A Note on Approximate Liftings

TL;DR

This paper establishes approximate lifting theorems in C*-algebra and von Neumann algebra contexts, including results on semiprojective algebras, trace-preserving homomorphisms, and a generalization of Lin's theorem for almost commuting operators.

Contribution

It introduces new approximate lifting results for semiprojective C*-algebras and von Neumann algebras, extending classical theorems and providing new structural insights.

Findings

Semiprojective C*-algebras can be glued with partial isometries to form larger semiprojective algebras.

Lifting theorems for trace-preserving *-homomorphisms into ultraproducts are established.

A tracial ultraproduct of C*-algebras is shown to be a von Neumann algebra, generalizing Lin's theorem.

Abstract

In this paper, we prove approximate lifting results in the C$^{\ast}$-algebra and von Neumann algebra settings. In the C$^{\ast}$-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 \cite{sakai} by showing that a tracial ultraproduct of C*-algebras is a von Neumann algebra, which yields a generalization of Lin's theorem \cite{Lin} on almost commuting selfadjoint operators with respect to $\Vert\cdot\Vert_{p}$ on any unital C*-algebra with trace.