# Banach-Jordan algebra

Jordan–Banach algebra

A Jordan algebra over the field of real or complex numbers, endowed with a complete norm $|.|$ satisfying

\begin{equation*} \| x \circ y \| \leq \| x \| \| y \| \end{equation*}

for all $x$, $y$ in the algebra. Since an (associative) Banach algebra is a Banach–Jordan algebra under the Jordan product $x \circ y : = ( x y + y x ) / 2$, the theory of Banach–Jordan algebras can be regarded as a generalization of that of Banach algebras. For forerunners in this last theory, see Banach algebra and [a5]. Pioneering papers on Banach–Jordan algebras are [a4], [a19] and [a13]. A relatively complete panoramic view of the results on Banach–Jordan algebras can be obtained by combining [a16], [a3] and [a7].

Spectral methods in Banach–Jordan algebras have been possible thanks to the concept of invertible element in a Jordan algebra with a unit, introduced by N. Jacobson and K. McCrimmon (see [a12] or Jordan algebra). From this concept, the spectrum $\operatorname { sp } ( J , x )$ of an arbitrary element $x$ of a Banach–Jordan algebra $J$ is defined as in the associative case, and the spectral radius formula $\operatorname { lim } \{ \| x ^ { n } \| ^ { 1 / n } \} = \operatorname { max } \{ | \lambda | : \lambda \in \operatorname { sp } ( J , x ) \}$ holds. In fact, Banach–Jordan algebras are "locally spectrally" associative. This means that each element in such an algebra $J$ can be imbedded in some closed associative subalgebra $J ^ { \prime }$ of $J$ satisfying $\operatorname { sp } ( J , x ) = \operatorname { sp } ( J ^ { \prime } , x )$ for every $x \in J ^ { \prime }$. Then, for a single element in a complex Banach–Jordan algebra, a holomorphic functional calculus follows easily.

A Jordan algebra is said to be semi-simple (or semi-primitive, as preferred by people working in pure algebra) whenever its Jacobson-type radical [a11] is zero (cf. also Jacobson radical). Refining spectral methods, B. Aupetit [a2] gave a Jacobson-representation-theory-free proof of Johnson's uniqueness-of-norm theorem for semi-simple Banach algebras, and extended the result to semi-simple Banach–Jordan algebras. The absence of representation theory in Aupetit's proof was relevant because, although semi-simple Jordan algebras can be expressed as subdirect products of Jordan algebras which are "primitive" (in a peculiar Jordan sense), primitive Jordan algebras were not well-understood at that time. Aupetit's methods have shown also useful in extending from Banach algebras to Banach–Jordan algebras many other relevant results (see again [a3]), as well as in obtaining a general non-associative variant of Johnson's theorem [a15]. Recently, using work of E.I. Zel'manov [a22] on Jordan algebras without any finiteness condition, primitive Banach–Jordan algebras have been described in detail [a8]. Such a description has allowed one to extend to Banach–Jordan algebras the Johnson–Sinclair theorem, stating that derivations on semi-simple Banach algebras (cf. also Derivation in a ring) are automatically continuous [a18].

$\operatorname{JB}$-algebras are defined as the real Banach–Jordan algebras $J$ satisfying $\| x \| ^ { 2 } \leq \| x ^ { 2 } + y ^ { 2 } \|$ for all $x , y \in J$. The basic theory of $\operatorname{JB}$-algebras, originally due to E.M. Alfsen, F.W. Shultz and E. Stormer [a1], is fully treated in [a10]. If $A$ is a $C ^ { * }$-algebra, then the self-adjoint part $A _ { \text{sa} }$ of $A$ is a $\operatorname{JB}$-algebra under the Jordan product. Closed subalgebras of $A _ { \text{sa} }$, for some $C ^ { * }$-algebra $A$, become relevant examples of $\operatorname{JB}$-algebras, and are called $\operatorname {JC}$-algebras. Through the consideration of $\text{JBW}$-algebras (i.e., $\operatorname{JB}$-algebras that are dual Banach spaces, cf. also Banach space), $\text{JBW}$-factors (i.e., prime $\text{JBW}$-algebras), and factor representations of a given $\operatorname{JB}$-algebra $J$ (i.e., $w ^ { * }$-dense range homomorphisms from $J$ to $\text{JBW}$-factors), the knowledge of arbitrary $\operatorname{JB}$-algebras is reasonably reduced to that of $\operatorname {JC}$-algebras and the exceptional $\operatorname{JB}$-algebra $H _ { 3 } ( \text{O} )$ of all Hermitian $( 3 \times 3 )$-matrices over the alternative division algebra $\mathbf{O}$ of real octonions.

$\operatorname{JB} ^ { * }$-algebras are defined as complex Banach–Jordan algebras $J$ endowed with a conjugate-linear algebra involution $*$ satisfying $\| U _ { X } ( x ^ { * } ) \| = \| x \| ^ { 3 }$ for every $x \in J$. Here, for $x \in J$, $U _ { x }$ denotes the operator on $J$ defined by $U _ { x } ( y ) := 2 x \circ ( x \circ y ) - x ^ { 2 } \circ y$ for every $y \in J$. Every $C ^ { * }$-algebra becomes a $\operatorname{JB} ^ { * }$-algebra under its Jordan product. $\operatorname{JB} ^ { * }$-algebras are closely related to $\operatorname{JB}$-algebras. Indeed, $\operatorname{JB}$-algebras are nothing but the self-adjoint parts of $\operatorname{JB} ^ { * }$-algebras [a20]. The one-to-one categorical correspondence between $\operatorname{JB}$-algebras and $\operatorname{JB} ^ { * }$-algebras derived from the above result completely reduces the $*$-theory of $\operatorname{JB} ^ { * }$-algebras to the theory of $\operatorname{JB}$-algebras. However, $\operatorname{JB} ^ { * }$-algebras are of interest on their own, mainly due to their connection with complex analysis (see [a6], [a17], and [a21]). Using Zel'manov's prime theorem, the structure theory of $\operatorname{JB}$- and $\operatorname{JB} ^ { * }$-algebras can be refined as follows (see [a9]). A $\operatorname{JB} ^ { * }$-algebra $J$ is primitive if and only if it is of one of the following types:

$J$ is the unique $\operatorname{JB} ^ { * }$-algebra whose self-adjoint part is $H _ { 3 } ( \text{O} )$.

There exists a complex Hilbert space $( H , ( \cdot | \cdot ) )$ of dimension $\geq 3$, with a conjugation $\sigma$ and a $\sigma$-invariant norm-one element $1$, such that $J = H$ as complex vector spaces, whereas the product $\circ$, the involution $*$, and the norm $|.|$ of $J$ are given by

\begin{equation*} x \circ y : = ( x | 1 ) y + ( y | 1 ) x - ( x | \sigma ( y ) ) 1, \end{equation*}

\begin{equation*} x ^ { * } : = 2 ( 1 | x ) 1 - \sigma ( x ) , \| x \| ^ { 2 } : = ( x | x ) + ( ( x | x ) ^ { 2 } - | ( x | \sigma ( x ) ) | ^ { 2 } ) ^ { 1 / 2 }, \end{equation*}

respectively.

There exists a primitive $C ^ { * }$-algebra $A$ such that $J$ is a closed self-adjoint Jordan subalgebra of the $C ^ { * }$-algebra $M ( A )$, of multipliers of $A$, containing $A$.

There exists a primitive $C ^ { * }$-algebra with a $*$-involution $\tau$ such that $J$ is a closed self-adjoint Jordan subalgebra of $M ( A )$ contained in the $\tau$-Hermitian part of $M ( A )$ and containing the $\tau$-Hermitian part of $A$.

From the point of view of analysis, the Jordan identity $x \circ ( y \circ x ^ { 2 } ) = ( x \circ y ) \circ x ^ { 2 }$ (which, together with the commutativity, is characteristic of Jordan algebras) can be regarded as a theorem instead of as an axiom. Indeed, if a unital complete normed non-associative complex algebra $A$ is subjected to the geometric conditions that, through the Vidav–Palmer theorem, characterize $C ^ { * }$-algebras in the associative setting, then $A$ under the product $x \circ y : = ( x y + y x ) / 2$ and a suitable involution becomes a $\operatorname{JB} ^ { * }$-algebra [a14].