I am wondering on whether there are any references discussing how path integrals relate to the algebraic approach for quantum field theory. More specifically, in the algebraic approach, states are normalized linear functionals acting on an algebra of operators $\mathcal{A}$. Expectation values then take the form $\omega(A)$, for $A \in \mathcal{A}$. In the path integral approach, we have expectation values often being written as $\int e^{i S} A \mathcal{D}\phi$, where $\phi$ is the quantum field and $S$ its action. I'd like references discussing how these two ways of writing expectation values relate to each other in the sense of whether any algebraic state can be written as a path integral, which sorts of caveats appear when dealing with path integrals, and so on.

Ideally, I'd like for the discussion to include QFT in curved spacetime, but references in flat spacetime are also welcome if more general situations are not possible. I'm interested in both Euclidean and Lorentzian signature, so references in either case are welcome.