Spectral Collapse

Following recent work of B . Barnes, we explore various conditions which guarantee inverse closure of a closed subalgebra of a unital Banach algebra. Noting the connection between inverse closure and collapse of the spectral lattice, we then salvage the hull‐boundary relations which remain when the given subalgebra is not necessarily inverse closed. The results of the paper are then applied to determine the extent of spectral collapse in the special case of seminormal elements of a B ∗ ‐algebra.