Residual finiteness for central pushouts

Chirvasitu, Alexandru

doi:10.1090/proc/15368

Residual finiteness for central pushouts
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by Alexandru Chirvasitu

Proc. Amer. Math. Soc. 149 (2021), 2551-2559

DOI: https://doi.org/10.1090/proc/15368

Published electronically: March 23, 2021

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Abstract:

We prove that pushouts $A*_CB$ of residually finite-dimensional (RFD) $C^*$-algebras over central subalgebras are always residually finite-dimensional provided the fibers $A_p$ and $B_p$, $p\in \mathrm {spec}~C$ are RFD, recovering and generalizing results by Korchagin and Courtney-Shulman. This then allows us to prove that certain central pushouts of amenable groups have RFD group $C^*$-algebras. Along the way, we discuss the problem of when, given a central group embedding $H\le G$, the resulting $C^*$-algebra morphism is a continuous field: this is always the case for amenable $G$ but not in general.

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