Why do quantum observables form an associative algebra in some contexts?

Question

In elementary quantum mechanics, we learn that quantum observables are self-adjoint operators that act on the Hilbert space of states.

However, in more advanced context, we talk of local operators, which are supposed to be quantum observables supported locally, forms an algebra. 

First, there is the Weyl algebra appearing in deformation quantization. It is the universal enveloping algebra of the Heisenberg Lie algebra, and obtained by deforming the the space of classical observables.

Second, in CFT, we assume operator product expansion. By multiplying two fields, we obtain another.

I would be grateful to anybody who could clear things out for me.

Answer 1

The usage of the term 'observable' in algebraic quantum field teory is more general than the textbook usage based on von Neumann measurements. it formalizes the common practice in quantum field theory to use this label for arbitrary operators with a common nuclear domain, which form an algebra.

Thsi is consistent with the fact that POVM measurements (generalizing von Neumann measurements) are unrelated to spectral theory and need neither self-adjointness nor even Hermiticity. For a discussion of the latter point, see my paper 'Quantum mechanics via quantum tomography'.