tracial states on a finite dimensional $C^*$algebra

Question

If $A$ is a finite dimensional $C^*$ algebra,how many tracial states on $A$ ,is it countable or uncountable?How to construct a tracial state on $A$? Can anyone give me some hints?Thanks.

Answer 1

A finite-dimensional C$^*$-algebra is of the form $$ A=\bigoplus_{j=1}^m M_{k_j}(\mathbb C). $$ The number of states is indeed uncountable. Traces are precisely "convex combinations" of the traces in each block. That is, any trace on $A$ is of the form $$ \phi(\bigoplus_{j=1}^m x_j)=\sum_{j=1}^m t_j\, \tau_{k_j}(x_j), $$ where $\tau_{k_j}$ is the tracial state on $M_{k_j}(\mathbb C)$ and $t_1,\ldots, t_m$ are convex coefficients.

For the simplest example, let $A=\mathbb C^2$. Then all states are tracial, and they are precisely of the form $$ \phi_t(x,y)=tx+(1-t)y,\qquad\qquad t\in[0,1]. $$