Since their inception in the 1930s by von Neumann, operator algebras have been used to shed light on many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two has been sought since their emergence in the late 1960s. We connect these seemingly separate types of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and $C^{*}$-algebras with additional $C^{*}$-algebraic structure. Our approach naturally applies to algebras arising from $C^{*}$-correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs.

We investigate the functors from modules to modules that occur as the summands of tensor powers and the functors from modules to Hopf algebras that occur as natural coalgebra summands of tensor algebras. The main results provide some explicit natural coalgebra summands of tensor algebras. As a consequence, we obtain some decompositions of Lie powers over the general linear groups.

A non-commutative multivariable analogue of Parrott’s generalization of the Sz.-Nagy–Foia\c{s} commutant lifting theorem is obtained. This yields Tomita-type commutant results and interpolation theorems (e.g. Sarason, Nevanlinna–Pick, Carathéodory) for $F_n^\infty\,\bar{\otimes}\,\M$, the weakly-closed algebra generated by the spatial tensor product of the non-commutative analytic Toeplitz algebra $F_n^\infty$ and an arbitrary von Neumann algebra $\M$. In particular, we obtain interpolation theorems for bounded analytic functions from the open unit ball of $\mathbb{C}^n$ into a von Neumann algebra.

A variant of the non-commutative Poisson transform is used to extend the von Neumann inequality to tensor algebras, and to provide a generalization of the functional calculus for contractive sequences of operators on Hilbert spaces. Commutative versions of these results are also considered.

AMS 2000 Mathematics subject classification: Primary 47L25; 47A57; 47A60. Secondary 30E05

We identify the C*-envelopes of certain quotients of tensor algebras over C*-correspondences.

We develop a notion of Morita equivalence for general C$^{\ast}$-correspondences over C$^{\ast}$-algebras. We show that if two correspondences are Morita equivalent, then the tensor algebras built from them are strongly Morita equivalent in the sense developed by Blecher, Muhly and Paulsen. Also, the Toeplitz algebras are strongly Morita equivalent in the sense of Rieffel, as are the Cuntz--Pimsner algebras. Conversely, if the tensor algebras are strongly Morita equivalent, and if the correspondences are aperiodic in a fashion that generalizes the notion of aperiodicity for automorphisms of C$^{\ast}$-algebras, then the correspondences are Morita equivalent. This generalizes a venerated theorem of Arveson on algebraic conjugacy invariants for ergodic, measure-preserving transformations. The notion of aperiodicity, which also generalizes the concept of full Connes spectrum for automorphisms, is explored; its role in the ideal theory of tensor algebras and in the theory of their automorphisms is investigated. 1991 Mathematics Subject Classification: 46H10, 46H20, 46H99, 46M99, 47D15, 47D25.

Our objective in this sequel to $[18]$ is to develop extensions, to representations of tensor algebras over ${{C}^{*}}$-correspondences, of two fundamental facts about isometries on Hilbert space: The Wold decomposition theorem and Beurling’s theorem, and to apply these to the analysis of the invariant subspace structure of certain subalgebras of Cuntz-Krieger algebras.