The Stable Rank of Some Free Product C -Algebras

It is proved that the reduced group C∗-algebra C∗ red(G) has stable rank one (i.e. its group of invertible elements is a dense subset) if G is a discrete group arising as a free product G1 ∗G2 where |G1| ≥ 2 and |G2| ≥ 3. This follows from a more general result where it is proved that if (A, τ) is the reduced free product of a family (Ai, τi), i ∈ I, of unital C∗-algebras Ai with normalized faithful traces τi, and if the family satisfies the Avitzour condition (i.e. the traces, τi, are not too lumpy in a specific sense), then A has stable rank one. Introduction It is an open problem if every finite, simple C-algebra has stable rank one. Recall that a unital C-algebra A is said to have stable rank one if the group of invertible elements in A is a norm dense subset of A. The notion of stable rank was introduced by M. Rieffel in [11] with the purpose of establishing what one might call non-stable K-theory results for certain concrete C-algebras, most notably the irrational rotation C-algebras. On a more speculative note, stable rank (which associates a number in {1, 2, 3, . . .}∪{∞} to every C-algebra) should measure the (non-commutative) dimension of the C-algebra, with stable rank equal to one corresponding to dimension 0 or 1. (It has later turned out that different definitions of dimensions, that agree in the “commutative” case, generalize to dimension concepts for C-algebras which do not agree.) 1 2 Ken Dykema, Uffe Haagerup, Mikael Rørdam Some of the non-stableK-theory results, obtained in [11] and [13], for unital C-algebras A of stable rank one are as follows. The three relations on projections in A (or in a matrix algebra over A), Murray–von Neumann equivalence, unitary equivalence and homotopy equivalence, are the same. Moreover, if p, q are projections in A (or in a matrix algebra over A), such that [p]0 = [q]0 in K0(A), then p and q are equivalent with respect to either of the three relations mentioned above. Also, the natural group homomorphism U(A)/U0(A) → K1(A), where U(A) is the group of unitary elements in A and U0(A) its connected component containing the unit of A, is an isomorphism. Another property of C-algebras of stable rank one can be found in [7]. It is observed in that paper that if A is a C-algebra of stable rank one, then each normal element in A can be approximated by normal elements in A with 1-dimensional spectrum. This again is used to prove that there exists a function f : R → R (independent of A) which is continuous at 0 and with f(0) = 0, such that dist ( a,N(A) ) ≤ f ( ‖aa− aa‖ ) for every a ∈ A with ‖a‖ ≤ 1. (Here N(A) denotes the set of normal elements in A.) The main result of this paper (Theorem 3.8) states that the reduced free product of any pair of unital C-algebras A1 and A2 with faithful normalized traces τ1 and τ2 has stable rank one if there exist unitary elements u ∈ A1, v, w ∈ A2 such that 0 = τ1(u) = τ2(v) = τ2(w) = τ2(w v) (the Avitzour condition). This result applies in particular to the reduced group C-algebras C red(G) when G = G1 ∗ G2 for some groups G1 and G2 satisfying |G1| ≥ 2 and |G2| ≥ 3 (see Corollary 3.9). It follows in particular that C red(Fn), 2 ≤ n ≤ ∞, and the Choi algebras C red(Zn ∗ Zm), n ≥ 2 and m ≥ 3, have stable rank one. Our result answers a question of Marc Rieffel [12], showing that every projective module over C red(Fn), 2 ≤ n ≤ ∞, is free. Indeed, if p is a projection in a matrix algebra over C red(Fn), then by [10] we have [p]0 = k · [1]0 in K0(C red(Fn)) for some k ∈ N. Now, if q is the free module over C red(Fn) of dimension k (viewed as a projection in the same matrix algebra over C red(Fn) as p belongs to), then [p]0 = [q]0. Since C ∗ red(Fn) has stable rank one, it follows that p and q are equivalent, and hence p is free. For comparison, it has been known for some time that the group ring, CFn, is a (left and right) free ideal ring, (fir), see [5], and hence every submodule of a free module over CFn is free. That CFn is a fir follows because, from [3] Corollary 2.12, the free product (also called coproduct) of firs is The Stable Rank of Some Free Products 3 a fir, and because CZ is a principal ideal domain, thus a fir. It also follows (see [10]) that U(C red(Fn))/U0(C ∗ red(Fn)) is naturally isomorphic to Z . In particular, with λ : Fn → C red(Fn) the left regular representation, λ(g) is connected in U(C red(Fn)) to 1 if and only if g belongs to the commutator subgroup of Fn. The strategy of the proof of Theorem 3.8 follows in parts (Lemmas 3.3 and 3.4) the work of the second named author in [8]. A crucial ingredient of the proof is the result of the third named author [14], that if A is a C–algebra whose group of invertibles is not dense in A, then there is an element, b ∈ A of norm 1 whose distance to the invertibles is equal to 1. In order to emphasize the main ideas of the proof, we will first, in Section 1, go through the proof that the invertibles are dense in C red(F2). In Section 2 we give some preliminaries for the proof of the more general result that the reduced free product of two unital C– algebras satisfying the Avitzour conditions has stable rank one, and in Section 3 we prove this theorem. In Section 4 we derive some conditions under which the Avitzour condition is satisfied (see Proposition 4.1). In particular, the Avitzour condition is satisfied if A1 and A2 both contain unital abelian subalgebras which are non-atomic with respect to the traces τ1 and τ2. Section 5 contains a brief discussion of the structure of more general reduced free products. 1 The proof of a special case Theorem 1.1 The reduced group C–algebra C red (F2) has stable rank one. Proof: Write F2 = 〈a, b〉, i.e. F2 is freely generated by a and b, and write A = C red(F2). Thus A is generated by the set of left translators, {λg | g ∈ F2}. Let τ denote the canonical, faithful, tracial state on A. We then have the inner product 〈w, z〉 = τ(zw) and we denote ‖z‖2 = 〈z, z〉1/2. Note that (λg)g∈F2 is an orthonormal basis for (the closure of) A with respect to this inner product. Suppose for contradiction that A has stable rank strictly greater that 1. Then by Theorem 2.6 of [14], there must be x ∈ A such that ‖x‖ = 1 and the distance in norm from x to the invertibles of A, (denoted GL(A)), is 1. But then ‖x‖2 < 1 because ‖x‖2 = 1 would imply that x be unitary. Hence we can find a finite linear combination of left translators, y = ∑n j=1 αjλgj , such that ‖y‖2 is strictly less than the distance from y to GL(A). Taking k ∈ N and considering each bgjb as a reduced word in a, b and their inverses, we see that there is k large enough so that for every j, bgjb −k when reduced begins and ends with b or b. Thus we see that there is no cancellation when we multiply (abgjb a)(abgj′b a), 4 Ken Dykema, Uffe Haagerup, Mikael Rørdam for any j and j. This shows that when u = λabk and v = λb−ka we have ‖(uyv)‖2 = ‖uyv‖2 , m ∈ N. (1.1) In [8] it was shown that for finite linear combinations of left translators from F2, the operator norm can be bounded in terms of the two–norm. Indeed from Lemma 1.5 of that paper it follows that if z = ∑p j=1 βjλhj and if N is the maximum of the lengths of the words hj, (as reduced words in a, b and their inverses), then ‖z‖ ≤ 2(N + 1)‖z‖2. Now we can apply the estimates from (1.1) to obtain an upper bound on the spectral radius of uyv, namely, lettingN be the maximum of the lengths of the words abg1b a, . . . , abgnb a, we have r(uyv) = lim sup m→∞ ‖(uyv)‖ ≤ lim sup m→∞ (2(mN + 1)‖uyv‖2 ) = ‖uyv‖2 = ‖y‖2. But the distance from uyv to GL(A) is clearly no greater than the spectral radius of uyv and the distance from y to GL(A) is equal to the distance from uyv to GL(A), so the inequality r(uyv) ≤ ‖y‖2 gives a contradiction to the choice of y. 2 Preliminaries for the general case 2.1 Standard orthonormal basis Let A be a unital C-algebra with a faithful normalized trace τ . Consider the corresponding Euclidean structure: 〈a, b〉 = τ(ba), a, b ∈ A, ‖a‖2 = 〈a, a〉 1 2 , a ∈ A. A subset X of A will be called a standard orthonormal basis for A ifX is an orthonormal set with respect to this Euclidean structure, if the linear span of X is a dense -subalgebra of A (with respect to the C-norm), and if 1 ∈ X. The set difference X\{1} will often be The Stable Rank of Some Free Products 5 denoted by X. Lemma Assume A is a separable C-algebra and that F ⊆ A is a finite orthonormal set containing 1. Then there exists a (countable) standard orthonormal basis for A which contains F . Proof: Choose a dense subset {a1, a2, a3, . . . } of A. Set X0 = F . Construct inductively finite orthonormal sets X0 ⊆ X1 ⊆ X2 ⊆ · · · , satisfying (i) an ∈ spanXn, (ii) spanXn is self-adjoint, (iii) x, y ∈ Xn−1 ⇒ xy ∈ spanXn, for all n ≥ 1, as follows. Suppose Xn−1 has been constructed. Let Vn be the finite dimensional subspace of A spanned by an, Xn−1, Xn−1 · Xn−1 and the adjoints of those elements. Then choose Xn to be an orthonormal basis for Vn that extends Xn−1. It is now easily verified that X = ⋃∞ n=0 Xn is a standard orthonormal basis for A. 2.2 Reduced free products Let (Ai, τi), i ∈ I, be a family of unital C-algebras Ai with faithful normalized traces τi. To each such family one can associate the reduced free product C-algebra (A, τ) = ∗ i∈I (Ai, τi), where A is a unital C-algebra and τ is a normalized faithful trace on A ([15], see also [16]). By construction, Ai is a sub-C -algebra of A, and τ extends τi for each i ∈ I. Elements in A of the form w = a1a2a3 · · ·an, where aj ∈ Ai(j), τ(aj) = 0, and i(1) 6= i(2), i(2) 6= i(3), . . . , i(n − 1) 6= i(n), are said to be reduced words of (block-) length n, and a1, a2, . . . , an are said to be the letters of the word w. (It turns out that the block length is well defined.) The unit 1 in A is said to be a reduced word of length 0. For each reduced word w of length n ≥ 1 we have τ(w) = 0. The linear span of all reduced words in A is a norm dense -subalgebra of A. 6 Ken Dykema, Uffe Haagerup, Mikael Rørdam Suppose now that Xi is a standard orthonormal basis for Ai. For each n ≥ 1, let Yn be the set of all reduced words x1x2x3 · · ·xn where xj ∈ X i(j), and i(1) 6= i(2), i(2) 6= i(3), . . . , i(n− 1) 6= i(n). Set Y0 = {1} and set ∗ i∈I Xi = ∞ ⋃ n=0 Yn. ¿From the construction of the reduced free products (see [15]) it is easily seen that ∗ i∈IXi is an orthonormal set (with respect to the Euclidean structure on A arising from τ). The linear span of i∈IXi is a -algebra (because each spanXi is a -algebra). The closure of the linear span of i∈IXi contains all reduced words in A, and is therefore equal to A. This shows that i∈IXi is a standard orthonormal basis for A. 3 The main result As in Section 2, let (Ai, τi), i ∈ I, be a family of unital C-algebras Ai with faithful normalized traces τi, and with standard orthonormal bases Xi ⊆ Ai. Let (A, τ) = ∗ i∈I (Ai, τi) be the reduced free product C-algebra, and let Y = ∗i∈IXi be the standard orthonormal basis for A defined in 2.2. Let En : spanY → spanYn be the orthogonal projection. We shall in the first lemma of this section describe the element En(vw), where v ∈ Yk and w ∈ Yl for some k, l and n. As in Section 2 we shall equip A with the Euclidean structure 〈a, b〉 = τ(ba), ‖a‖2 = 〈a, a〉 1 2 , a, b ∈ A. Lemma 3.1 Let v ∈ Yk, let w ∈ Yl and let n ≥ 0 be given. (i) Assume |k − l| < n ≤ k + l. Let q be the integer satisfying k + l − n = 2q or The Stable Rank of Some Free Products 7 k + l − n = 2q + 1 (which entails that 0 ≤ q < min{k, l}). Write v = v1xv2, v1 ∈ Yk−q−1, x ∈ X i , v2 ∈ Yq, w = w2yw1, w1 ∈ Yl−q−1, y ∈ X j , w2 ∈ Yq. It follows that En(vw) = { 〈v2w2, 1〉v1xyw1, if i 6= j 0, if i = j if k + l − n is even, and En(vw) = { ∑ u∈X i 〈v2w2, 1〉〈xy, u〉v1uw1, if i = j 0, if i 6= j if k+ l−n is odd. (Observe that 〈xy, u〉 6 = 0 for at most finitely many u ∈ X i because xy ∈ spanXi.) (ii) Assume n = |k − l|. Put q = min{k, l}, so that k + l − n = 2q, and write v = v1v2, v1 ∈ Yk−q, v2 ∈ Yq, w = w2w1, w1 ∈ Yl−q, w2 ∈ Yq. It follows that v1 = 1 or w1 = 1, and En(vw) = 〈v2w2, 1〉v1w1. (iii) If n < |k − l| or if n > k + l, then En(vw) = 0. Proof: We prove (i), (ii) and (iii) simultaneously by induction on min{k, l}. If min{k, l} = 0, then v = 1 or w = 1, and either n = |k − l| and q = 0, or n < |k − l|, or n > k + l. The claims are trivial in all six cases. Consider now the case where min{k, l} ≥ 1. Write v = vx and w = yw with v ∈ Yk−1, w ∈ Yl−1, x ∈ X s and y ∈ X t . If s 6= t, then vw is reduced, and so En(vw) = { vw, if n = k + l 0, if n 6= k + l. This formula agrees with (iii). If n = k + l, then q = 0 in (i) and (ii), which entails v2 = w2 = 1 and thereby 〈v2w2, 1〉 = 1. If |k − l| ≤ n < k + l, then q ≥ 1 in (i) and (ii), and v2w2 is a reduced word (because s 6= t). Hence 〈v2w2, 1〉 = 0. In either event, the expression for En(vw) displayed above agrees with the formulae in (i) and (ii). 8 Ken Dykema, Uffe Haagerup, Mikael Rørdam Suppose now that s = t. Then vw = 〈xy, 1〉vw + ∑ u∈X s 〈xy, u〉vuw. Hence En(vw) is as claimed in the lemma when n ≥ k + l − 1. Consider now the case where |k− l| ≤ n < k+ l−1. Then q ≥ 1, and in the notation of (i) and (ii) we can write v2 = v ′ 2x ′ and w2 = y w 2 for some v ′ 2, w ′ 2 ∈ Yq−1. Hence v = v1xv 2 and w = w 2yw1. Now, En(vw) = 〈xy, 1〉En(vw), and En(v w) is by the induction hypothesis given by the formulae in (i) and (ii). Since 〈v2w2, 1〉 = τ(v 2xyw 2) = τ ( v 2〈xy, 1〉1 · w 2 ) + τ ( v 2(x y − 〈xy, 1〉1)w 2 ) = τ ( v 2〈xy, 1〉1 · w 2 ) = 〈xy, 1〉〈v 2w 2, 1〉, the formulae for En(vw) in (i) and (ii) are verified. Finally, if n < |k − l|, then n < |(k − 1)− (l − 1)|, whence En(vw) = 〈xy, 1〉En(vw) = 0. Definition 3.2 For every a ∈ spanY and for every i ∈ I define Fi(a) to be the set of all x ∈ X i that appear as letters in words w ∈ Y in the support of a. (An element w ∈ Y is said to lie in the support of a if 〈a, w〉 6 = 0.) Since the support of a is finite, it follows that each Fi(a) is finite and that Fi(a) 6= ∅ only for finitely many i ∈ I. Set K(a) = max i∈I ( ∑