Free products with amalgamation over central 𝐶*-subalgebras

Courtney, Kristin; Shulman, Tatiana

doi:10.1090/proc/14746

Free products with amalgamation over central $\mathrm {C}^*$-subalgebras
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by Kristin Courtney and Tatiana Shulman

Proc. Amer. Math. Soc. 148 (2020), 765-776

DOI: https://doi.org/10.1090/proc/14746

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.

References

R. J. Archbold, On residually finite-dimensional $C^*$-algebras, Proc. Amer. Math. Soc. 123 (1995), no. 9, 2935–2937. MR 1301006, DOI 10.1090/S0002-9939-1995-1301006-9
Scott Armstrong, Ken Dykema, Ruy Exel, and Hanfeng Li, On embeddings of full amalgamated free product $C^*$-algebras, Proc. Amer. Math. Soc. 132 (2004), no. 7, 2019–2030. MR 2053974, DOI 10.1090/S0002-9939-04-07370-8
Gilbert Baumslag, On the residual finiteness of generalised free products of nilpotent groups, Trans. Amer. Math. Soc. 106 (1963), 193–209. MR 144949, DOI 10.1090/S0002-9947-1963-0144949-8
M. B. Bekka, On the full $C^*$-algebras of arithmetic groups and the congruence subgroup problem, Forum Math. 11 (1999), no. 6, 705–715. MR 1725593, DOI 10.1515/form.1999.021
M. B. Bekka and N. Louvet, Some properties of $C^*$-algebras associated to discrete linear groups, $C^*$-algebras (Münster, 1999) Springer, Berlin, 2000, pp. 1–22. MR 1796907
V. Belyaev, R. Grigorchuk, and P. Shumyatsky, On just-infiniteness of locally finite groups and their $C^*$-algebras, Bull. Math. Sci. 7 (2017), no. 1, 167–175. MR 3625854, DOI 10.1007/s13373-016-0091-4
Nathanial P. Brown and Kenneth J. Dykema, Popa algebras in free group factors, J. Reine Angew. Math. 573 (2004), 157–180. MR 2084586, DOI 10.1515/crll.2004.058
Man Duen Choi, The full $C^{\ast }$-algebra of the free group on two generators, Pacific J. Math. 87 (1980), no. 1, 41–48. MR 590864, DOI 10.2140/pjm.1980.87.41
A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
Kristin Courtney and Tatiana Shulman, Elements of $C^\ast$-algebras attaining their norm in a finite-dimensional representation, Canad. J. Math. 71 (2019), no. 1, 93–111. MR 3928257, DOI 10.4153/cjm-2017-040-x
Kenneth R. Davidson, $C^*$-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012, DOI 10.1090/fim/006
Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
Søren Eilers, Tatiana Shulman, and Adam P. W. Sørensen, $\mathrm {C}^*$-stability of discrete groups, preprint, arXiv:1808.06793.
Ruy Exel and Terry A. Loring, Finite-dimensional representations of free product $C^*$-algebras, Internat. J. Math. 3 (1992), no. 4, 469–476. MR 1168356, DOI 10.1142/S0129167X92000217
Tobias Fritz, Tim Netzer, and Andreas Thom, Can you compute the operator norm?, Proc. Amer. Math. Soc. 142 (2014), no. 12, 4265–4276. MR 3266994, DOI 10.1090/S0002-9939-2014-12170-8
Rostislav Grigorchuk, Magdalena Musat, and Mikael Rørdam, Just-infinite $C^*$-algebras, Comment. Math. Helv. 93 (2018), no. 1, 157–201. MR 3777128, DOI 10.4171/CMH/432
Donald W. Hadwin, An operator-valued spectrum, Indiana Univ. Math. J. 26 (1977), no. 2, 329–340. MR 428089, DOI 10.1512/iumj.1977.26.26025
Don Hadwin, A lifting characterization of RFD $\rm C^*$-algebras, Math. Scand. 115 (2014), no. 1, 85–95. MR 3250050, DOI 10.7146/math.scand.a-18004
Don Hadwin and Tatiana Shulman, Stability of group relations under small Hilbert-Schmidt perturbations, J. Funct. Anal. 275 (2018), no. 4, 761–792. MR 3807776, DOI 10.1016/j.jfa.2018.05.006
Graham Higman, A finitely related group with an isomorphic proper factor group, J. London Math. Soc. 26 (1951), 59–61. MR 38347, DOI 10.1112/jlms/s1-26.1.59
M. S. Khan and Sidney A. Morris, Free products of topological groups with central amalgamation. II, Trans. Amer. Math. Soc. 273 (1982), no. 2, 417–432. MR 667154, DOI 10.1090/S0002-9947-1982-0667154-7
Eberhard Kirchberg, On nonsemisplit extensions, tensor products and exactness of group $C^*$-algebras, Invent. Math. 112 (1993), no. 3, 449–489. MR 1218321, DOI 10.1007/BF01232444
Anton Korchagin, Amalgamated free products of commutative $C^*$-algebras are residually finite-dimensional, J. Operator Theory 71 (2014), no. 2, 507–515. MR 3214649, DOI 10.7900/jot.2012jul03.1986
Qihui Li and Junhao Shen, A note on unital full amalgamated free products of RFD $C^\ast$-algebras, Illinois J. Math. 56 (2012), no. 2, 647–659. MR 3161345
Alexander Lubotzky and Yehuda Shalom, Finite representations in the unitary dual and Ramanujan groups, Discrete geometric analysis, Contemp. Math., vol. 347, Amer. Math. Soc., Providence, RI, 2004, pp. 173–189. MR 2077037, DOI 10.1090/conm/347/06272
A. Malcev, On isomorphic matrix representations of infinite groups, Rec. Math. [Mat. Sbornik] N.S. 8 (50) (1940), 405–422 (Russian, with English summary). MR 0003420
Calvin C. Moore, Groups with finite dimensional irreducible representations, Trans. Amer. Math. Soc. 166 (1972), 401–410. MR 302817, DOI 10.1090/S0002-9947-1972-0302817-8
Calvin C. Moore and Jonathan Rosenberg, Groups with $T_{1}$ primitive ideal spaces, J. Functional Analysis 22 (1976), no. 3, 204–224. MR 0419675, DOI 10.1016/0022-1236(76)90009-4
Gilles Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294, Cambridge University Press, Cambridge, 2003. MR 2006539, DOI 10.1017/CBO9781107360235
Nico Spronk and Peter Wood, Diagonal type conditions on group $C^*$-algebras, Proc. Amer. Math. Soc. 129 (2001), no. 2, 609–616. MR 1800241, DOI 10.1090/S0002-9939-00-05788-9
Elmar Thoma, Eine Charakterisierung diskreter Gruppen vom Typ I, Invent. Math. 6 (1968), 190–196 (German). MR 248288, DOI 10.1007/BF01404824