© Cambridge University Press, 2019. In this paper we consider the algebraic crossed product induced by a homeomorphism on the Cantor set, where is an arbitrary field with involution and denotes the -algebra of locally constant -valued functions on. We investigate the possible Sylvester matrix rank functions that one can construct on by means of full ergodic -invariant probability measures on. To do so, we present a general construction of an approximating sequence of -subalgebras which are embeddable into a (possibly infinite) product of matrix algebras over. This enables us to obtain a specific embedding of the whole -algebra into, the well-known von Neumann continuous factor over, thus obtaining a Sylvester matrix rank function on by restricting the unique one defined on. This process gives a way to obtain a Sylvester matrix rank function on, unique with respect to a certain compatibility property concerning the measure, namely that the rank of a characteristic function of a clopen subset must equal the measure of.
- 16D70 (Secondary)
- 16E50 (Primary)
- 2010 Mathematics Subject Classification