Dual algebra theory can be said to have begun in earnest with S. Brown's (positive) solution of the invariant subspace problem for subnormal operators (1978).

In this first talk, designed for a general audience, after a quick introduction to the invariant subspace problem, we will recall the definitions and fundamental concepts of Dual algebra theory as they have been subsequently identified and further developed. Special emphasis will be given to the connexion of factorization properties (in the predual of a dual algebra) with the existence of invariant subspaces. We will give an outline of the use of certain approximating sets (an essential feature of the theory) in a standard context and conclude with a quick survey of significant results including the 1986 Brown-Chevreau-Pearcy theorem asserting the existence of invariant subspaces for contractions whose spectrum contains the unit circle.