Multiplicity one for pairs of Prasad--Takloo-Bighash type

Let $E\mathrm{/}F$ be a quadratic extension of non-archimedean local fields of characteristic different from $2$. Let $A$ be an $F$-central simple algebra of even dimension so that it contains $E$ as a subfield, set $G=A^{\times }$ and $H$ for the centralizer of $E^{\times }$ in $G$. Using a Galois descent argument, we prove that all double cosets $HgH\subset G$ are stable under the anti-involution $g\mapsto g^{-1}$, reducing to Guo’s result for $F$-split $G$ which we extend to fields of positive characteristic different from $2$. We then show, combining global and local results, that $H$-distinguished irreducible representations of $G$ are self-dual and this implies that $(G,H)$ is a Gelfand pair: $$di{m}_{\mathbb{C}}(Ho{m}_H(\pi ,\mathbb{C}))\le 1$$ for all smooth irreducible representations $\pi$ of $G$. Finally we explain how to obtain the the multiplicity one statement in the archimedean case using the criteria of Aizenbud and Gourevitch, and we then show self-duality of irreducible distinguished representations in the archimedean case too.