We examine the ranks of operators in semi-finite C∗-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple C∗ -algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray-von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are satisfied by not-necessarily-unital approximately subhomogeneous algebras of slow dimension growth. Combined with results of the first author, this shows that slow dimension growth coincides with Z-stability for approximately subhomogeneous algebras.
The authors would like to thank George Elliott and Victor Kaftal for helpful comments.
The first author was supported by DFG (SFB 878). The second author was supported by NSF grant DMS-0969246 and the 2011 AMS Centennial Fellowship. The first author would also like to thank the Department of Mathematics at Purdue University for hosting him, during which time part of this article was written.
- Approximately subhomogeneous C∗-algebras
- Cuntz semigroup
- Dimension functions
- Nuclear C∗-algebras
- Slow dimension growth
- Stably projectionless C∗-algebras