Toward the construction of a C^* algebra in string field theory

In string field theory there is a fundamental object, the algebra of string field states \A, that must be understood better from a mathematical point of view. In particular we are interested in finding, if possible, a C^* structure over it, or possibly over a subalgebra U \subset \A. In this paper we define a * operation on \A, and then using a particular description of Witten's star product, the Moyal's star product, we find an appropriate pre C^* algebra S(R^{2n}) on a finite dimensional manifold, where the finite dimensionality is obtained with a cutoff procedure on the string oscillators number. Then we show that using an inductive limit we obtain a pre C^* algebra S \subset \A that can be completed to a C^* algebra.