ x\|} = {\|x\|^2}$ (compare the $B^*$-identity or $C^*$-identity of a $C^*$-algebra);
• positivity: ${\|x \circ x \|} \leq {\|x \circ x + y \circ y\|}$.

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.

## Equivalence to $C^*$-algebras

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:

• norm: ${\|a + \mathrm{i} b\|} \coloneqq \sqrt{{\|a\|^2} + {\|b\|^2}}$,
• addition: $(a + \mathrm{i} b) + (c + \mathrm{i} d) \coloneqq (a + c) + \mathrm{i} (b + d)$,
• opposite: $-(a + \mathrm{i} b) \coloneqq (-a) + \mathrm{i} (-b)$,
• zero: $0 \coloneqq 0 + \mathrm{i} 0$,
• scalar multiplication: $(x + \mathrm{i} y) (a + \mathrm{i} b) = (x a - y b) + \mathrm{i} (x b + y a)$ (for $x + \mathrm{i} y$ a complex number),
• involution: $(a + \mathrm{i} b)^* \coloneqq a + \mathrm{i} (-b)$,
• multiplication: $(a + \mathrm{i} b) (c + \mathrm{i} d) \coloneqq (a \circ c + a \bullet d + b \bullet c - b \circ d) + \mathrm{i} (-a \bullet c + a \circ d + b \circ c + b \bullet d)$.

If the Jordan product of the JLB-algebra has an identity $1$, then so does the $C^*$-algebra:

• $1 \coloneqq 1 + \mathrm{i} 0$.

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):

• norm, addition, opposite, zero, scalar multiplication (by real numbers only): by restriction,
• Jordan product: $a \circ b \coloneqq \frac{1}{2} a b + \frac{1}{2} b a$,
• Lie product: $a \bullet b \coloneqq \frac{1}{2} \mathrm{i} a b - \frac{1}{2} \mathrm{i} b a$.

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.

## References

A definition is in section 1.1 of

A brief remark is on p. 80 of

Revised on September 27, 2014 09:16:05 by Toby Bartels (98.19.41.214)