Equivalence of Projections as Gauge Equivalence on Noncommutative Space

Projections play crucial roles in the ADHM construction on noncommutative ${\mathbb{R}}^4$. In this article a framework for the description of equivalence relations between projections is proposed. We treat the equivalence of projections as gauge equivalence'' on noncommutative space. We find an interesting application of this framework to the study of U(2) instanton on noncommutative $\R^4$: A zero winding number configuration with a hole at the origin is gauge equivalent’’ to the noncommutative analog of the BPST instanton. Thus the ``gauge transformation’’ in this case can be understood as a noncommutative resolution of the singular gauge transformation in ordinary ${\mathbb{R}}^4$.