Resumen de Topological quivers as multiplicity free relations

Berndt Brenken

• 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 OR(E) is the unaugmented C∗-algebra associated with E. For a topological quiver G an associated multiplicity free quiver, or topological relation, G1 is introduced. The Cuntz-Pimsner algebra OR(E) of the correspondence E associated with G is contained in the algebra OR(E1) for the correspondence E1 associated with G1 if the source map for the quiver is proper on an appropriate codomain. The unaugmented Cuntz-Pimsner algebras for G and G1 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.