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This memoir is devoted to the study of the structure of irreducible triangular algebras generated by a maximal abelian algebra and an ordered semigroup of unitary operators acting on the maximal abelian algebra.
The motivation for this work is a construction proposed by Kadison and Singer for building new maximal triangular algebras out of triangular algebras and their ideals. This leads to the study of diagonal-disjoint ideals of next algebras. This memoir answers longstanding question…