A note on approximate liftings

Abstract

In this paper, we prove approximate lifting results in the C ∗ -algebra and von Neumann algebra settings. In the C∗ -algebra setting, we show that two (weakly) semiprojective unital C*-algebras, each generated by n projections, can be glued together with partial isometries to define a larger (weakly) semiprojective algebra. In the von Neumann algebra setting, we prove lifting theorems for trace-preserving *-homomorphisms from abelian von Neumann algebras or hyperfinite von Neumann algebras into ultraproducts. We also extend a classical result of S. Sakai [16] by showing that a tracial ultraproduct of C*-algebras is a von Neumann algebra, which yields a generalization of Lin’s theorem [12] on almost commuting selfadjoint operators with respect to ‖ ·‖p on any unital C*-algebra with trace. Mathematics subject classification (2000): 46L05, 46L10.