Title: Continuum Limits of Discrete Quantum Systems and their Algebras of Observables

Recently, new perspectives linking quantum theory to the theory of stochastic systems have been proposed, primarily focusing on finite-dimensional and discrete quantum systems. We are interested in developing these perspectives so that they directly apply to the continuum case. Interesting questions arise about continuum limits for algebras of observables. We present results indicating how careful continuum limits can be taken for such algebras and some relationships with the standard theory of von Neumann algebra.