Naderi, Fouad2026-09-282026-09-282026-09-032026-09-25http://hdl.handle.net/1993/40171Let A be an operator algebra and X be a dual Banach A-bimodule. A bounded linear map D: A → X is called a derivation if D(ab) = D(a)b + aD(b) for all a, b ∈ A. The algebra is said to be amenable if for each derivation D from A into any dual Banach A-bimodule X, there is an x ∈ X such that D(a) = ax − xa for all a ∈ A. If for each derivation D, there is a net (or sequence) (xᵢ) in X such that D(a) = limᵢ (axᵢ − xᵢa) for all a ∈ A, either in the norm or weak* topology of X, then A is called approximately (sequentially) amenable. Clearly, amenability implies approximate amenability; however, the converse does not hold in general. A question posed by Ghahramani, Zhang, and others asks whether this converse implication holds for C*-algebras. In this thesis, we explore generalized amenability in C*-algebras, von Neumann algebras, quantum groups, and certain operator algebras arising in non-commutative function theory. We first show that, for certain locally compact quantum groups, approximate amenability implies amenability. This allows us to recover and extend some classical results of Ghahramani et al. Secondly, we study the sequential form of Ghahramani et al.'s question. We show that every sequentially amenable C*-algebra is amenable. We establish this by considering the enveloping von Neumann algebra of the given C*-algebra and a generalized version of Connes amenability for von Neumann algebras. Our proof relies on the theories of Alain Connes and Tomita–Takesaki for von Neumann algebras, spatial normal tensor products (laid out in Appendix B), lifting derivations via Hilbert W*-modules, and the weak* decomposition and Radon–Nikodym theorem for quantum expectations (developed in Appendix A). Afterwards, we introduce a non-commutative function-theoretic approach to study approximate amenability. This point of view leads to generalizations of several classical results. In particular, we study the Cuntz–Toeplitz C*-algebra E_d, the universal C*-algebra generated by d non-commuting isometries. We show that E_d is not a compact quantum group, while it possesses a compact quantum semigroup structure. However, we prove that it is an amenable Banach algebra. Finally, in Appendix A, we study weak* continuity and weak* singularity for quantum expectations, and we introduce a weak* decomposition for them. Using the Grothendieck–Haagerup inequality together with the notion of KMS states, we show that our weak* decomposition agrees with that of Arveson–Gheondea–Kavruk. This enables us to compute the Radon–Nikodym derivative for quantum expectations, which is then used in Chapter 4 to prove the equivalence between sequential Connes amenability and Connes amenability for finite von Neumann algebras.engC*-algebrasvon Neumann algebrasquantum groupsHilbert weak* modulesnon-commutative function theoryquantum expectations and weak* decomposition of themamenabilitybalanced normal tensor productslifting derivationsunbounded operatorsTomita-Takesaki theory