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\section*{Jordan-Banach algebra}

\hypertarget{jordanbanach_algebras}{}\section*{{Jordan--Banach algebras}}\label{jordanbanach_algebras}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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

\newline | associative [[Banach algebra|Banach]] $*$-algebra | Jordan--Banach algebra | [[Jordan–Lie–Banach algebra]] | | $C^*$-[[C-star-algebra|algebra]] | $JB$-algebra | $JLB$-[[JLB-algebra|algebra]] | | [[von Neumann algebra]] | $JBW$-algebra | $JLBW$-[[JLBW-algebra|algebra]] |

Just as a [[Jordan algebra]] that happens to be associative is the same thing as an [[associative algebra|associative]] $*$-[[star-algebra|algebra]] with trivial [[involution]] (aka simply an associative algebra) that happens to be [[commutative algebra|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 $C^*$-[[C-star-algebra|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 \emph{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).

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{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]].

\begin{defn}
\label{}\hypertarget{}{}
The Banach space $A$ becomes a \textbf{Jordan--Banach algebra} if it is equipped with a [[binary operation]] $A \times A \to A$, called the \emph{Jordan multiplication} and often written infix as $\circ$, satisfying these identities:

\begin{itemize}%
\item [[short map|shortness]]: ${\|x \circ y\|} \leq {\|x\|} {\|y\|}$,
\item [[bilinear map|bilinearity]]: $(a x + y) \circ (b u + v) = a b (x \circ u) + a (x \circ v) + b (y \circ u) + y \circ v$,
\item commutativity: $x \circ y = y \circ x$,
\item Jordan identity: $(x \circ y) \circ (x \circ x) = x \circ (y \circ (x \circ x))$.

\end{itemize}
\end{defn}
For motivation of the first two, see [[Banach algebra]]; for motivation of the last three, see [[Jordan algebra]].

\begin{defn}
\label{}\hypertarget{}{}
The Jordan--Banach algebra $A$ is \textbf{unital} if the Jordan multiplication has an [[identity element]] in its unit ball, usually denoted $1$:

\begin{itemize}%
\item $1 \circ x = x$,
\item $x \circ 1 = x$,
\item ${\|1\|} \leq 1$.

\end{itemize}
\end{defn}
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 ring|trivial algebra]], and this should be allowed.

\begin{defn}
\label{}\hypertarget{}{}
The Jordan--Banach algebra $A$ is a \textbf{$JB$-algebra} if it satisfies these identities:

\begin{itemize}%
\item \textbf{$B$-identity}: ${\|x \circ x\|} = {\|x\|^2}$,
\item \emph{[[formally real algebra|positivity]]}: ${\|x \circ x\|} \leq {\|x \circ x + y \circ y\|}$.

\end{itemize}
\end{defn}
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 $C^*$-[[C-star-algebra|algebras]].) Conversely, given shortness of multiplication (or even of squaring), these two identities may be combined into the single inequality

\begin{displaymath}
{\|x\|}^2 \leq {\|x \circ x + y \circ y\|} .
\end{displaymath}
Also, positivity implies that $A$ is [[formally real algebra|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-associative algebra|power-associativity]] (which it implies regardless) in a direct definition of $JB$-algebra.

\begin{defn}
\label{}\hypertarget{}{}
The $JB$-algebra $A$ is a \textbf{$JBW$-algebra} if it is unital and if its [[underlying]] Banach space has a [[predual]] $A_*$.

\end{defn}
See [[von Neumann algebra]] for motivation of the predual. (Is it unique? Should be!)

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{[[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 \textbf{unital} if $T(1) = 1$.

\end{defn}
To get the right notion of [[isomorphism]], the [[morphisms]] in the [[category]] of Jordan--Banach algebras should be the \emph{[[short linear map|short]]} homomorphisms (see \href{Ban#morphisms}{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.

\begin{defn}
\label{}\hypertarget{}{}
Given a Jordan--Banach algebra $A$ and a [[Hilbert space]] $H$, a \textbf{[[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 \textbf{[[faithful representation|faithful]]} if it's an [[injective function]].

\end{defn}
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 \textbf{[[special Jordan algebra|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).

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{$JC$-algebra} is $JB$-algebra that has a faithful representation on a Hilbert space. A \textbf{$JW$-algebra} is a $JBW$-algebra that's also a $JC$-algebra.

\end{defn}
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 \emph{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 \textbf{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.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The main example of a $JB$-algebra is the algebra of [[self-adjoint operators]] in a $C^*$-[[C-star-algebra|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 ring|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 algebra|formally real]]; conversely, all of the formally real Jordan algebras in [[finite-dimensional space|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 \href{Jordan+algebra#frc}{Jordan algebra\#frc} for the classification of the finite-dimensional formally real Jordan algebras.)

\hypertarget{properties}{}\subsection*{{Properties}}\label{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 algebra|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 function|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 algebra|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.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[JBW-algebraic quantum mechanics]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Basic stuff is at

\begin{itemize}%
\item \href{https://en.wikipedia.org/wiki/Jordan_operator_algebra}{Jordan operator algebra} on the English Wikipedia;

\end{itemize}
and most of that appears to be from

\begin{itemize}%
\item Harald Hanche-Olsen and Erling St\o{}rmer (1984); \emph{Jordan operator algebras}; Monographs and Studies in Mathematics 21 (Pitman); \href{http://www.math.ntnu.no/~hanche/joa/}{web},

\end{itemize}
which I have only begun to read.

There is also

\begin{itemize}%
\item Harald Upmeier (1987); \emph{Jordan Algebras in Analysis, Operator Theory, and Quantum Mechanics}; CBMS Regional Conference Series in Mathematics 67 (AMS),

\end{itemize}
of which a lot is already on pages 1--4 (the only ones that Google Books would show me).

\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[Jordan algebra]]
\item \textbf{Jordan--Banach algebra}
\item [[Jordan–Lie–Banach algebra]]

\end{itemize}
[[!redirects Jordan Banach algebra]] [[!redirects Jordan Banach algebras]] [[!redirects Jordan-Banach algebra]] [[!redirects Jordan-Banach algebras]] [[!redirects Jordan–Banach algebra]] [[!redirects Jordan–Banach algebras]] [[!redirects Jordan--Banach algebra]] [[!redirects Jordan--Banach algebras]]

[[!redirects JB-algebra]] [[!redirects JB-algebras]] [[!redirects JB algebra]] [[!redirects JB algebras]]

[[!redirects JBW-algebra]] [[!redirects JBW-algebras]] [[!redirects JBW algebra]] [[!redirects JBW algebras]]

[[!redirects JC-algebra]] [[!redirects JC-algebras]] [[!redirects JC algebra]] [[!redirects JC algebras]]

[[!redirects JW-algebra]] [[!redirects JW-algebras]] [[!redirects JW algebra]] [[!redirects JW algebras]]

[[!redirects B-identity]]



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