12.28 (1992)

Let $G$ be a group. A function $f : G \to \mathbb{C}$ is called
$\qquad$ 1) normed if $f(1) = 1$;
$\qquad$ 2) central if $f(gh) = f(hg)$ for all $g, h \in G$;
$\qquad$ 3) positive-definite if $\sum_{k, l} f(g_k^{-1}g_l)\bar{c}_k c_l \geqslant 0$ for any $g_1, \dots, g_n \in G$ and any $c_1, \dots, c_n \in \mathbb{C}$.

Classify the infinite simple locally finite groups $G$ which possess functions satisfying 1)–3). The simple Chevalley groups are known to have no such functions, while such functions exist on locally matrix (or stable) classical groups over finite fields. The question is motivated by the theory of $C^*$-algebras, see § 9 in (A. M. Vershik, S. V. Kerov, J. Sov. Math., 38 (1987), 1701–1733).