The Structure of Spin Systems

A spin system is a sequence of self-adjoint unitary operators $U_1,U_2,...$ acting on a Hilbert space $H$ which either commute or anticommute, $U_iU_j=\pm U_jU_i$ for all $i,j$; it is is called irreducible when $\{U_1,U_2,...\}$ is an irreducible set of operators. There is a unique infinite matrix $(c_{ij})$ with $0,1$ entries satisfying $$U_iU_j=(-1)^{c_{ij}}U_jU_i, \qquad i,j=1,2,....$$ Every matrix $(c_{ij})$ with $0,1$ entries satisfying $c_{ij}=c_{ji}$ and $c_{ii}=0$ arises from a nontrivial irreducible spin system, and there are uncountably many such matrices. Infinite dimensional irreducible representations exist when the commutation matrix $(c_{ij})$ 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 $M_2(\Bbb C)$, and we classify the irreducible spin systems associated with a given matrix $(c_{ij})$ up to approximate unitary equivalence. That follows from a structural result. The $C^*$-algebra generated by the universal spin system $u_1,u_2,...$ of $(c_{ij})$ decomposes into a tensor product $C(X)\otimes\Cal A$, where $X$ is a Cantor set (possibly finite) and $\Cal A$ is either the CAR algebra or a finite tensor product of copies of $M_2(\Bbb C)$. The Bratteli diagram technology of AF algebras is not well suited to spin systems. Instead, we work out elementary properties of the $\Bbb Z_2$-valued "symplectic" form $$\omega(x,y) =\sum_{p,q=1}^\infty c_{pq}x_qy_p,$$ $x,y$ ranging over the free infninite dimensional vector space over the Galois field $\Bbb Z_2$, and show that one can read off the structure of $C(X)\otimes\Cal A$ from properties of $\omega$.