Linear-topological classification of matroid C*-algebras

We classify, up to a linear-topological isomorphism, all matroid C*-algebras (i.e. direct limits of a sequence of finite dimensional matrix algebras). There are two isomorphism classes: one is represented by LC(12), the C*-algebra of all compact operators on the Hilbert space 12, and the other - by the Fermion algebra F= RM1 M2. In particular, any UHF algebra is isomorphic to F as a Banach space. We also show that LC(12) is isometric to a 1-complemented subspace of F, but F is not isomorphic to a subspace of a quotient space of LC(L2).