Topological quivers as multiplicity free relations - Brenken, Berndt (Mathematica Scandinavica, 2010)
For a C^*-correspondence E over a C^*-algebra A the restricted correspondence R(E) over the ideal I=E,E of A is introduced. The Cuntz-Pimsner algebra O_R(E) is the unaugmented C^*-algebra associated with E. For a topological quiver G an associated multiplicity free quiver, or topological relation, G^1 is introduced. The Cuntz-Pimsner algebra O_R(E) of the correspondence E associated with G is contained in the algebra O_R(E^1) for the correspondence E^1 associated with G^1 if the source map for the quiver is proper on an appropriate codomain. The unaugmented Cuntz-Pimsner algebras for G and G^1 are isomorphic if the left action for the correspondence R(E) is by compact adjointable maps and if the kernel for the left action is complemented in I. There are counter examples if either condition fails.