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Residual finiteness for central pushouts


Author: Alexandru Chirvasitu
Journal: Proc. Amer. Math. Soc. 149 (2021), 2551-2559
MSC (2020): Primary 46L09, 20E26, 22D10, 18A30
DOI: https://doi.org/10.1090/proc/15368
Published electronically: March 23, 2021
MathSciNet review: 4246805
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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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Additional Information

Alexandru Chirvasitu
Affiliation: Department of Mathematics, University of Buffalo, Buffalo, New York 14260-2900
MR Author ID: 868724
Email: achirvas@buffalo.edu

Keywords: $C^*$-algebra, amenable group, pushout, residually finite, residually finite-dimensional, Fell topology
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
Article copyright: © Copyright 2021 American Mathematical Society