Let $\mathcal{A}$ be a unital $C^*$-algebra and let $\tau$ be any \emph{tracial state} on $\mathcal{A}$.

Set $\rk_\tau(B)=\lim_{k\to\infty}\tau(|B|^{1/k})$ for every $B\in \Mat_{n,m}(\mathcal{A})$.

Then $\rk_\tau$ is a \emph{Sylvester rank function} defined on rectangular matrices over $\mathcal{A}$.

Let $G$ be a discrete amenable group which admits a trace preserving action $\alpha$ on $(\mathcal{A},\tau)$.

Denote by $C_c(G,\mathcal{A})$, the \emph{group ring} of $G$ with coefficients in $\mathcal{A}$.

In this talk, we'll give two natural Sylvester rank functions on $C_c(G,\mathcal{A})$ and prove that they are equal.

This is a joint work with Prof. Hanfeng Li.