Measures on the Lattice of Closed Inner Ideals in a Spin Triple

Two elements J and K of the complete lattice IŽA. of weak -closed inner ideals in a JBW -triple A are said to be centrally orthogonal if there exists a weak -closed ideal I in A such that A2ŽJ. A2ŽI. and A2ŽK. A0ŽI., and are said to be rigidly collinear when A2ŽJ. A1ŽK. and A2ŽK. A1ŽJ., where, for j equal to 0, 1, or 2, AjŽI., AjŽJ., and AjŽK., are the components in the generalized Peirce decomposition of A relative to the weak -closed inner ideals I, J, and K, respectively. A measure m on IŽA. is a mapping from IŽA. to such that, if J and K are either centrally orthogonal or rigidly collinear, then mŽJ K. mŽJ. mŽK.. A complex Hilbert space A endowed with a conjugation possesses a triple product and norm with respect to which it forms a JBW -triple, known as a spin triple. In this paper the structure of the complete lattice IŽA. of closed inner ideals in a spin triple A and the measures on it are investigated. It is shown that, if the dimension of A is greater than 5, then there are no non-zero measures on IŽA.. When the dimension of A is 5, non-zero measures exist and, up to multiplication by a constant, a unique measure exists that is invariant under automorphisms of A. When the dimension of A is 4, then A is triple isomorphic to the W -algebra of 2 2 complex matrices. In this case results of Bunce and Wright are used to show that there is an uncountable number of measures on IŽA.. The situation when the dimension of A is less than 4 is also described.