I'm currently studying the basics of Banach and $C^*$-algebras. Almost all the proofs i've seen so far are very simple but some of them are extremely tricky (in my opinion). This tricky interplay between the bits of analysis and algebra make some of the proofs seem unilluminating and disconnected from the flow of the theory.
Is there an approach to operator algebras that uses category theory to simplify and "trivilize" the elementary theory?
Ideally it would use input from analysis (category of banach spaces, hilbert spaces, topological vector spaces) and algebra (category of associative algebras over $\mathbb{C}$ for a start) only when absolutely (and obviously) necessary.
While I cannot give an authoritative answer that such an approach does not exist, it would really surprise me if it did.
That "tricky interplay" between algebra and analysis is precisely what makes operator algebras interesting. In particular, the close relation between an algebraic notion (the spectral radius) and a topological one (the norm) is at the cornerstone of the theory. Every key theorem uses that interplay in a very smart (rather than "tricky") way.