The state space is a vector space and observables are constructed from linear operators. However the state space is infinite dimensional. The algebraic theory just gives infinite matrices but to construct observables you need to be able to take limits. Different types of limits give different classes of operator algebras eg C^* algebras , von Neumann algebras etc.

Experts propose the axiomatic quantum field theory (AQFT) to systematize quantum field theories (QFTs). It is fundamentally grounded in traditions of C*-algebras and von Neumann algebras for simulating quantum systems (Haag, 1996). AQFT, by definition, reveals the infinite variables within the quantum environment and analyzes how quantum field theories are seen in this paradigm and how states and observables are expressed in quantum field theory. In summary, AQFT suggests that all operator algebras capture the locality and causality principles that relate to relativistic QFT. For example, there will always be an algebraic structure that expresses local values in individual spatial regions that cannot be explained with ordinary field functions. Specifically, local values in any spatial region form a net that connects to other variable regions for the field.

This depends on algebraic structure that we can mathematically validate on quantum regions (Araki, 1999). Certain von Neumann algebras are associated with infinite Hilbert spaces in QFT. They can also be used to quantify variables and type III, making the theory consistent even in very large subsystems (Takesaki, 2003). Some von Neumann algebras have algebraic structures that provide a more finite interaction of the field operators. As a result, we need to find a more precise understanding of the vacuum and the entities of the infinite variables. These factors, in many-variable spacetime, allow us to represent transitions and thermal states with algebraic structures (Doplicher et al., 1971). The Tomita–Takesaki theory also clarifies how to include the phase, which fits better with the modular definition of the algebraic process (Doplicher et al., 1971). This is strikingly analogous to the steps we use in operator algebras in QFT.

In fact, the modular theories describe a kind of algebra that can represent any algebraic structure by constructing bi-similarity structures (Haag, 1996). In this sense, a bijective simorphism of states on the observable algebra will Zariski-analyze the algebraic structures (Doplicher et al., 1971). This gives us an analysis of second-quantization of experimentals that is very different from other experimental approaches to the quantum field theory Lagrangian or path integration, which are hard to exploit and, consequently, analyses the QFT model could use a targeted method that is based on the state that is now studied (Haag, 1996). In sum, operator algebras in the AQFT model are extremely effective.

References

Araki, H. (1999). Mathematical theory of quantum fields. Oxford University Press.

Doplicher, S., Haag, R., & Roberts, J. E. (1971). Local observables and particle statistics. Communications in Mathematical Physics, 23(3), 199-230.

Haag, R. (1996). Local quantum physics: Fields, particles, Algebras (2nd ed.). Springer.

Takesaki, M. (2003). Theory of operator algebras. (1) Springer.