Sylvester rank functions for amenable normal extensions

摘要

We introduce a notion of amenable normal extension Sof a unital ring Rwith a finite approximation system F, encompassing the amenable algebras over a field of Gromov and Elek, the twisted crossed product by an amenable group, and the tensor product with a field extension. It is shown that every Sylvester matrix rank function rk of Rpreserved by S has a canonical extension to a Sylvester matrix rank function rkF for S. In the case of twisted crossed product by an amenable group, and the tensor product with a field extension, it is also shown that rkFdepends on rkcontinuously. When an amenable group has a twisted action on a unital C*-algebra preserving a tracial state, we also show that two natural Sylvester matrix rank functions on the algebraic twisted crossed product constructed out of the tracial state coincide. Published by Elsevier Inc.