Nondegenerate representations by compact operators

A nondegenerate star-algebra of compact operators decomposes its Hilbert space into irreducible closed reducing subspaces, each equivalence class occurring finitely many times. Pass to its norm closure. A nonzero compact positive element supplies a finite-rank spectral projection; a minimal projection in its finite-dimensional corner has scalar corner algebra. A cyclic vector in its range generates an irreducible reducing subspace. A maximal orthogonal family exhausts the space by nondegeneracy. Infinitely many equivalent copies would give orthogonal unit vectors whose images under some compact algebra element have equal nonzero norms, contradicting compactness.