Our main goal here is to show that many of essential results in quantized functional analysis rely on the algebraic structure of the unital ring $B(H)$ of bounded operators on a Hilbert space $H$. The wide spectrum of structures on this ring is the main motivation for investigating the role of algebraic structure of $B(H)$ in different major results in this field. Our strategy for dealing with this general problem is finding the right category, containing operator algebras, in which a specific result remains true. The authors and their collaborators, have approached this problem from three directions, a survey of which is presented here. In the first approach, major theorems of quantized functional analysis such as Arveson's extension theorem, Ruan's theorem and Choi-Effros characterization of operator systems were proved in the much larger category of unital $*$-algebras. Moreover we unify all generalizations of the notion of operator systems. The second approach is devoted to investigating existence of projections properties in the category of $*$-algebras and constructing some noncommutative topology results. In particular some characterizations of Rickart $*$-algebras and some other types of $*$-algebras in terms of topological properties, were proved. In the third approach we work in the category of Baer $*$-rings ,that is, $*$-rings which only possess the lattice structure of projections of $B(H)$ but not necessarily the other structures. In this part major decomposition theorems of Wold, Nagy-Foias-Langer and Halmos-Wallen were proved in the purely algebraic setting.