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.