Complete Order Equivalence of Spin Unitaries

This paper is a study of linear spaces of matrices and linear maps on matrix algebras that arise from \emph{spin systems}, or \emph{spin unitaries}, which are finite sets $\mathcal S$ of selfadjoint unitary matrices such that any two unitaries in $\mathcal S$ anticommute. We are especially interested in linear isomorphisms between these linear spaces of matrices such that the matricial order within these spaces is preserved; such isomorphisms are called complete order isomorphisms, which might be viewed as weaker notion of unitary similarity. The main result of this paper shows that all $m$-tuples of anticommuting selfadjoint unitary matrices are equivalent in this sense, meaning that there exists a unital complete order isomorphism between the unital linear subspaces that these tuples generate. We also show that the C$^*$-envelope of any operator system generated by a spin system of cardinality $2k$ or $2k+1$ is the simple matrix algebra $\mathcal M_{2^k}(\mathbb C)$. As an application of the main result, we show that the free spectrahedra determined by spin unitaries depend only upon the number of the unitaries, not upon the particular choice of unitaries, and we give a new, direct proof of the fact [Helton-Klep-McCullough-Schweighofer, "Dilations, linear matrix inequalities, the matrix cube problem and beta distributions", Mem. Amer. Math. Soc. 257(2019)] that the spin ball $\mathfrak B_m^{\rm spin}$ and max ball $\mathfrak B_m^{\rm max}$ coincide as matrix convex sets in the cases $m=1,2$. We also derive analogous results for countable spin systems and their C$^*$-envelopes.