  The classification of topological phases of matter is fundamental to
understand and characterize the properties of quantum materials. In this paper
we study phases of matter in one-dimensional open quantum systems. We define
two mixed states to be in the same phase if both states can be transformed into
the other by a shallow circuit of local quantum channels. We aim to understand
the phase diagram of matrix product density operators that are renormalization
fixed points. These states arise, for example, as boundaries of two-dimensional
topologically ordered states. We first construct families of such states based
on C*-weak Hopf algebras, the algebras whose representations form a fusion
category. More concretely, we provide explicit local fine-graining and local
coarse-graining quantum channels for the renormalization procedure of these
states. Finally, we prove that those arising from C*-Hopf algebras are in the
trivial phase.
