Gauge Invariant Uniqueness Theorem for Corners of k-graphs

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