On functional classes in higher-spin theory

The factorization over the ideal in the $d$-dimensional Vasiliev theory requires an accurate description of the higher-spin algebra $\mathcal{A}$ and its twisted adjoint module $\mathcal{A}^{tw}$ as certain functional spaces. It is shown that the conditions of $\mathcal{A}$ to be closed under action of the Moyal star product, and of $\mathcal{A}^{tw}$ to be a star-product $\mathcal{A}$-module along with some auxiliary conditions lead to properly defined free-level factorization procedure. The conditions on admissible functions have simple algebraic form; in the case of $\mathcal{A}$ the analogous functional space has been considered by Soloviev \cite{Soloviev:2013iba}. The relation of the obtained functional classes to the locality problem in the Vasiliev theory is discussed. In particular it is observed that certain non-local expressions diverge due to properties of $\mathcal{A}^{tw}$ while the spin-local expressions are well-defined.