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In the past five years some remarkable new ideas and techniques concerning the structure of the preduals of certain operator algebras have entered the theory of operators on Hilbert space. This has led to new invariant subspace theorems, a new dilation theory, and new techniques for proving the reflexivity of operators. (See the bibliography for a list of pertinent articles.) In the present paper, which is a natural successor to [ 1 l] and [9], this theory of the structure of preduals of operator algebras is carried forward, and several additional interesting results are found. In particular, as corollaries of our main theorems in Section 2 we obtain a new invariant subspace theorem (Theorem 3.8), and some new and quite surprising propositions (Coroilaries 3.3 and 3.4) concerning the invariant-subspace lattice of the much studied Bergman shift operator. The results herein were 369

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