Let $k$ be an algebraically closed field and let $X$, $Y$ be varieties over $k$. Let us denote by $\mathcal{O}(X)$ and $\mathcal{O}(Y)$ the $k$-algebra of regular functions on $X$ and $Y$ respectively. There exists a natural homomorphism of $k$-algebras: $$ \theta \colon \mathcal{O}(X) \otimes_k \mathcal{O}(Y) \to \mathcal{O}(X \times Y) \, , \quad f \otimes g \mapsto \left( (x,y) \mapsto f(x)f(y) \right) \, . $$ It is well-known, that $\theta$ is an isomorphism in case $X$ and $Y$ are affine. Is it true that $\theta$ is an isomorphism if $X$ and $Y$ are just quasi-affine (i.e. not necessarily affine)? Is this true for arbitrary varieties $X$ and $Y$? Any proof or counter-example or any reference to a text book would be perfect.

Let $k$ be an algebraically closed field and let $X$, $Y$ be varieties over $k$. Let us denote by $\mathcal{O}(X)$ and $\mathcal{O}(Y)$ the $k$-algebra of regular functions on $X$ and $Y$ respectively. There exists a natural homomorphism of $k$-algebras: $$ \theta \colon \mathcal{O}(X) \otimes_k \mathcal{O}(Y) \to \mathcal{O}(X \times Y) \, , \quad f \otimes g \mapsto \left( (x,y) \mapsto f(x)g(y) \right) \, . $$ It is well-known, that $\theta$ is an isomorphism in case $X$ and $Y$ are affine. Is it true that $\theta$ is an isomorphism if $X$ and $Y$ are just quasi-affine (i.e. not necessarily affine)? Is this true for arbitrary varieties $X$ and $Y$? Any proof or counter-example or any reference to a text book would be perfect.