The space of Schwartz functions on the plane is denoted by $\mathcal{S}$. The usual multiplication and the convolution multiplication on $\mathcal{S}$ are denoted by $m_1$ and $m_2$, respectively.

The Fourier transform $\mathcal{F}$ on $\mathcal{S}$ give a bijective correspondence between the $m_1$-subalgebras of $\mathcal{S}$ and $m_2$- subalgebras of $\mathcal{S}$.

We say that a subset $A$ of $\mathcal{S}$ separates compact subsets of $\mathbb{R}^2$ if for every disjoint compact set $K_1,K_2, \ldots,K_n$, there exist a function $f\in A$ such that $f(K_i) \cap f(K_j)$ is null, for $i\neq j$.

Is it true to say that the Fourier transform gives a bijective correspondence between separating $m_1$-subalgebras of $\mathcal{S}$ and its $m_2$ separating subalgebras?

The question is motivated by the following post: