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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j1300302.png" />-triples were introduced by W. Kaup [[#References|[a8]]] in connection with the study of bounded symmetric domains in complex Banach spaces. A definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j1300303.png" />-triples involving holomorphy is as follows: A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j1300305.png" />-triple is a complex [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j1300306.png" /> such that the open unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j1300307.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j1300308.png" /> is homogeneous under its full group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j1300309.png" /> of biholomorphic automorphisms (and hence is symmetric, cf. [[Symmetric space|Symmetric space]]). The main result in [[#References|[a8]]] states that to every abstract bounded symmetric domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003010.png" /> in a complex Banach space there exists a unique (up to linear isometry) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003011.png" />-triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003012.png" /> whose open unit ball is biholomorphically equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003013.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003014.png" /> is always a real Banach Lie group (cf. also [[Lie group, Banach|Lie group, Banach]]) acting in a natural way on various spaces of holomorphic functions as well as on various submanifolds of the unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003015.png" /> (in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003016.png" /> has finite dimension, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003017.png" /> is semi-simple and also has finite dimension — the induced [[Unitary representation|unitary representation]] on Bergman space, cf. also [[Bergman spaces|Bergman spaces]], is of special interest in [[Harmonic analysis|harmonic analysis]]).
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An equivalent, but more algebraic definition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003018.png" />-triples is as follows: The complex Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003019.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003021.png" />-triple if it carries a (necessarily unique) ternary product (called triple product) or [[trilinear mapping]]
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<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/j/j130/j130030/j13003022.png" /></td> </tr></table>
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$\operatorname{JB} ^ { * }$-triples were introduced by W. Kaup [[#References|[a8]]] in connection with the study of bounded symmetric domains in complex Banach spaces. A definition of $\operatorname{JB} ^ { * }$-triples involving holomorphy is as follows: A $\operatorname{JB} ^ { * }$-triple is a complex [[Banach space|Banach space]] $E$ such that the open unit ball $B$ of $E$ is homogeneous under its full group $G$ of biholomorphic automorphisms (and hence is symmetric, cf. [[Symmetric space|Symmetric space]]). The main result in [[#References|[a8]]] states that to every abstract bounded symmetric domain $D$ in a complex Banach space there exists a unique (up to linear isometry) $\operatorname{JB} ^ { * }$-triple $E$ whose open unit ball is biholomorphically equivalent to $D$. The group $G$ is always a real Banach Lie group (cf. also [[Lie group, Banach|Lie group, Banach]]) acting in a natural way on various spaces of holomorphic functions as well as on various submanifolds of the unit sphere in $E$ (in case $E$ has finite dimension, $G$ is semi-simple and also has finite dimension — the induced [[Unitary representation|unitary representation]] on Bergman space, cf. also [[Bergman spaces|Bergman spaces]], is of special interest in [[Harmonic analysis|harmonic analysis]]).
  
satisfying the following properties for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003024.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003025.png" />:
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An equivalent, but more algebraic definition for $\operatorname{JB} ^ { * }$-triples is as follows: The complex Banach space $E$ is a $\operatorname{JB} ^ { * }$-triple if it carries a (necessarily unique) ternary product (called triple product) or [[trilinear mapping]]
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003026.png" /> is symmetric complex bilinear in the outer variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003028.png" /> and conjugate linear in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003029.png" />;
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\begin{equation*} E \times E \times E \rightarrow E , ( x , y , z ) \mapsto \{ x y z \} \end{equation*}
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003030.png" /> (the Jordan triple identity);
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satisfying the following properties for all $a, b , x , y , z \in E$ and $a \square b ^ { * } : E \rightarrow E$ defined by $z \mapsto \{ a b z \}$:
  
iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003031.png" />, as a [[Linear operator|linear operator]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003032.png" />, is Hermitian and has spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003033.png" /> (cf. also [[Hermitian operator|Hermitian operator]]; [[Spectrum of an operator|Spectrum of an operator]]);
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i) $\{ x y z \}$ is symmetric complex bilinear in the outer variables $x$, $z$ and conjugate linear in $y$;
  
iv) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003034.png" /> (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003036.png" />-condition).
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ii) $[ a \square b ^ { * } , x \square y ^ { * } ] = \{ a b x \} \square y ^ { * } - x \square \{ y a b \}^{*}$ (the Jordan triple identity);
  
The sesquilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003037.png" /> may be considered as an operator-valued product on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003038.png" />. It satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003039.png" /> (not an elementary fact!) and condition iv) is analogous to the characteristic property of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003040.png" />-algebras (cf. also [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003041.png" />-algebra]]). On the other hand, by iii), the above mapping may also be considered as a positive Hermitian operator-valued form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003042.png" />, thus giving a natural orthogonality relation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003043.png" />.
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iii) $a \square a ^ { * }$, as a [[Linear operator|linear operator]] on $E$, is Hermitian and has spectrum $\geq 0$ (cf. also [[Hermitian operator|Hermitian operator]]; [[Spectrum of an operator|Spectrum of an operator]]);
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iv) $\| a \square a ^ { * } \| = \| a \| ^ { 2 }$ (the $C ^ { * }$-condition).
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The sesquilinear mapping $( a , b ) \mapsto a \square b ^ { * }$ may be considered as an operator-valued product on $E$. It satisfies $\| a \square b ^ { * } \| \leq \| a \| \cdot \|  b \|$ (not an elementary fact!) and condition iv) is analogous to the characteristic property of $C ^ { * }$-algebras (cf. also [[C*-algebra|$C ^ { * }$-algebra]]). On the other hand, by iii), the above mapping may also be considered as a positive Hermitian operator-valued form on $E$, thus giving a natural orthogonality relation on $E$.
  
 
Some examples are:
 
Some examples are:
  
1) Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003044.png" />-algebra (more precisely, the underlying complex Banach space). The triple product is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003045.png" />.
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1) Every $C ^ { * }$-algebra (more precisely, the underlying complex Banach space). The triple product is given by $\{ x y z \} = ( x y ^ { * } z + z y ^ { * } x ) / 2$.
  
2) Every closed (complex) subtriple of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003046.png" />-algebra. These are also called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003048.png" />-triples. These triples were originally introduced and intensively studied by L.A. Harris [[#References|[a5]]] under the name <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003050.png" />-algebra (cf. also [[Banach–Jordan algebra|Banach–Jordan algebra]]).
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2) Every closed (complex) subtriple of a $C ^ { * }$-algebra. These are also called $\operatorname{JC} ^ { * }$-triples. These triples were originally introduced and intensively studied by L.A. Harris [[#References|[a5]]] under the name $J ^ { * }$-algebra (cf. also [[Banach–Jordan algebra|Banach–Jordan algebra]]).
  
3) Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003052.png" />-algebra (i.e. Jordan <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003054.png" />-algebra, [[#References|[a15]]]), with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003055.png" />. In particular, the famous exceptional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003056.png" />-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003057.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003058.png" /> (which is not a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003059.png" />-triple) of all Hermitian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003060.png" />-matrices over the complex octonian algebra.
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3) Every $\operatorname{JB} ^ { * }$-algebra (i.e. Jordan $C ^ { * }$-algebra, [[#References|[a15]]]), with $\{ x y z \} = x \circ ( y ^ { * } \circ z ) + z \circ ( y ^ { * } \circ x ) - ( x \circ z ) \circ y ^ { * }$. In particular, the famous exceptional $27$-dimensional $\operatorname{JB} ^ { * }$-algebra $\mathcal{H} _ { 3 } ( \mathbf{O} ^ { c } )$ (which is not a $\operatorname{JC} ^ { * }$-triple) of all Hermitian $( 3 \times 3 )$-matrices over the complex octonian algebra.
  
4) Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003062.png" />-triple, i.e. a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003063.png" />-triple having a (necessarily unique) pre-dual. Among these are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003064.png" />-closed subtriples of von Neumann algebras as well as the Cartan factors, which are the building blocks of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003065.png" />-triples of type I (in analogy to the von Neumann algebras of type I, cf. also [[Von Neumann algebra|von Neumann algebra]]).
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4) Every $\operatorname {JBW} ^ { * }$-triple, i.e. a $\operatorname{JB} ^ { * }$-triple having a (necessarily unique) pre-dual. Among these are the $w ^ { * }$-closed subtriples of von Neumann algebras as well as the Cartan factors, which are the building blocks of the $\operatorname {JBW} ^ { * }$-triples of type I (in analogy to the von Neumann algebras of type I, cf. also [[Von Neumann algebra|von Neumann algebra]]).
  
The class of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003066.png" />-triples is invariant under taking arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003068.png" />-sums, quotients by closed triple ideals, ultrapowers, biduals [[#References|[a1]]], as well as contractive projections [[#References|[a9]]]. Notice that the range of a contractive projection on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003069.png" />-algebra in general does not have the structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003070.png" />-algebra, but always is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003071.png" />-triple. The Gel'fand–Naimark theorem of Y. Friedman and B. Russo [[#References|[a3]]] states that each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003072.png" />-triple can be realized as a subtriple of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003073.png" />-sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003074.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003075.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003076.png" />-triple of all bounded linear operators on a suitable complex [[Hilbert space|Hilbert space]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003077.png" /> is the exceptional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003078.png" />-algebra of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003079.png" />-valued continuous functions on a suitable compact [[Topological space|topological space]]. By [[#References|[a6]]], the classification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003080.png" />-triples can be achieved modulo the classification of von Neumann algebras. Furthermore, in [[#References|[a11]]] all prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003082.png" />-triples have been classified using Zel'manov techniques.
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The class of all $\operatorname{JB} ^ { * }$-triples is invariant under taking arbitrary $l_ { \infty }$-sums, quotients by closed triple ideals, ultrapowers, biduals [[#References|[a1]]], as well as contractive projections [[#References|[a9]]]. Notice that the range of a contractive projection on a $C ^ { * }$-algebra in general does not have the structure of a $C ^ { * }$-algebra, but always is a $\operatorname{JC} ^ { * }$-triple. The Gel'fand–Naimark theorem of Y. Friedman and B. Russo [[#References|[a3]]] states that each $\operatorname{JB} ^ { * }$-triple can be realized as a subtriple of an $l_ { \infty }$-sum $A \oplus B$ where $A$ is the $\operatorname {JBW} ^ { * }$-triple of all bounded linear operators on a suitable complex [[Hilbert space|Hilbert space]] and $B$ is the exceptional $\operatorname{JB} ^ { * }$-algebra of all $\mathcal{H} _ { 3 } ( \mathbf{O} ^ { c } )$-valued continuous functions on a suitable compact [[Topological space|topological space]]. By [[#References|[a6]]], the classification of $\operatorname {JBW} ^ { * }$-triples can be achieved modulo the classification of von Neumann algebras. Furthermore, in [[#References|[a11]]] all prime $\operatorname{JB} ^ { * }$-triples have been classified using Zel'manov techniques.
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003083.png" />-triples form a large class of complex Banach spaces whose geometry can be described algebraically. Examples of this are:
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The $\operatorname{JB} ^ { * }$-triples form a large class of complex Banach spaces whose geometry can be described algebraically. Examples of this are:
  
A bijective linear operator between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003084.png" />-triples is an isometry if and only if it respects the Jordan triple product.
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A bijective linear operator between $\operatorname{JB} ^ { * }$-triples is an isometry if and only if it respects the Jordan triple product.
  
The M-ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003085.png" /> are precisely the closed triple ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003086.png" />.
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The M-ideals in $E$ are precisely the closed triple ideals of $E$.
  
The open unit ball of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003087.png" /> is the largest convex subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003088.png" /> containing the origin such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003089.png" /> the Bergman operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003090.png" /> is invertible, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003091.png" /> is an extreme point of the closed unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003092.png" /> if and only if the Bergman operator associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003093.png" /> is the zero operator.
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The open unit ball of $E$ is the largest convex subset $B \subset E$ containing the origin such that for every $a \in B$ the Bergman operator $z \mapsto z - 2 \{ a a z \} + \{ a \{ a z a \} a \}$ is invertible, and $a \in E$ is an extreme point of the closed unit ball in $E$ if and only if the Bergman operator associated to $a$ is the zero operator.
  
Real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003095.png" />-triples were studied in [[#References|[a7]]]; these are the real forms of (complex) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003097.png" />-triples. In general, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003098.png" />-triple may have many non-isomorphic real forms. An important class of real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003099.png" />-triples is obtained from the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j130030101.png" />-algebras, compare [[#References|[a4]]].
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Real $\operatorname{JB} ^ { * }$-triples were studied in [[#References|[a7]]]; these are the real forms of (complex) $\operatorname{JB} ^ { * }$-triples. In general, a $\operatorname{JB} ^ { * }$-triple may have many non-isomorphic real forms. An important class of real $\operatorname{JB} ^ { * }$-triples is obtained from the class of $\operatorname{JB}$-algebras, compare [[#References|[a4]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Dineen,  "Complete holomorphic vector fields on the second dual of a Banach space"  ''Math. Scand.'' , '''59'''  (1986)  pp. 131–42</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.M. Edwards,  K. McCrimmon,  G.T. Rüttimann,  "The range of a structural projection"  ''J. Funct. Anal.'' , '''139'''  (1996)  pp. 196–224</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Y. Friedman,  B. Russo,  "The Gelfand–Naimark theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j130030102.png" />-triples"  ''Duke Math. J.'' , '''53'''  (1986)  pp. 139–148</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Hanche-Olsen,  E. Størmer,  "Jordan operator algebras" , ''Mon. Stud. Math.'' , '''21''' , Pitman  (1984)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L.A. Harris,  "Bounded symmetric homogeneous domains in infinite dimensional spaces" , ''Lecture Notes in Math.'' , '''364''' , Springer  (1973)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Horn,  E. Neher,  "Classification of continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j130030103.png" />-triples"  ''Trans. Amer. Math. Soc.'' , '''306'''  (1988)  pp. 553–578</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.M. Isidro,  W. Kaup,  A. Rodríguez,  "On real forms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j130030104.png" />-triples"  ''Manuscripta Math.'' , '''86'''  (1995)  pp. 311–335</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  W. Kaup,  "A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces"  ''Math. Z.'' , '''183'''  (1983)  pp. 503–529</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  W. Kaup,  "Contractive projections on Jordan <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j130030105.png" />-algebras and generalizations"  ''Math. Scand.'' , '''54'''  (1984)  pp. 95–100</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  O. Loos,  "Bounded symmetric domains and Jordan pairs"  ''Math. Lectures. Univ. California at Irvine''  (1977)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  A. Moreno,  A. Rodríguez,  "On the Zelmanovian classification of prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j130030106.png" />-triples"  ''J. Algebra'' , '''226'''  (2000)  pp. 577–613</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  B. Russo,  "Stucture of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j130030107.png" />-triples" , ''Proc. Oberwolfach Conf. Jordan Algebras, 1992'' , de Gruyter  (1994)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  H. Upmeier,  "Jordan algebras in analysis, operator theory and quantum mechanics" , ''Regional Conf. Ser. Math.'' , '''67''' , Amer. Math. Soc.  (1987)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  H. Upmeier,  "Symmetric Banach manifolds and Jordan <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j130030108.png" />-algebras" , ''Math. Studies'' , '''104''' , North-Holland  (1985)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  J.D.M. Wright,  "Jordan <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j130030109.png" />-algebras"  ''Michigan Math. J.'' , '''24'''  (1977)  pp. 291–302</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  S. Dineen,  "Complete holomorphic vector fields on the second dual of a Banach space"  ''Math. Scand.'' , '''59'''  (1986)  pp. 131–42</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  C.M. Edwards,  K. McCrimmon,  G.T. Rüttimann,  "The range of a structural projection"  ''J. Funct. Anal.'' , '''139'''  (1996)  pp. 196–224</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  Y. Friedman,  B. Russo,  "The Gelfand–Naimark theorem for $J B ^ { * }$-triples"  ''Duke Math. J.'' , '''53'''  (1986)  pp. 139–148</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  H. Hanche-Olsen,  E. Størmer,  "Jordan operator algebras" , ''Mon. Stud. Math.'' , '''21''' , Pitman  (1984)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  L.A. Harris,  "Bounded symmetric homogeneous domains in infinite dimensional spaces" , ''Lecture Notes in Math.'' , '''364''' , Springer  (1973)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  G. Horn,  E. Neher,  "Classification of continuous $J B W ^ { x }$-triples"  ''Trans. Amer. Math. Soc.'' , '''306'''  (1988)  pp. 553–578</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  J.M. Isidro,  W. Kaup,  A. Rodríguez,  "On real forms of $J B ^ { * }$-triples"  ''Manuscripta Math.'' , '''86'''  (1995)  pp. 311–335</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  W. Kaup,  "A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces"  ''Math. Z.'' , '''183'''  (1983)  pp. 503–529</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  W. Kaup,  "Contractive projections on Jordan $C ^ { * }$-algebras and generalizations"  ''Math. Scand.'' , '''54'''  (1984)  pp. 95–100</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  O. Loos,  "Bounded symmetric domains and Jordan pairs"  ''Math. Lectures. Univ. California at Irvine''  (1977)</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  A. Moreno,  A. Rodríguez,  "On the Zelmanovian classification of prime $J B ^ { * }$-triples"  ''J. Algebra'' , '''226'''  (2000)  pp. 577–613</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  B. Russo,  "Stucture of $J B ^ { * }$-triples" , ''Proc. Oberwolfach Conf. Jordan Algebras, 1992'' , de Gruyter  (1994)</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  H. Upmeier,  "Jordan algebras in analysis, operator theory and quantum mechanics" , ''Regional Conf. Ser. Math.'' , '''67''' , Amer. Math. Soc.  (1987)</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  H. Upmeier,  "Symmetric Banach manifolds and Jordan $C ^ { * }$-algebras" , ''Math. Studies'' , '''104''' , North-Holland  (1985)</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  J.D.M. Wright,  "Jordan $C ^ { * }$-algebras"  ''Michigan Math. J.'' , '''24'''  (1977)  pp. 291–302</td></tr></table>

Revision as of 15:30, 1 July 2020

$\operatorname{JB} ^ { * }$-triples were introduced by W. Kaup [a8] in connection with the study of bounded symmetric domains in complex Banach spaces. A definition of $\operatorname{JB} ^ { * }$-triples involving holomorphy is as follows: A $\operatorname{JB} ^ { * }$-triple is a complex Banach space $E$ such that the open unit ball $B$ of $E$ is homogeneous under its full group $G$ of biholomorphic automorphisms (and hence is symmetric, cf. Symmetric space). The main result in [a8] states that to every abstract bounded symmetric domain $D$ in a complex Banach space there exists a unique (up to linear isometry) $\operatorname{JB} ^ { * }$-triple $E$ whose open unit ball is biholomorphically equivalent to $D$. The group $G$ is always a real Banach Lie group (cf. also Lie group, Banach) acting in a natural way on various spaces of holomorphic functions as well as on various submanifolds of the unit sphere in $E$ (in case $E$ has finite dimension, $G$ is semi-simple and also has finite dimension — the induced unitary representation on Bergman space, cf. also Bergman spaces, is of special interest in harmonic analysis).

An equivalent, but more algebraic definition for $\operatorname{JB} ^ { * }$-triples is as follows: The complex Banach space $E$ is a $\operatorname{JB} ^ { * }$-triple if it carries a (necessarily unique) ternary product (called triple product) or trilinear mapping

\begin{equation*} E \times E \times E \rightarrow E , ( x , y , z ) \mapsto \{ x y z \} \end{equation*}

satisfying the following properties for all $a, b , x , y , z \in E$ and $a \square b ^ { * } : E \rightarrow E$ defined by $z \mapsto \{ a b z \}$:

i) $\{ x y z \}$ is symmetric complex bilinear in the outer variables $x$, $z$ and conjugate linear in $y$;

ii) $[ a \square b ^ { * } , x \square y ^ { * } ] = \{ a b x \} \square y ^ { * } - x \square \{ y a b \}^{*}$ (the Jordan triple identity);

iii) $a \square a ^ { * }$, as a linear operator on $E$, is Hermitian and has spectrum $\geq 0$ (cf. also Hermitian operator; Spectrum of an operator);

iv) $\| a \square a ^ { * } \| = \| a \| ^ { 2 }$ (the $C ^ { * }$-condition).

The sesquilinear mapping $( a , b ) \mapsto a \square b ^ { * }$ may be considered as an operator-valued product on $E$. It satisfies $\| a \square b ^ { * } \| \leq \| a \| \cdot \| b \|$ (not an elementary fact!) and condition iv) is analogous to the characteristic property of $C ^ { * }$-algebras (cf. also $C ^ { * }$-algebra). On the other hand, by iii), the above mapping may also be considered as a positive Hermitian operator-valued form on $E$, thus giving a natural orthogonality relation on $E$.

Some examples are:

1) Every $C ^ { * }$-algebra (more precisely, the underlying complex Banach space). The triple product is given by $\{ x y z \} = ( x y ^ { * } z + z y ^ { * } x ) / 2$.

2) Every closed (complex) subtriple of a $C ^ { * }$-algebra. These are also called $\operatorname{JC} ^ { * }$-triples. These triples were originally introduced and intensively studied by L.A. Harris [a5] under the name $J ^ { * }$-algebra (cf. also Banach–Jordan algebra).

3) Every $\operatorname{JB} ^ { * }$-algebra (i.e. Jordan $C ^ { * }$-algebra, [a15]), with $\{ x y z \} = x \circ ( y ^ { * } \circ z ) + z \circ ( y ^ { * } \circ x ) - ( x \circ z ) \circ y ^ { * }$. In particular, the famous exceptional $27$-dimensional $\operatorname{JB} ^ { * }$-algebra $\mathcal{H} _ { 3 } ( \mathbf{O} ^ { c } )$ (which is not a $\operatorname{JC} ^ { * }$-triple) of all Hermitian $( 3 \times 3 )$-matrices over the complex octonian algebra.

4) Every $\operatorname {JBW} ^ { * }$-triple, i.e. a $\operatorname{JB} ^ { * }$-triple having a (necessarily unique) pre-dual. Among these are the $w ^ { * }$-closed subtriples of von Neumann algebras as well as the Cartan factors, which are the building blocks of the $\operatorname {JBW} ^ { * }$-triples of type I (in analogy to the von Neumann algebras of type I, cf. also von Neumann algebra).

The class of all $\operatorname{JB} ^ { * }$-triples is invariant under taking arbitrary $l_ { \infty }$-sums, quotients by closed triple ideals, ultrapowers, biduals [a1], as well as contractive projections [a9]. Notice that the range of a contractive projection on a $C ^ { * }$-algebra in general does not have the structure of a $C ^ { * }$-algebra, but always is a $\operatorname{JC} ^ { * }$-triple. The Gel'fand–Naimark theorem of Y. Friedman and B. Russo [a3] states that each $\operatorname{JB} ^ { * }$-triple can be realized as a subtriple of an $l_ { \infty }$-sum $A \oplus B$ where $A$ is the $\operatorname {JBW} ^ { * }$-triple of all bounded linear operators on a suitable complex Hilbert space and $B$ is the exceptional $\operatorname{JB} ^ { * }$-algebra of all $\mathcal{H} _ { 3 } ( \mathbf{O} ^ { c } )$-valued continuous functions on a suitable compact topological space. By [a6], the classification of $\operatorname {JBW} ^ { * }$-triples can be achieved modulo the classification of von Neumann algebras. Furthermore, in [a11] all prime $\operatorname{JB} ^ { * }$-triples have been classified using Zel'manov techniques.

The $\operatorname{JB} ^ { * }$-triples form a large class of complex Banach spaces whose geometry can be described algebraically. Examples of this are:

A bijective linear operator between $\operatorname{JB} ^ { * }$-triples is an isometry if and only if it respects the Jordan triple product.

The M-ideals in $E$ are precisely the closed triple ideals of $E$.

The open unit ball of $E$ is the largest convex subset $B \subset E$ containing the origin such that for every $a \in B$ the Bergman operator $z \mapsto z - 2 \{ a a z \} + \{ a \{ a z a \} a \}$ is invertible, and $a \in E$ is an extreme point of the closed unit ball in $E$ if and only if the Bergman operator associated to $a$ is the zero operator.

Real $\operatorname{JB} ^ { * }$-triples were studied in [a7]; these are the real forms of (complex) $\operatorname{JB} ^ { * }$-triples. In general, a $\operatorname{JB} ^ { * }$-triple may have many non-isomorphic real forms. An important class of real $\operatorname{JB} ^ { * }$-triples is obtained from the class of $\operatorname{JB}$-algebras, compare [a4].

References

[a1] S. Dineen, "Complete holomorphic vector fields on the second dual of a Banach space" Math. Scand. , 59 (1986) pp. 131–42
[a2] C.M. Edwards, K. McCrimmon, G.T. Rüttimann, "The range of a structural projection" J. Funct. Anal. , 139 (1996) pp. 196–224
[a3] Y. Friedman, B. Russo, "The Gelfand–Naimark theorem for $J B ^ { * }$-triples" Duke Math. J. , 53 (1986) pp. 139–148
[a4] H. Hanche-Olsen, E. Størmer, "Jordan operator algebras" , Mon. Stud. Math. , 21 , Pitman (1984)
[a5] L.A. Harris, "Bounded symmetric homogeneous domains in infinite dimensional spaces" , Lecture Notes in Math. , 364 , Springer (1973)
[a6] G. Horn, E. Neher, "Classification of continuous $J B W ^ { x }$-triples" Trans. Amer. Math. Soc. , 306 (1988) pp. 553–578
[a7] J.M. Isidro, W. Kaup, A. Rodríguez, "On real forms of $J B ^ { * }$-triples" Manuscripta Math. , 86 (1995) pp. 311–335
[a8] W. Kaup, "A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces" Math. Z. , 183 (1983) pp. 503–529
[a9] W. Kaup, "Contractive projections on Jordan $C ^ { * }$-algebras and generalizations" Math. Scand. , 54 (1984) pp. 95–100
[a10] O. Loos, "Bounded symmetric domains and Jordan pairs" Math. Lectures. Univ. California at Irvine (1977)
[a11] A. Moreno, A. Rodríguez, "On the Zelmanovian classification of prime $J B ^ { * }$-triples" J. Algebra , 226 (2000) pp. 577–613
[a12] B. Russo, "Stucture of $J B ^ { * }$-triples" , Proc. Oberwolfach Conf. Jordan Algebras, 1992 , de Gruyter (1994)
[a13] H. Upmeier, "Jordan algebras in analysis, operator theory and quantum mechanics" , Regional Conf. Ser. Math. , 67 , Amer. Math. Soc. (1987)
[a14] H. Upmeier, "Symmetric Banach manifolds and Jordan $C ^ { * }$-algebras" , Math. Studies , 104 , North-Holland (1985)
[a15] J.D.M. Wright, "Jordan $C ^ { * }$-algebras" Michigan Math. J. , 24 (1977) pp. 291–302
How to Cite This Entry:
JB*-triple. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=JB*-triple&oldid=49905
This article was adapted from an original article by Wilhelm Kaup (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article