## Residual finiteness for central pushouts

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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.## References

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## Additional Information

**Alexandru Chirvasitu**- Affiliation: Department of Mathematics, University of Buffalo, Buffalo, New York 14260-2900
- MR Author ID: 868724
- Email: achirvas@buffalo.edu
- Received by editor(s): April 28, 2020
- Received by editor(s) in revised form: September 7, 2020, and September 29, 2020
- Published electronically: March 23, 2021
- Additional Notes: This work was partially supported by NSF grants DMS-1801011 and DMS-2001128
- Communicated by: Adrian Ioana
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 2551-2559 - MSC (2020): Primary 46L09, 20E26, 22D10, 18A30
- DOI: https://doi.org/10.1090/proc/15368
- MathSciNet review: 4246805