Gauge Invariant Uniqueness Theorem for Corners of k-graphs

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