Let $V$ be a vector space (over $\mathbb C$, but I don't think it matters), and $m: V\otimes V \to V$ a "multiplication" that is associative and commutative (but I do not demand that it is unital). Is it possible that $m$ is an isomorphism $V\otimes V \overset \sim \to V$? Yes: $V$ can be zero-dimensional, or $V$ can be one-dimensional and $m$ non-zero.
Since $0$ and $1$ are the only finite solutions to $v^2 = v$, any other example must have $\dim V = \infty$. But there are many $\infty$s and many possible maps, and although I am sure that there are some examples, I am having trouble writing one down. Hence:
What is an example of an infinite-dimensional vector space $V$ and an isomorphism $m: V\otimes V \overset\sim\to V$ that is associative and commutative? Or is this impossible?