Explore the intricacies of non-commutative analytic Toeplitz algebra and free semigroup algebras in this 54-minute lecture by Ken Davidson from the University of Waterloo. Delivered as part of the Focus Program on Analytic Function Spaces and their Applications at the Fields Institute, delve into topics such as weak operator topology, vector crv and c0, Popescu's decomposition, row isometry, and functional calculus. Examine finitely correlated representations, invariant subspaces, and inner functions. Investigate homomorphisms, surjective maps, and the concept of free semigroups. Analyze absolute continuity, decomposition methods, and L1 decomposition results. Gain a deeper understanding of these advanced mathematical concepts and their applications in analytic function spaces.