A characterization of the Leinert property

Let G be a discrete group and denote by λG its left regular representation on `2(G). Denote further by Fn the free group on n generators {g1, g2, . . . , gn} and λ its left regular representation. In this paper we show that a subset S = {t1, t2, . . . , tn} of G has the Leinert property if and only if for some real positive coefficients α1, α2, . . . , αn the identity ∥∥∥∥ n ∑ i=1 αi λG(ti) ∥∥∥∥ C∗ λ(G) = ∥∥∥∥ n ∑ i=1 αi λ(gi) ∥∥∥∥ C∗ λ(Fn) holds. Using the same method we obtain some metric estimates about abstract unitaries U1, U2, . . . , Un satisfying the similar identity ∥∥∥∥∑ni=1 Ui ⊗ Ui∥∥∥∥ min = 2 √ n− 1. Notation. Throughout in this paper G will denote a discrete group with unit element e and C∗ λ(G) the sub-C ∗-algebra of B(`2(G)) generated by its left regular representation λG. This algebra is equipped with the trace state τG(X) = 〈Xδe, δe〉. A subset {t1, t2, . . . , tn} of G is called free if it generates a copy of Fn, the free group on n generators. We shall denote the canonical generators of the free group by {g1, g2, . . . , gn}. When considering the free group we shall omit the subscript in λFn . We shall almost exclusively deal with finitely generated groups and repeatedly use the fact that given a generating set {h1, h2, . . . , hn} for the group G there is a (unique) quotient mapping q : Fn → G with q(gi) = hi ∀i ∈ {1, 2, . . . , n}. The following proposition collects some results from [K, Lemma 3.1 and Theorem 3]: Proposition 1. Let {h1, h2, . . . , hn} be a generating set of the group G and denote by α1, α2, . . . , αn positive real numbers. Then ∥∥∥∥ n ∑ i=1 αi (λG(hi) + λG(hi) ∗) ∥∥∥∥ C∗ λ(G) ≥ ∥∥∥∥ n ∑ i=1 αi (λ(gi) + λ(gi) ∗) ∥∥∥∥ C∗ λ(Fn) . Moreover in the case of equal coefficients, ∥∥∥∥ n ∑ i=1 (λG(hi) + λG(hi) ∗) ∥∥∥∥ C∗ λ(G) = ∥∥∥∥ n ∑ i=1 (λ(gi) + λ(gi) ∗) ∥∥∥∥ C∗ λ(Fn) = 2 √ 2n− 1 if and only if {h1, h2, . . . , hn} is a free set. Received by the editors February 22, 1996 and, in revised form, May 21, 1996. 1991 Mathematics Subject Classification. Primary 22D25; Secondary 43A05, 43A15, 60J15.