Abstract. A theory of real Jordan triples and real bounded symmetric domains in finite dimensions was developed by Loos. Upmeier has proposed a definition.
proceedings of the american mathematical society Volume 122, Number 1, September 1994
REAL BANACHJORDAN TRIPLES TRUONG C. DANG AND BERNARDRUSSO (Communicated by Palle E. T. Jorgensen)
Abstract. A theory of real Jordan triples and real bounded symmetric domains in finite dimensions was developed by Loos. Upmeier has proposed a definition of a real 7ß*-triple in arbitrary dimensions. These spaces include real Calgebras and JB*-triples considered as vector spaces over the reals and have the property that their open unit balls are real bounded symmetric domains. This, together with the observation that many of the more recent techniques in Jordan theory rely on functional analysis and algebra rather than holomorphy, suggests that it may be possible to develop a real theory and to explore its relationship with the complex theory. In this paper we employ a Banach algebraic approach to real Banach Jordan triples. Because of our recent observation on commutative 7fi*-triples (see §2), we can now propose a new definition of a real ./¿"-triple, which we call a J*B-triple. Our 7*5-triples include real C*-algebras and complex JB*triples. Our main theorem is a structure theorem of Gelfand-Naimark type for commutative 7*ß-triples.
1. Real Banach Jordan
triples
Definition 1.1. A Banach Jordan triple is a real or complex Banach space U equipped with a continuous bilinear (sesquilinear in the complex case) map
U x U 9 (x, y) * xDy £ S?iU) such that with {xyz} := xDy(z) we have (1) (2)
{xyz} = {zyx}, {x, y, {uvz}} + {u, {yxv}, z} = {{xyu} ,v,z}
+ {u,v,
{xyz}}.
A Banach Jordan triple U over C is said to be a JB*-triple if (a) for any x £ U the operator xDx from U to U (that is, xDx(y) = {xxy} , y £ U) is hermitian (i.e., exp z'ixDx is an isometry for all real t) with nonnegative spectrum, (b) the following norm condition holds
(3)
||xDx|| = ||x||2.
Received by the editors December 1, 1992; the contents of this paper were presented by the first author at the Great Plains Operator Theory Symposium, Texas A&M University, April 1991.
1991 Mathematics Subject Classification.Primary 46J99, 17C65. ©1994 American Mathematical Society
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136
T. C. DANG AND BERNARDRUSSO
We note that equations (1) and (2) are the defining algebraic identities for a Jordan triple system.
Our first result was originally stated with some extra hypothesis. The authors wish to thank Jonathan Arazy for pointing this out and for suggesting the
following proof. Theorem 1.2. Let U be a complex Banach Jordan triple. Suppose that (1) ||{xxx}|| = ||x||3; (2) ||{xyz}||{[/)(xDx) c [0, oo) for each x £ U.
Then U is a JB*-triple. Proof. We only need to show that xDx is hermitian for each x £ U. Since ô := z'xDx is a continuous derivation, a := etS is a continuous automorphism for each real t. Thus for each x £ U ||a(x)||3 = \\{a(x), a(x), a(x)}\\ = \\a({xxx})\\ < ||a|| ||x||\
and therefore, by iteration,
H^II^IHI^IIxH; that is, \\a\\ < 1.
D
The terminology in the next definition was motivated by [1], and the spectral
conditions were inspired by [7]. Definition 1.3. A J*B-triple is a real Banach space A equipped with a structure of a real Jordan triple system which satisfies (1) ||{xxx}|| = ||x||3; (2) ||{xyz}|| {xya}. In order to obtain the analogue of Theorem 2.2 we need to consider the complexification of A.
Let U :- Ac - c/>(A)+i(j)(A) be the complexification of A, and let 4>:A —>U be the natural embedding.
The space U becomes a complex commutative
Jordan triple system in the natural way, and 0 is a real-linear triple isomorphism into. Explicitly, U = A x A becomes a complex linear space under (a + iß)(x, y) = (ax-ßy,
ay + ßx),
a, ß £R,
x,y
£ A, and (y)e U,
IMIi= 11*11+ IMIRecall that (t/, Il*Hi) is a real Banach space, (U, || • ||2) is a complex Banach
space, r/>is an isometry in each norm, and (l/-\/2)|| • 111< Il• II2< Il' Hi• F01" T£^f(U)
=5?(U,
||-||2) let ||r||^{[/) denote the operator norm \\T\\^{U) =
sup
«5¿0, u€U
(||r«||2/||M||2).
Lemma 3.1. UDU c ^(U) and ||xDy||^(t/) < 23/2||x||2||y||2 for x,y Thus U is a commutative complex Banach Jordan triple. Proof. If a, b £ A and z = zx + iz2 £ U, then
£ U.
||(flDÔ)z||2 < \\(aDb)z\\x = \\{abzx}\\A+ \\{abz2}\\A< ||ay|ftyi*lli
^HûlUflôlUv^llzlb, so \\anb\\
