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Journal of Operator Theory

Volume 73, Issue 2, Spring 2015  pp. 417-424.

A note on strongly quasidiagonal groups

Authors:  Caleb Eckhardt
Author institution:Department of Mathematics, Miami University, Oxford, 45056, U.S.A.

Summary: Recently we showed that all solvable virtually nilpotent groups have strongly quasidiagonal $C^*$-algebras, while together with Carri\'on and Dadarlat we showed that most wreath products fail to have strongly quasidiagonal $C^*$-algebras. These two results raised the question of whether or not strong quasidiagonality could characterize virtual nilpotence among finitely generated groups. This note provides examples of groups of the form $\Z^3\rtimes \Z^2$ that are not virtually nilpotent yet have strongly quasidiagonal $C^*$-algebras. Moreover we show these examples are the ``simplest" possible by proving that a group of the form $\Z^d\rtimes \Z$ is virtually nilpotent if and only if its group $C^*$-algebra is strongly quasidiagonal.

Keywords: group $C^*$-algebras, quasidiagonality

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