Emulating equal operator using multiplication
Suppose that $f,g$ were functions satisfying $f(A) g(A) = 1$ and $f(A) g(B) = 0$ if $A \neq B$. Take any $A \neq B$. Then $f(A) g(B) = 0$, and so either $f(A) = 0$ or $g(B) = 0$. If $f(A) = 0$ then $f(A) g(A) = 0$, and if $g(B) = 0$ then $f(B) g(B) = 0$. So no such functions exist.