Starting from the abstract C* algebra $A$ of canonical commutative relations, a state $\rho$ over this algebra enables to construct a Hilbert space $A/I$ where $I$ is the ideal of the elements $a$ such that $\rho (a^* a) = 0 $. the vacuum in this space depends on the equivalence class of $\rho$. I read that in the case of infinite degrees of freedom, many inequivalent representations can be built by this construction. I wonder how the inequivalence of the representations depends on properties of these ideals (or not).


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