Measures on the lattice of closed inner ideals in a spin triple

[摘要] Two elements J and K of the complete lattice F(A) 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 A(2)(J) subset of or equal to A(2)(I) and A(2)(K) subset of or equal to A(0)(I), and are said to be rigidly collinear when A(2)(J) subset of or equal to A(1)(K) and A(2)(K) subset of or equal to A(1) (J), where, for j equal to 0, 1, or 2, A(j)(I), A(j)(J), and A(j)(K), 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 F(A) is a mapping from F(A) to C such that, if J and K are either centrally orthogonal or rigidly collinear, then m(J boolean OR K) = m(J) + m(K). 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 F(A) 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 F(A). 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 x 2 complex matrices. In this case results of Bunce and Wright are used to show that there is an uncountable number of measures on F(A). The situation when the dimension of A is less than 4 is also described. (C) 2000 Academic Press.

[发布日期] 2000-12-15

[关键词] spin triple;measure