THE STRUCTURE OF SPIN SYSTEMS

Abstract

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 called irreducible when {U1, U2, …} is an irreducible set of operators. There is a unique infinite matrix (cij) with 0, 1 entries satisfying [Formula: see text] 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. In cases where the commutation matrix (cij) is of "infinite rank" (these are the ones for which infinite dimensional irreducible representations exist), we show that the C*-algebra generated by an irreducible spin system is the CAR algebra, an infinite tensor product of copies of [Formula: see text], 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 [Formula: see text], where X is a Cantor set (possibly finite) and [Formula: see text] is either the CAR algebra or a finite tensor product of copies of [Formula: see text]. We describe the nature of this decomposition in terms of the "symplectic" properties of the ℤ2-valued form [Formula: see text]x, y ranging over the free infninite dimensional vector space over the Galois field ℤ2.