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''Jordan–Banach algebra''
 
''Jordan–Banach algebra''
  
A [[Jordan algebra|Jordan algebra]] over the field of real or complex numbers, endowed with a complete norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b1300201.png" /> satisfying
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A [[Jordan algebra|Jordan algebra]] over the field of real or complex numbers, endowed with a complete norm $|.|$ satisfying
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b1300202.png" /></td> </tr></table>
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\begin{equation*} \| x \circ y  \| \leq \| x \| \| y \| \end{equation*}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b1300203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b1300204.png" /> in the algebra. Since an (associative) [[Banach algebra|Banach algebra]] is a Banach–Jordan algebra under the Jordan product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b1300205.png" />, 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]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b1300206.png" /> of an arbitrary element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b1300207.png" /> of a Banach–Jordan algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b1300208.png" /> is defined as in the associative case, and the spectral radius formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b1300209.png" /> holds. In fact, Banach–Jordan algebras are  "locally spectrally"  associative. This means that each element in such an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002010.png" /> can be imbedded in some closed associative subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002012.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002013.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002014.png" />. Then, for a single element in a complex Banach–Jordan algebra, a holomorphic functional calculus follows easily.
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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]]].
 
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]]].
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002016.png" />-algebras are defined as the real Banach–Jordan algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002017.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002019.png" />. The basic theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002020.png" />-algebras, originally due to E.M. Alfsen, F.W. Shultz and E. Stormer [[#References|[a1]]], is fully treated in [[#References|[a10]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002021.png" /> is a [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002022.png" />-algebra]], then the self-adjoint part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002024.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002025.png" />-algebra under the Jordan product. Closed subalgebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002026.png" />, for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002027.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002028.png" />, become relevant examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002029.png" />-algebras, and are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002031.png" />-algebras. Through the consideration of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002033.png" />-algebras (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002034.png" />-algebras that are dual Banach spaces, cf. also [[Banach space|Banach space]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002036.png" />-factors (i.e., prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002037.png" />-algebras), and factor representations of a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002039.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002040.png" /> (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002041.png" />-dense range homomorphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002042.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002043.png" />-factors), the knowledge of arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002044.png" />-algebras is reasonably reduced to that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002045.png" />-algebras and the exceptional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002046.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002047.png" /> of all Hermitian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002048.png" />-matrices over the alternative [[Division algebra|division algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002049.png" /> of real octonions.
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$\operatorname{JB}$-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.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002051.png" />-algebras are defined as complex Banach–Jordan algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002052.png" /> endowed with a conjugate-linear algebra involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002053.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002054.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002055.png" />. Here, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002057.png" /> denotes the operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002058.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002059.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002060.png" />. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002061.png" />-algebra becomes a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002062.png" />-algebra under its Jordan product. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002063.png" />-algebras are closely related to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002064.png" />-algebras. Indeed, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002065.png" />-algebras are nothing but the self-adjoint parts of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002066.png" />-algebras [[#References|[a20]]]. The one-to-one categorical correspondence between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002067.png" />-algebras and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002068.png" />-algebras derived from the above result completely reduces the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002069.png" />-theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002070.png" />-algebras to the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002071.png" />-algebras. However, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002072.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002073.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002074.png" />-algebras can be refined as follows (see [[#References|[a9]]]). A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002075.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002076.png" /> is primitive if and only if it is of one of the following types:
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$\operatorname{JB} ^ { * }$-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:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002077.png" /> is the unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002078.png" />-algebra whose self-adjoint part is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002079.png" />.
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$J$ is the unique $\operatorname{JB} ^ { * }$-algebra whose self-adjoint part is $H _ { 3 } ( \text{O} )$.
  
There exists a complex Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002080.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002081.png" />, with a conjugation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002082.png" /> and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002083.png" />-invariant norm-one element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002084.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002085.png" /> as complex vector spaces, whereas the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002086.png" />, the involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002087.png" />, and the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002088.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002089.png" /> are given by
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002090.png" /></td> </tr></table>
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\begin{equation*} x \circ y : = ( x | 1 ) y + ( y | 1 ) x - ( x | \sigma ( y ) ) 1, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002091.png" /></td> </tr></table>
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\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.
 
respectively.
  
There exists a primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002092.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002093.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002094.png" /> is a closed self-adjoint Jordan subalgebra of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002095.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002096.png" />, of multipliers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002097.png" />, containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002098.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b13002099.png" />-algebra with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020100.png" />-involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020101.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020102.png" /> is a closed self-adjoint Jordan subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020103.png" /> contained in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020104.png" />-Hermitian part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020105.png" /> and containing the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020106.png" />-Hermitian part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020107.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020108.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020109.png" /> is subjected to the geometric conditions that, through the Vidav–Palmer theorem, characterize <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020110.png" />-algebras in the associative setting, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020111.png" /> under the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020112.png" /> and a suitable involution becomes a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020113.png" />-algebra [[#References|[a14]]].
+
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.
 
This article is dedicated to the memory of Eulalia Garcia Rus.
  
 
====References====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020114.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020115.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020116.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020117.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020118.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130020/b130020119.png" />-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>
+
<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>

Latest revision as of 16:46, 1 July 2020

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].

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

References

[a1] E.M. Alfsen, F.W. Shultz, E. Stormer, "A Gelfand–Neumark theorem for Jordan algebras" Adv. Math. , 28 (1978) pp. 11–56
[a2] B. Aupetit, "The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras" J. Funct. Anal. , 47 (1982) pp. 7–25
[a3] 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
[a4] 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
[a5] F.F. Bonsall, J. Duncan, "Complete normed algebras" , Springer (1973)
[a6] R.B. Braun, W. Kaup, H. Upmeier, "A holomorphic characterization of Jordan $C ^ { * }$-algebras" Math. Z. , 161 (1978) pp. 277–290
[a7] 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
[a8] M. Cabrera, A. Moreno, A. Rodriguez, "Zel'manov's theorem for primitive Jordan–Banach algebras" J. London Math. Soc. , 57 (1998) pp. 231–244
[a9] A. Fernandez, E. Garcia, A. Rodriguez, "A Zelmanov prime theorem for $J B ^ { * }$-algebras" J. London Math. Soc. , 46 (1992) pp. 319–335
[a10] H. Hanche-Olsen, E. Stormer, "Jordan operator algebras" , Monograph Stud. Math. , 21 , Pitman (1984)
[a11] L. Hogben, K. Mccrimmon, "Maximal modular inner ideals and the Jacobson radical of a Jordan algebra" J. Algebra , 68 (1981) pp. 155–169
[a12] N. Jacobson, "Structure and representations of Jordan algebras" , Colloq. Publ. , 37 , Amer. Math. Soc. (1968)
[a13] P.S. Putter, B. Yood, "Banach Jordan $*$-algebras" Proc. London Math. Soc. , 41 (1980) pp. 21–44
[a14] A. Rodriguez, "Nonassociative normed algebras spanned by hermitian elements" Proc. London Math. Soc. , 47 (1983) pp. 258–274
[a15] A. Rodriguez, "The uniqueness of the complete algebra norm topology in complete normed nonassociative algebras" J. Funct. Anal. , 60 (1985) pp. 1–15
[a16] 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
[a17] H. Upmeier, "Symmetric Banach manifolds and Jordan $C ^ { * }$-algebras" , North-Holland (1985)
[a18] A.R. Villena, "Continuity of derivations on Jordan–Banach algebras." Studia Math. , 118 (1996) pp. 205–229
[a19] C. Viola Devapakkiam, "Jordan algebras with continuous inverse" Math. Japon. , 16 (1971) pp. 115–125
[a20] J.D. M. Wright, "Jordan $C ^ { * }$-algebras" Michigan Math. J. , 24 (1977) pp. 291–302
[a21] M.A. Youngson, "Non unital Banach Jordan algebras and $C ^ { * }$-triple systems" Proc. Edinburgh Math. Soc. , 24 (1981) pp. 19–31
[a22] E. Zel'manov, "On prime Jordan algebras II" Sib. Math. J. , 24 (1983) pp. 89–104
How to Cite This Entry:
Banach-Jordan algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach-Jordan_algebra&oldid=50003
This article was adapted from an original article by Angel Rodriguez Palacios (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article