Abstract
The homotopy symmetric \(C^*\)-algebras are those separable \(C^*\)-algebras for which one can unsuspend in E-theory. We find a new simple condition that characterizes homotopy symmetric nuclear \(C^*\)-algebras and use it to show that the property of being homotopy symmetric passes to nuclear \(C^*\)-subalgebras and it has a number of other significant permanence properties. As an application, we show that if I(G) is the kernel of the trivial representation \(\iota :C^*(G)\rightarrow \mathbb {C}\) for a countable discrete torsion free nilpotent group G, then I(G) is homotopy symmetric and hence the Kasparov group KK(I(G), B) can be realized as the homotopy classes of asymptotic morphisms \([[I(G),B \otimes \mathcal {K}]]\) for any separable \(C^*\)-algebra B.
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M.D. was partially supported by NSF Grant #DMS–1362824. U.P. was partially supported by the SFB 878—“Groups, Geometry & Actions”.
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Dadarlat, M., Pennig, U. Deformations of nilpotent groups and homotopy symmetric \(C^*\)-algebras. Math. Ann. 367, 121–134 (2017). https://doi.org/10.1007/s00208-016-1379-0
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DOI: https://doi.org/10.1007/s00208-016-1379-0