The Structure of Spin Systems

A spin system is a sequence of self-adjoint unitary operators U1, U2, . . . acting on a Hilbert space H which either commute or anticommute, UiUj = ±UjUi for all i, j; it is is called irreducible when {U1, U2, . . . } is an irreducible set of operators. There is a unique infinite matrix (cij) with 0, 1 entries satisfying UiUj = (−1) cij UjUi, i, j = 1, 2, . . . . Every matrix (cij ) with 0, 1 entries satisfying cij = cji and cii = 0 arises from a nontrivial irreducible spin system, and there are uncountably many such matrices. Infinite dimensional irreducible representations exist when the commutation matrix (cij) is of “infinite rank”. In such cases we show that the C-algebra generated by an irreducible spin system is the CAR algebra, an infinite tensor product of copies of M2(C), and we classify the irreducible spin systems associated with a given matrix (cij ) up to approximate unitary equivalence. That follows from a structural result. The C∗-algebra generated by the universal spin system u1, u2, . . . of (cij ) decomposes into a tensor product C(X)⊗A, where X is a Cantor set (possibly finite) and A is either the CAR algebra or a finite tensor product of copies of M2(C). The Bratteli diagram technology of AF algebras is not well suited to spin systems. Instead, we work out elementary properties of the Z2-valued “symplectic” form