I am wondering if Kaplansky's "Ring of Operators" is worth studying if I'm interested in functional analysis (more specifically von Neumann algebras). Yes any vNa is a B. ring but my question is more if reading this book or studying this subject is fruitful for practical vNa research. Do people who actually do research in vNa or C* algebras ever appeal to the more general structure of Baer rings?
In particular, do Baer rings play a significant role in the (modern) study of von Neumann algebras?
The answer is definitely no. Kaplansky's attemp to study von Neumann algebras from a completely algebraic point of view was a very interesting exercise, but it didn't really bear fruit. Until the 70s people were publishing papers showing that certain property occurred (or didn't) in an AW$^*$-algebra. These days, only a handful of people (mostly from those times) pay attention to AW$^*$-algebras. And I know of no instance where a paper in von Neumann algebras builds on something done on AW$^*$-algebras. Even if someone would prove the main open conjecture (that all AW$^*$-algebras are monotone complete), I don't think that would attract much attention.
Shorter answer: the whole Baer-AW$^*$ effort always played catchup to the theory of von Neumann algebras, and was eventually left in the dust.