Bounded symmetric domains and generalized operator algebras

Jordan C*-algebras go back to Kaplansky, see [20]. Let J be a complex Banach Jordan algebra, that is, a complex Banach space with commutative bilinear product x◦y satisfying x◦(x2◦y) = x2◦(x◦y) as well as ||x◦y|| ≤ ||x||·||y||, Bounded symmetric domains and generalized operator algebras 51 and suppose that on J is given a (conjugate linear) isometric algebra involution x 7→ x∗. Then J is called a JB*-algebra if ||{xxx}|| = ||x||3 for every x ∈ J , where the ‘triple product’ {xyz} on J is given by (3.6) {xyz} := x ◦ (z ◦ y∗) + z ◦ (x ◦ y∗)− (x ◦ z) ◦ y∗ . Every JC*-algebra is a JB*-algebra with respect to the Jordan product x ◦ y = (xy+yx)/2, but not every JB*-algebra can be obtained this way. The most prominent counter-example can be described as follows: Let O be the standard real Cayley algebra, that is the unique (non-associative) real division algebra of dimension 8. Then O comes with an algebra involution x 7→ x whose fixed point set is a subfield isomorphic to IR. Let V := H3(O) be the space of all hermitian 3×3−matrices over O, which obviously is a real vector space of dimension 27. With respect to x◦y = (xy+yx)/2 the space V becomes a real Jordan algebra with the unit matrix 1 as identity. Now the formal complexification J := V C := V ⊕ iV is a complex Jordan algebra by extending the Jordan product in a complex bilinear way. Also, (x + iy)∗ := x − iy defines an algebra involution on J . As in every unital Jordan algebra, to every z ∈ J there exists a (commutative) associative subalgebra of J containing z and 1 . This implies that all powers of z and hence also exp(z) ∈ J are defined (usual exponential power series). There exists a unique norm on J such that the corresponding closed unit ball is the convex hull of the ‘generalized unit circle’ exp(iV ). With respect to this norm the Jordan *-algebra J is a JB*-algebra not isomorphic to any JC*-algebra (J is called an exceptional JB*-algebra). The open unit ball D of every JB*-algebra also is homogeneous. We do not go into details here since there also exists an abstract analog to JC*-triples that encloses the JB*-algebras. These are the JB*-triples that will be discussed in the next section.