Return to Revisions

In your inequality you need to evaluate the $\pi$'s somewhere! Unless I am mistaken unpacking Fell (1962), Theorem 2.2 and Remark following, $[\pi_j]\to[\pi]$ means that for every choice of an $\varepsilon>0$, a compact set $K\subset G$, an integer $n$, and vectors $u_1,\dots,u_n\in\mathscr H_\pi$, there exist a $j$ and $v_1,\dots,v_n\in\mathscr H_{\pi_j}$ such that $$ \left|\langle u_i,\pi(g)u_i\rangle-\langle v_i,\pi_j(g)v_i\rangle\right|<\varepsilon $$ for all $i\in\{1,\dots,n\}$ and all $g\in K$. In short, positive-definite functions associated with $\pi$ are uniform-on-compacta limits of positive-definite functions associated with the $\pi_j$. (See also Dixmier (1977), 3.4.10 for $\mathrm C^*$-algebras and 18.1.5 for groups.)