JB*-triple

$\operatorname{JB} ^ { * }$-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