Let $K$ be a global field, $A,B$ be two finite dimensional $K$ algebras. My question is, if $A \otimes_K K_v$ can be embedded in $B \otimes_K K_v$ for all places of $K$, can $A$ be embedded in $B$?Here shows the answer is negative. But is it true if $B$ is a central simple algebra over $K$?
Motivation: by density theorem, the result is true for Galois field extensions of $K$.
If $A$ is $M_2(K)$ and $B$ is a quaternion algebra over $K$, then $A \otimes_K K_v$ embeds in $B \otimes_K K_v$ for all but finitely many places of $K$, but certainly $A$ does not embed in $B$.