A gauge invariant uniqueness theorem for corners of higher rank graph algebras

For a finitely aligned k-graph Λ with X a set of vertices in Λ, we define a universal C*-algebra called C* (Λ, X) generated by partial isometries. We show that C* (Λ, X) is isomorphic to the corner PXC*(Λ) PX, where PX is the sum of vertex projections in X. We then prove a version of the Gauge Invariant Uniqueness theorem for C*(Λ, X) and then use the theorem to prove various results involving fullness, simplicity and Morita equivalence as well as results relating to application in symbolic dynamics.