Let $H$ be a discrete hypergroup. Suppose I have a matrix $A=(A_{x,y})$ indexed over $H$ with nonnegative entries which defines a bounded operator on $\ell^2(H)$. When does there exist $f\in\ell^1(H)$ such that $A_{x,y}=\langle f*\delta_x,\delta_y\rangle$, i.e., $A$ is the matrix of transition probabilities for a random walk given by convolution with $f$?

A necessary condition is that $A$ commutes with $\ell^1(H)$ convolution on the right. Is this sufficient?