Sylvester matrix rank functions on crossed products

In this paper we consider the algebraic crossed product A: = C-K (X) (sic)(T) Z induced by a homeomorphism T on the Cantor set X, where K is an arbitrary field with involution and C-K(X) denotes the K-algebra of locally constant K-valued functions on X. We investigate the possible Sylvester matrix rank functions that one can construct on A by means of full ergodic T-invariant probability measures mu on X. To do so, we present a general construction of an approximating sequence of *-subalgebras A(n) which are embeddable into a (possibly infinite) product of matrix algebras over K. This enables us to obtain a specific embedding of the whole *-algebra A into M-K, the well-known von Neumann continuous factor over K, thus obtaining a Sylvester matrix rank function on A by restricting the unique one defined on M-K. This process gives a way to obtain a Sylvester matrix rank function on A, unique with respect to a certain compatibility property concerning the measure mu, namely that the rank of a characteristic function of a clopen subset U subset of X must equal the measure of U.