On Normed E∞-rings in Genuine Equivariant Cp-spectra

Genuine equivariant homotopy theory is equipped with a multitude of coherently commutative multiplication structures generalizing the classical notion of an E infinity-algebra. In this paper we study the Cp-E infinity-algebras of Nardin-Shah with respect to a cyclic group Cp of prime order. We show that many of the higher coherences inherent to parametrized algebras collapse; in particular, they may be described more simply and conceptually in terms of ordinary E infinity-algebras as a diagram category which we call normed algebras. Our main result provides a relatively straightforward criterion for identifying Cp-E infinity-algebra structures. We visit some applications of our result to real trace theories.