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Free products with amalgamation over central $ \mathrm{C}^*$-subalgebras

Authors: Kristin Courtney and Tatiana Shulman
Journal: Proc. Amer. Math. Soc. 148 (2020), 765-776
MSC (2010): Primary 46L05; Secondary 47A67
Published electronically: September 20, 2019
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Abstract: Let $ A$ and $ B$ be $ \mathrm {C}^*$-algebras whose quotients are all RFD (residually finite dimensional), and let $ C$ be a central $ \mathrm {C}^*$-subalgebra in both $ A$ and $ B$. We prove that the full amalgamated free product $ A*_C B$ is then RFD. This generalizes Korchagin's result that amalgamated free products of commutative $ \mathrm {C}^*$-algebras are RFD. When applied to the case of trivial amalgam, our methods recover the result of Exel and Loring for separable $ \mathrm {C}^*$-algebras. As corollaries to our theorem, we give sufficient conditions for amalgamated free products of maximally almost periodic (MAP) groups to have RFD $ \mathrm {C}^*$-algebras and hence to be MAP.

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Additional Information

Kristin Courtney
Affiliation: Mathematical Institute, WWU Münster, Einsteinstr. 62, Münster, Germany

Tatiana Shulman
Affiliation: Department of Mathematical Physics and Differential Geometry, Institute of Mathematics of Polish Academy of Sciences, Warsaw, Poland

Keywords: Amalgamated products, RFD
Received by editor(s): December 14, 2018
Received by editor(s) in revised form: June 8, 2019
Published electronically: September 20, 2019
Additional Notes: The research of the first-named author was supported by the Deutsche Forschungsgemeinschaft (SFB 878 Groups, Geometry & Actions).
The research of the second-named author was supported by the Polish National Science Centre grant under the contract number DEC- 2012/06/A/ST1/00256, by the grant H2020-MSCA-RISE-2015-691246-QUANTUM DYNAMICS and Polish Government grant 3542/H2020/2016/2, and from the Eric Nordgren Research Fellowship Fund at the University of New Hampshire.
Communicated by: Adrian Ioana
Article copyright: © Copyright 2019 American Mathematical Society