Isomorphic Loday functors of non-homeomorphic spaces

Each commutative algebra $A$ gives rise to a representation ${\mathcal{L}}_A$, which we call the Loday functor of $A$, of the category $\mathrm{\Omega }$ of finite sets and surjective maps. In this paper we present two (infinite-dimensional) non-isomorphic algebras over $\mathbb{C}$ with isomorphic Loday functors – the algebras of continuous functions on the Möbius strip and on the cylinder.