Suppose $A$ is a $C^*$algebra, $\tau$ is a tracial functional on $M_k(A)$, then it produces a function $d_{\tau}:M_k(A)^{+}\to [0,\infty) $ defined by $d_{\tau}(a)=\lim_{n}\tau(a^{\frac{1}{n}})$.
The sequence in the right side is increasing and bounded above,so the limit exists.
My question is: what is the limit?
The answer depends on the algebra $A$ and the tracial state $\tau$. The general idea is that $d_\tau(a)$ measures the "size" of $a$, relative to $\tau$ if $A$ has multiple traces. I provide a couple of basic examples which might help to make this more clear.
As pointed out in the comments, if $A=\mathbb C$ and $\tau$ is the unique tracial state on $M_k(\mathbb C)$ then $d_\tau(a)=\operatorname{rank}(a)/k$.
If instead $A=C(X)$ for some compact Hausdorff space $X$, $k=1$, and $\tau$ is given by some probability measure $\mu\in M(X)$ (by one of the Riesz representation theorems, all tracial states on $C(X)$ arise in this way), then we have $d_\tau(f)=\mu(f^{-1}(0,\infty)).$