UC San Diego

The classical Tannaka reconstruction theorem \cite{tannaka1939dualitatssatz} allows one to recover a compact group $G$, up to isomorphism, from the monoidal category of finite dimensional representations of $G$ over $\mathbb{C}$, $\text{Rep}_{\mathbb{C}}(G)$, as the tensor preserving automorphisms of the forgetful functor $\text{Rep}_{\mathbb{C}}(G) \longrightarrow \text{Vec}_{\mathbb{C}}$. Now let $G$ be a profinite group, $K$ a finite extension of $\mathbb{Q}_p$ and $\text{Ban}_G(K)$ the category of $K$-Banach space representations (of $G$). $\text{Ban}_G(K)$ can be equipped with a tensor product bi-functor $\hat\otimes_K$ and forgetful functor $\omega : \text{Ban}_G(K) \longrightarrow \text{Ban}(K)$. Using an anti-equivalence of categories (\cite{schneider2002banach}[Thm 2.3]) between $\text{Ban}_G(K)$ and the category of Iwasawa $G$-modules, we prove that a profinite group $G$ can be recovered from $\text{Ban}_G(K)$ as the group of tensor preserving automorphisms of $\omega$, in particular $G \cong \text{Aut}^\otimes(\omega)$.