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''Jordan–Banach algebra''

A [[Jordan algebra|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|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|Banach algebra]] and [[#References|[a5]]]. Pioneering papers on Banach–Jordan algebras are [[#References|[a4]]], [[#References|[a19]]] and [[#References|[a13]]]. A relatively complete panoramic view of the results on Banach–Jordan algebras can be obtained by combining [[#References|[a16]]], [[#References|[a3]]] and [[#References|[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 [[#References|[a12]]] or [[Jordan algebra|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 [[#References|[a11]]] is zero (cf. also [[Jacobson radical|Jacobson radical]]). Refining spectral methods, B. Aupetit [[#References|[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 [[#References|[a3]]]), as well as in obtaining a general non-associative variant of Johnson's theorem [[#References|[a15]]]. Recently, using work of E.I. Zel'manov [[#References|[a22]]] on Jordan algebras without any finiteness condition, primitive Banach–Jordan algebras have been described in detail [[#References|[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|Derivation in a ring]]) are automatically continuous [[#References|[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 [[#References|[a1]]], is fully treated in [[#References|[a10]]]. If $A$ is a [[C*-algebra|$C ^ { * }$-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|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|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 [[#References|[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 [[#References|[a6]]], [[#References|[a17]]], and [[#References|[a21]]]). Using Zel'manov's prime theorem, the structure theory of $\operatorname{JB}$- and $\operatorname{JB} ^ { * }$-algebras can be refined as follows (see [[#References|[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 [[#References|[a14]]].

This article is dedicated to the memory of Eulalia Garcia Rus.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  E.M. Alfsen,   F.W. Shultz,   E. Stormer,   "A Gelfand–Neumark theorem for Jordan algebras"  ''Adv. Math.'' , '''28'''  (1978)  pp. 11–56</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  B. Aupetit,   "The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras"  ''J. Funct. Anal.'' , '''47'''  (1982)  pp. 7–25</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  B. Aupetit,   "Recent trends in the field of Jordan–Banach algebras"  J. Zemánek (ed.) , ''Functional Analysis and Operator Theory'' , '''30''' , Banach Center Publ.  (1994)  pp. 9–19</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  V.K. Balachandran,   P.S. Rema,   "Uniqueness of the complete norm topology in certain Banach Jordan algebras"  ''Publ. Ramanujan Inst.'' , '''1'''  (1969)  pp. 283–289</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  F.F. Bonsall,   J. Duncan,   "Complete normed algebras" , Springer  (1973)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  R.B. Braun,   W. Kaup,   H. Upmeier,   "A holomorphic characterization of Jordan $C ^ { * }$-algebras"  ''Math. Z.'' , '''161'''  (1978)  pp. 277–290</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  M. Cabrera,   A. Moreno,   A. Rodriguez,   "Normed versions of the Zel'manov prime theorem: positive results and limits"  A. Gheondea (ed.)  R.N. Gologan (ed.)  D. Timotin (ed.) , ''Operator Theory, Operator Algebras and Related Topics (16th Internat. Conf. Operator Theory, Timisoara (Romania) July, 2-10, 1996)'' , The Theta Foundation, Bucharest  (1997)  pp. 65–77</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  M. Cabrera,   A. Moreno,   A. Rodriguez,   "Zel'manov's theorem for primitive Jordan–Banach algebras"  ''J. London Math. Soc.'' , '''57'''  (1998)  pp. 231–244</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  A. Fernandez,   E. Garcia,   A. Rodriguez,   "A Zelmanov prime theorem for $J B ^ { * }$-algebras"  ''J. London Math. Soc.'' , '''46'''  (1992)  pp. 319–335</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  H. Hanche-Olsen,   E. Stormer,   "Jordan operator algebras" , ''Monograph Stud. Math.'' , '''21''' , Pitman  (1984)</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  L. Hogben,   K. Mccrimmon,   "Maximal modular inner ideals and the Jacobson radical of a Jordan algebra"  ''J. Algebra'' , '''68'''  (1981)  pp. 155–169</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  N. Jacobson,   "Structure and representations of Jordan algebras" , ''Colloq. Publ.'' , '''37''' , Amer. Math. Soc.  (1968)</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  P.S. Putter,   B. Yood,   "Banach Jordan $*$-algebras"  ''Proc. London Math. Soc.'' , '''41'''  (1980)  pp. 21–44</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  A. Rodriguez,   "Nonassociative normed algebras spanned by hermitian elements"  ''Proc. London Math. Soc.'' , '''47'''  (1983)  pp. 258–274</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  A. Rodriguez,   "The uniqueness of the complete algebra norm topology in complete normed nonassociative algebras"  ''J. Funct. Anal.'' , '''60'''  (1985)  pp. 1–15</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  A. Rodriguez,   "Jordan structures in analysis"  W. Kaup (ed.)  K. McCrimmon (ed.)  H.P. Petersson (ed.) , ''Jordan Algebras (Proc. Conf. Oberwolfach, Germany, August, 9-15, 1992)'' , de Gruyter  (1994)  pp. 97–186</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  H. Upmeier,   "Symmetric Banach manifolds and Jordan $C ^ { * }$-algebras" , North-Holland  (1985)</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  A.R. Villena,   "Continuity of derivations on Jordan–Banach algebras."  ''Studia Math.'' , '''118'''  (1996)  pp. 205–229</td></tr><tr><td valign="top">[a19]</td> <td valign="top">  C. Viola Devapakkiam,   "Jordan algebras with continuous inverse"  ''Math. Japon.'' , '''16'''  (1971)  pp. 115–125</td></tr><tr><td valign="top">[a20]</td> <td valign="top">  J.D. M. Wright,   "Jordan $C ^ { * }$-algebras"  ''Michigan Math. J.'' , '''24'''  (1977)  pp. 291–302</td></tr><tr><td valign="top">[a21]</td> <td valign="top">  M.A. Youngson,   "Non unital Banach Jordan algebras and $C ^ { * }$-triple systems"  ''Proc. Edinburgh Math. Soc.'' , '''24'''  (1981)  pp. 19–31</td></tr><tr><td valign="top">[a22]</td> <td valign="top">  E. Zel'manov,   "On prime Jordan algebras II"  ''Sib. Math. J.'' , '''24'''  (1983)  pp. 89–104</td></tr></table>