C* and W* Algebras

I recently came across C and W algebras which are basically just formalizations of the spaces we do quantum mechanics in.

Formally, we start with the following:

Definition: A algebra is a (complex) Banach space (complete normed vector space) equipped with a unary operator (involution) which satisfies the following:

1. It is conjugate linear:
   

2. It is antihomomorphic:
   

3. C* Property —

A

algebra is just a Algebra such that there exists a Banach space which is its dual.

The intuition I understand up till now is to consider these as the Hilbert space of state vectors where we do classical QM.