On normed $\mathbb{E}_\infty$-rings in genuine equivariant $C_p$-spectra

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