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 $C^*$ 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:

   $$(a \lambda + b)^* = a^* \bar \lambda+ b^*$$

2. It is antihomomorphic:

   $$(a b)^* = b^* a^*$$

3. C* Property-

   $$\|a * a\| = \|a\|^2$$

A $W^*$ algebra is just a $C^*$ Algebra such that there exists a banach space which is it’s dual.

The intution I understand uptill now is to consider these as the hilbert space of state vectors where we do classical QM.