  For any densely defined, lower semi-continuous trace \tau on a C*-algebra A
with mutually commuting C*-subalgebras A_1, A_2, ... A_n, and a convex function
f of n variables, we give a short proof of the fact that the function (x_1,
x_2, ..., x_n) --> \tau (f(x_1, x_2, ..., x_n)) is convex on the space
\bigoplus_{i=1}^n (A_i)_{self-adjoint}. If furthermore the function f is
log-convex or root-convex, so is the corresponding trace function. We also
introduce a generalization of log-convexity and root-convexity called
\ell-convexity, show how it applies to traces, and give a few examples. In
particular we show that the trace of an operator mean is always dominated by
the corresponding mean of the trace values.
