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An invariant for certain operator algebras.

It is shown that the so-called principal function invariant, which is associated in a unitarily invariant way to operators with trace class self commutator TT(*) - T(*)T, is invariant under trace class perturbations of T and is an extension of the index of T-z to the whole plane. The connection of the principal function, under additional hypothesis, with the determination of the maximal ideal space of the C(*) algebra generated by T is discussed, and it is shown that the principal function, even when it takes noninteger values, plays a role in establishing the existence of invariant subspaces for T and in determining the point spectrum of T.