View source for JB*-triple

<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]]).

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]]

<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>

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" />:

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" />;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130030/j13003030.png" /> (the Jordan triple identity);

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]]);

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).

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" />.

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" />.

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]]).

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.

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]]).

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.

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:

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.

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" />.

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.

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]]].

====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>