Closed Convex Hulls of Unitary Orbits in {C^{*}_{r}(mathbb{F}_{infty})}

Let ({C^*_r(mathbb{F}_{infty})}) be the reduced C*-algebra of the free group on infinitely many generators. Say that ({a, b in C^*_r(mathbb{F}_{infty})_{SA}}). Then ({a}) is majorized by ({b}) if and only if ({a in overline{Conv(U(b))}.}) In particular, ({tau(b)1 in overline{Conv(U(b))}.}) Moreover, in the above results, we provide uniform bounds for the number of unitary conjugates needed for a given approximation. In the above, ({Conv(U(b))}) is the convex hull of the unitary orbit of ({b}) in ({C^*_r(mathbb{F}_{infty})}).