  Let $E/F$ 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 $H g H\subset G$ 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: \[dim_{\mathbb{C}}(Hom_{H}(\pi,\mathbb{C}))\leq 1\] 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.
