One-to-one correspondance between maximal sets of antisymmetry and maximal projections of antisymmetry

Let X be a compact Hausdorff space and A a uniform algebra on X. Let if be an isometric unital representation that maps A into bounded linear operators on a Hilbert space. This research investigated that there is a one-to-one correspondence between the collection of maximal sets of antisymmetry for A and that of maximal projections of antisymmetry for π (A) under the extension of π if π satisfies a certain regularity property.