# A Note on Approximate Liftings - Mathematics > Operator Algebras

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Abstract: In this paper, we prove approximate lifting results in the C$^{\ast}$-algebraand von Neumann algebra settings. In the C$^{\ast}$-algebra setting, we showthat two weakly semiprojective unital C*-algebras, each generated by $n$projections, can be glued together with partial isometries to define a largerweakly semiprojective algebra. In the von Neumann algebra setting, we provelifting theorems for trace-preserving *-homomorphisms from abelian von Neumannalgebras or hyperfinite von Neumann algebras into ultraproducts. We also extenda classical result of S. Sakai \cite{sakai} by showing that a tracialultraproduct of C*-algebras is a von Neumann algebra, which yields ageneralization of Lin-s theorem \cite{Lin} on almost commuting selfadjointoperators with respect to $\Vert\cdot\Vert {p}$ on any unital C*-algebra withtrace.