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

Journal article

Two elements J and K of the complete lattice I 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 A₂ (J)andsubseteq; A₂(I) and A₂(K) andsubseteq; A₀(I) , and are said to be rigidly collinear when A₂(J)andsubseteq; A₁(K) and A₂(K) andsubseteq; A₁(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 I(A) is a mapping from I(A) to C such that, if J and K are either centrally orthogonal or rigidly collinear, then

m(J andvee; K) andequiv; 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 I(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 I(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 I(A). The situation when the dimension of A is less than 4 is also described.

Authors: Edwards, C; Ruttimann, G

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
• Volume: 252
• Issue: 2
• Pages: 649-674
• Publication date: 2000-12-15
• DOI: 10.1006/jmaa.2000.7059
• ISSN: 0022-247X
• Keywords: measure; spin triple