Non-compact generation of the equivariant E-theory ∞-category

Establish that the stable ∞-category E representing equivariant E-theory for separable G–C*-algebras is not compactly generated; explicitly, prove that E cannot be generated under countable colimits by its compact objects.

Background

The paper proves that the equivariant E-theory category E is compactly assembled, a weaker condition than compact generation. The authors argue that E is unlikely to be compactly generated; for example, if it were generated by certain compact objects induced from finite subgroups, it would force the Baum–Connes assembly map for the maximal crossed product to be an equivalence, contradicting known results for some groups with property T.

Clarifying whether E is or is not compactly generated would refine the structural understanding of E and its relationship to other constructions in noncommutative topology and K-theory.