On L-Open and L-Closed C*-Algebras and the Construction of C*-Diagonals in C*-Algebras

This thesis broadly focuses on certain lifting problems related to the stability of relations and the existence of a particular rich abelian subalgebra of an inductive limit 𝒞*-algebra of 1-dimensional noncommutative 𝒞𝒲-complexes. We consider the newly introduced notions of 𝓁-open and 𝓁-closed 𝒞*-algebras by Blackadar. These 𝒞*-algebras derive their definitions and properties from the space of *-homomorphisms from the algebra to another 𝒞*-algebra, equipped with the point-norm topology. We characterize 𝓁-open and 𝓁-closed 𝒞*-algebras and use these characterizations to resolve some questions posed by Blackadar. Additionally, we explore the relationships of these notions with other 𝒞*-algebraic concepts, such as extension theory and the homotopy lifting property, which is the dual concept of the classical homotopy extension property. A 1-dimensional noncommutative 𝒞𝒲-complex exemplifies an 𝓁-open 𝒞*-algebra which serves as a building block for many important examples of stably finite classifiable 𝒞*-algebras. A 𝒞*-diagonal is an abelian 𝒞*-subalgebra with the unique extension property and certain regularity conditions. This study investigates the unique extension property of an abelian 𝒞*-subalgebra within a 1-dimensional NCCW complex, along with the limitations and implications of approximating *-homomorphisms between two such complexes. Furthermore, leveraging Leonel's classification result, which extends beyond simple 𝒞*-algebras, we establish the existence of 𝒞*-diagonals in the inductive limits of some 1-dimensional NCCW complexes.