Example of a non-algebraic $\ell^2$-function in two variables Let's call an $\ell^2$-function $\mathbb{N} \times \mathbb{N} \to \mathbb{C}$ algebraic if it is in the image of the natural algebra homomorphism $\ell^2(\mathbb{N}) \otimes \ell^2(\mathbb{N}) \to \ell^2(\mathbb{N} \times \mathbb{N})$, where on the left hand side we consider the usual, non-completed tensor product. In other words, $f(m,n)$ is algebraic iff it may be written as $\sum_{i=1}^{k} g_i(m) h_i(n)$ for some $k \in \mathbb{N}$ and $\ell^2$-functions $g_i,h_i$. Probably there are abstract reasons for the existence of non-algebraic functions. But I would like to know an explicit example of an $\ell^2$-function together with a concise and complete proof that it is not algebraic. For example:
Question. Can you give a proof that the $\ell^2$-function $(n,m) \mapsto \dfrac{1}{2^{n \cdot m}}$ is not algebraic?

An algebraic function has bounded rank (either as a tensor, or in this case just as a matrix). So let $f(n,m)$ be $1/2^n$ if $n=m$ and zero otherwise.
For all $p$ the composition $\{1,\cdots,p\}\times\{1,\cdots,p\}\to \mathbb{N}\times \mathbb{N}\to \mathbb{C}$ is a $p\times p$ matrix of full rank. If $f$ is algebraic then this matrix is just $\sum_{i=1}^k g_i(m)h_i(n)$ which has rank at most $k$.
