I've been thinking about the algebra of observables in spin systems a lot recently and, with only a relatively standard background in abstract algebra, wonder if some of the objects I've been considering have been considered by mathematicians.

The construction I'm interested in is as follows. Let $A$ be the algebra of observables on a lattice system with Hamiltonian $H$. The Hamiltonian defines a set of eigenstates; let us take $\langle \cdot\rangle:A \to \mathbb{R}$ to be the linear functional on the algebra furnished by the ground state. This is guaranteed to exist and also has some guaranteed properties: in particular, $\langle H O\rangle = \langle H\rangle \langle O \rangle$.

The algebra $A$ has a number of relations between elements given by the multiplication table and in many cases by the Lie bracket between operators. I am wondering about the structure of the image of the algebra under the map $\langle \cdot \rangle$, which I'll call $\langle A \rangle$. The algebraic relations on $A$ descend to relations on this 'expected value algebra'; if $O_1 O_2 = O_3$ in $A$, then $\langle O_1O_2 \rangle = \langle O_3\rangle$ in $\langle A \rangle$.

If the ground state is translation invariant, then we also have equivalences like $\langle O(n) O)(n+m)\rangle = \langle O(0)O(m)\rangle$, so many elements of $A$ are mapped to the same element of $\langle A \rangle$.

Finally, we have the relations $\langle H O\rangle = \langle H\rangle \langle O \rangle$, which will create even more identifications in the image of $A$ under $\langle \cdot \rangle$.

So the question is, given a characterization of $A$ and of the map $\langle \cdot \rangle$, what is the structure of the expected value algebra $\langle A \rangle$? In the case that $A$ is the Pauli algebra on $N$ qubits, $A$ has 4^N elements; how many unique elements are in $\langle A \rangle$? What is a basis for the algebra? Does this type of algebraic structure have a name?

I am imagining that the 'constraints' inherited from the symmetry of the state or the relation $\langle H O\rangle = \langle H\rangle \langle O \rangle$ define ideals in $A$, and I am asking about the quotient of $A$ by these ideals. So perhaps there is an idea of constructing Grobner bases for the ideals that is relevant here?

I am hoping that someone has some ideas about references, definitions, or similar constructions in the mathematical literature that might be useful. This paper (https://arxiv.org/pdf/2310.13111) of Raamsdonk is vaguely related to the question, but not exactly.