Invertible subalgebras

We introduce invertible subalgebras of local operator algebras on lattices. An invertible subalgebra is defined to be one such that every local operator can be locally expressed by elements of the inveritible subalgebra and those of the commutant. On a two-dimensional lattice, an invertible subalgebra hosts a chiral anyon theory by a commuting Hamiltonian, which is believed not to be possible on any full local operator algebra. We prove that the stable equivalence classes of $\mathsf d$-dimensional invertible subalgebras form an abelian group under tensor product, isomorphic to the group of all $\mathsf d + 1$ dimensional QCA modulo blending equivalence and shifts. In an appendix, we consider a metric on the group of all QCA on infinite lattices and prove that the metric completion contains the time evolution by local Hamiltonians, which is only approximately locality-preserving. Our metric topology is strictly finer than the strong topology.