nLab Jordan-Banach algebra

Idea

Jordan–Banach algebras, $JB$-algebras, and the like fill out the following grand analogy:

associative $*$-algebra	Jordan algebra	Jordan–Lie algebra
associative Banach $*$-algebra	Jordan–Banach algebra	Jordan–Lie–Banach algebra
$C^*$-algebra	$JB$-algebra	$JLB$-algebra
von Neumann algebra	$JBW$-algebra	$JLBW$-algebra

Just as a Jordan algebra that happens to be associative is the same thing as an associative $*$-algebra with trivial involution (aka simply an associative algebra) that happens to be commutative, the analogous result holds in the lower rows.

One can also consider Jordan $*$-algebras and the like, but the interesting thing is that important results about $C^*$-algebras have analogues already for $JB$-algebras. Instead of an involution, we can add a compatible Lie algebra structure to a Jordan algebra; then even without assuming associativity or commutativity, a Jordan–Lie algebra over the real numbers is the same thing as an associative $*$-algebra over the complex numbers, up to equivalence of categories, and this extends to lower rows.

The right column is discussed at Jordan–Lie–Banach algebra; here we discuss the middle column (assuming the top row and left column as known).

Definitions

Let $A$ be a Banach space, typically over the real numbers, but potentially over any topological field (or possibly even more general). We will generally assume the real numbers, and some theorems may rely on this, or at least on the divisibility of $2$ in the ground field.

Definition

The Banach space $A$ becomes a Jordan–Banach algebra if it is equipped with a binary operation $A \times A \to A$, called the Jordan multiplication and often written infix as $\circ$, satisfying these identities:

• shortness: ${\|x \circ y\|} \leq {\|x\|} {\|y\|}$,
• bilinearity: $(a x + y) \circ (b u + v) = a b (x \circ u) + a (x \circ v) + b (y \circ u) + y \circ v$,
• commutativity: $x \circ y = y \circ x$,
• Jordan identity: $(x \circ y) \circ (x \circ x) = x \circ (y \circ (x \circ x))$.

For motivation of the first two, see Banach algebra; for motivation of the last three, see Jordan algebra.

Definition

The Jordan–Banach algebra $A$ is unital if the Jordan multiplication has an identity element in its unit ball, usually denoted $1$:

• $1 \circ x = x$,
• $x \circ 1 = x$,
• ${\|1\|} \leq 1$.

People might state the last clause as ${\|1\|} = 1$, which follows (using shortness of the multiplication) from the existence of any $x \ne 0$. However, ${\|1\|} = 0$ in the trivial algebra, and this should be allowed.

Definition

The Jordan–Banach algebra $A$ is a $JB$-algebra if it satisfies these identities:

• $B$-identity: ${\|x \circ x\|} = {\|x\|^2}$,
• positivity: ${\|x \circ x\|} \leq {\|x \circ x + y \circ y\|}$.

Shortness of the multiplication follows from the $B$-identity (via the polarization identities and the triangle identity), so it may be left out of a direct definition of $JB$-algebras; the same goes for the norm of $1$ in the unital case. (Compare the analogous results for $C^*$-algebras.) Conversely, given shortness of multiplication (or even of squaring), these two identities may be combined into the single inequality

$${\|x\|}^2 \leq {\|x \circ x + y \circ y\|} .$$

Also, positivity implies that $A$ is formally real: if $x^2 + y^2 = 0$, then $x, y = 0$ (and so on for any number of terms). Given this, the Jordan identity is equivalent to power-associativity (which it implies regardless) in a direct definition of $JB$-algebra.

Definition

The $JB$-algebra $A$ is a $JBW$-algebra if it is unital and if its underlying Banach space has a predual $A_*$.

See von Neumann algebra for motivation of the predual. As with $C^*$-algebras, the predual (if it exists) is unique up to unique isomorphism.

Definition

A homomorphism from a Jordan–Banach algebra $A$ to a Jordan–Banach algebra $B$ is a bounded linear map $T\colon A \to B$ of Banach spaces (everywhere defined) such that $T(x \circ y) = T(x) \circ T(y)$ always holds. If $A$ and $B$ are unital, then the homomorphism is unital if $T(1) = 1$.

To get the right notion of isomorphism, the morphisms in the category of Jordan–Banach algebras should be the short homomorphisms (see Ban#morphisms for discussion). However, if $A$ and $B$ are $JB$-algebras, then every homomorphism is short (as with $C^*$-algebras); in fact, we do not even have to assume that $T$ is bounded. Similarly, a morphism of unital Jordan–Banach algebras should be unital.

Definition

Given a Jordan–Banach algebra $A$ and a Hilbert space $H$, a representation $\pi$ of $A$ on $H$ is a homomorphism from $A$ to the algebra of Hermitian operators on $H$ (which is a $JB$-algebra, in fact a $JBW$-algebra, under the symmetrized product, as described in the examples). Such a representation is faithful if it's an injective function.

Again, we should really look only at the short representations (which is automatic for a $JB$-algebra) and look especially at the unital representations of a unital Jordan–Banach algebra.

Recall that a Jordan algebra is special if it has a faithful representation on a vector space (which follows if it has any injective homomorphism to the Jordanization of any associative algebra).

Definition

A $JC$-algebra is $JB$-algebra that has a faithful representation on a Hilbert space. A $JW$-algebra is a $JBW$-algebra that's also a $JC$-algebra.

This follows if the $JB$-algebra has any injective homomorphism to the algebra of Hermitian operators in any $C^*$-algebra, by the theorem that every abstract $C^*$-algebra may be made concrete. But this theorem does not itself have an analogue for $JB$-algebras; the Albert algebra $\mathfrak{h}_3(\mathbb{O})$ is a $JB$-algebra (even a $JBW$-algebra) that is not a $JC$-algebra.

This is probably unfortunate terminology (compare ‘$B^*$-algebra’ vs ‘$C^*$-algebra’ and ‘$W^*$-algebra’ vs ‘von Neumann algebra’); it would probably be better just to call such $JB$-algebras special (but somebody might think that this just means that the underlying Jordan algebra is special, which is weaker). We do need some term, however, thanks to the Albert algebra.

Examples

The main example of a $JB$-algebra is the algebra of self-adjoint operators in a $C^*$-algebra. For a $JBW$-algebra, try the algebra of self-adjoint operators in a von Neumann algebra. In particular, the algebra of Hermitian operators on a Hilbert space is a $JBW$-algebra, in fact a $JW$-algebra. Still more particularly, the trivial algebra (which is the algebra of self-adjoint operators on the zero Hilbert space) is a $JW$-algebra (although it won't fit definitions by authors who require ${\|1\|} = 1$).

Every $JB$-algebra is formally real; conversely, all of the formally real Jordan algebras in finite dimensions are $JBW$-algebras, and all of the special ones are $JW$-algebras. This leaves the Albert algebra $\mathfrak{h}_3(O)$ as the basic example of a $JBW$-algebra that is not a $JW$-algebra. (See Jordan algebra#frc for the classification of the finite-dimensional formally real Jordan algebras.)

Properties

$JB$-algebras have nice properties like those of $C^*$-algebras, and $JBW$-algebras have nicer properties like those of von Neumann algebras. They are generally proved in the analogous ways. Some properties are different, however.

An associative $JB$-algebra is the same thing as a commutative $C^*$-algebra with trivial involution, which (over the real numbers) is in turn the same thing as the algebra of (real-valued) continuous maps vanishing at infinity on a local compactum (which is a compactum iff the algebra is unital, and then every continuous map vanishes at infinity).

Like any Jordan algebra, a $JB$-algebra $A$ is power-associative, so each element $x$ generates an associative (and of course commutative) subalgebra? and hence a local compactum. In a unital $JB$-algebra, each element generates an associative unital subalgebra and hence a compactum $Spec(x)$. Any continuous map $f\colon Spec(x) \to \mathbb{R}$ therefore defines an element $f(x)$ of $A$. More generally, any associative unital subalgebra $X$ generates a compactum $Spec(X)$, its spectrum, and any continuous map on $Spec(X)$ defines an element of $A$ (in fact belonging to $X$). Thus we have a functional calculus on $JB$-algebras. In a $JBW$-algebra, we may instead interpret the spectrum as a localizable measure space, with $X$ identified as the algebra of essentially bounded measurable functions (modulo almost equality) on the spectrum, so that the functional calculus extends to measurable functions.

Since every $JB$-algebra $A$ is formally real, it comes equipped with a partial order: $x \leq y$ iff $y - x$ is a sum of squares.

The order-theoretic structure in quantum mechanics fixes the $JB$-algebra structure of a $C^*$-algebra, but not the $JLB$-algebra structure.

Related concepts

• JBW-algebraic quantum mechanics

References

Basic stuff is at

• Jordan operator algebra on the English Wikipedia;

and most of that appears to be from

• Harald Hanche-Olsen and Erling Størmer (1984); Jordan operator algebras; Monographs and Studies in Mathematics 21 (Pitman); web,

which I have only begun to read.

There is also

• Harald Upmeier (1987); Jordan Algebras in Analysis, Operator Theory, and Quantum Mechanics; CBMS Regional Conference Series in Mathematics 67 (AMS),

of which a lot is already on pages 1–4 (the only ones that Google Books would show me).

Related pages

• Jordan algebra
• Jordan–Banach algebra
• Jordan–Lie–Banach algebra