  Motivated by Williams' problem of measuring novel differences between shift
equivalence (SE) and strong shift equivalence (SSE), we introduce three
equivalence relations that provide new ways to obstruct SSE while merely
assuming SE.
  Our shift equivalence relations arise from studying graph C*-algebras, where
a variety of intermediary equivalence relations naturally arise. As a
consequence we realize a goal sought after by Muhly, Pask and Tomforde, measure
a delicate difference between SSE and SE in terms of Pimsner dilations for
C*-correspondences of adjacency matrices, and use this distinction to refute a
proof from a previous paper.

Operator Algebras;Dynamical Systems