A spin system is a sequence of self-adjoint unitary operators
$U_1,U_2,\dots$ 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 $\{U_1,U_2,\dots\}$ is an irreducible
set of operators.
There is a unique infinite
matrix $(c_{ij})$ with $0,1$ entries satisfying
$$
U_iU_j=(-1)^{c_{ij}}U_jU_i, \qquad i,j=1,2,\dots.
$$
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 $u_1,u_2,\dots$ 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 $\omega$.