ISRAELJOURNALOF MATHEMATICS, Vol.37, Nos. 1-2, 1980 COMPLEMENTED IDEALS IN THE DISK ALGEBRA BY P. G. CASAZZA,t R. W. PENGRA AND C. SUNDBERG ABSTRACT The complemented ideals in the disk algebra are characterized. The projection operators onto the ideals and the complements of the ideals are identified. 1. Let A denote the disk algebra. This is the set of analytic functions on the open unit disk in the complex plane which are continuous on the closed unit disk. It is a Banach space under the supremum norm. If K is a compact subset of the unit circle with Lebesgue measure zero, let AK denote the ideal in A consisting of functions which vanish on K. The most general closed ideals in A have the form (1) Jv = {g'F: g E AK} where F is an inner function continuous on the complement of K in the closed disk. *t When the set K is predetermined we will say that Jr is the ideal generated by F. A sequence of points {z.} in the unit disk will be called a Carleson sequence if the measure which assigns, for each n, a mass of 1 - [ z, j2 to the point z, (taking multiplicities into account) is a Carleson measure. That is, if there is a constant C so that whenever 0 < h < 1 and x real, (2) Z z~ESh(x) where ( a - Iz.12)_-< ch, t Research partially supported by a National Academy of Sciences Fellowship to the USSR during the 1977-78 academic year. This author wishes to thank S. Khrushchev, V. Peller, S. Kisljakov, I. Vasoonin, and B. Mitiagin for their many helpful comments and suggestions. ,t A convenient reference for the basic properties of the disk algebra is [7]. Received October 5, 1979 and in revised form December 23, 1979 76