THE ATIYAH CONJECTURE AND ARTINIAN RINGS

PETER A. LINNELL AND THOMAS SCHICK

Abstract. Let G be a group such that its finite subgroups have bounded order, let d denote the lowest common multiple of the orders of the finite subgroups of G, and let K be a subfield of ℂ that is closed under complex conjugation. Let U(G) denote the algebra of unbounded operators affiliated to the group von Neumann algebra N (G), and let D(KG, U(G)) denote the division closure of KG in U(G); thus D(KG, U(G)) is the smallest subring of U(G) containing KG that is closed under taking inverses. Suppose n is a positive integer, and α ∈ M_{n}(KG). Then α induces a bounded linear map α: ℓ^{2}(G)^{n} → ℓ^{2}(G)^{n}, and ker α has a well-defined von Neumann dimension dim_{N (G)}(ker α). This is a nonnegative real number, and one version of the Atiyah conjecture states that d dim_{N (G)}(ker α) ∈ ℤ. Assuming this conjecture, we shall prove that if G has no nontrivial finite normal subgroup, then D(KG, U(G)) is a d × d matrix ring over a skew field. We shall also consider the case when G has a nontrivial finite normal subgroup, and other subrings of U(G) that contain KG.

1. Introduction

In this paper ℕ will denote the positive integers {1, 2, … }, all rings will have a 1, subrings will have the same 1, and if n ∈ ℕ, then M_{n}(R) will indicate the n × n matrices over the ring R and GL_{n}(R) the invertible matrices in M_{n}(R). Let G be a group, let ℓ^{2}(G) denote the Hilbert space with orthonormal basis the elements of G, and let B(ℓ^{2}(G)) denote the bounded linear operators on ℓ^{2}(G). Thus we can write elements a ∈ ℓ^{2}(G) in the form ∑_{g∈G} a_{g}g, where a_{g} ∈ ℂ and ∑_{g∈G} |a_{g}|^{2} < ∞. Then ℂG acts faithfully on the left of ℓ^{2}(G) as bounded linear operators via the left regular representation, so we may consider ℂG as a subalgebra of B(ℓ^{2}(G)). The weak closure of ℂG in B(ℓ^{2}(G)) is the group von Neumann algebra N (G) of G. Also if n ∈ ℕ, then M_{n}(ℂG) acts as bounded linear operators on ℓ^{2}(G)^{n} and the weak closure of this ring in B(ℓ^{2}(G)^{n}) is M_{n}(N (G)). Let 1 indicate the element of ℓ^{2}(G) which is 1 at the identity of G and zero elsewhere. Then the map θ ↦ θ1: N (G) → ℓ^{2}(G) is an injection, so we may regard N (G) as a subspace of ℓ^{2}(G). We can now define tr: N (G) → ℂ by tr(a) = a_{1}. For α ∈ M_{n}(N (G)), we can extend this definition by setting tr(α) = ∑_{i=1}^{n} tr(α_{ii}), where α_{ij} are the entries of α. A useful property is that if α is a positive operator, then tr(α) ≥ 0. Also we can use tr to give any right N (G)-module M a well defined dimension dim_{N (G)} M, which in general is a non-negative real number or ∞ [10, §6.1]. If e is a projection in M_{n}(N (G)), then dim_{N (G)} e M_{n}(N (G)) = tr(e). Furthermore if α ∈ M_{n}(N (G)), so α is a Hilbert space map ℓ^{2}(G)^{n} → ℓ^{2}(G)^{n}, then since ℓ^{2}(G)^{n} is a right N (G)-module, dim_{N (G)} ker α is well defined and is equal to dim_{N (G)}{β ∈ N (G) | αβ = 0}. Finally N (G) has an involution which sends an operator to its adjoint; if a = ∑_{g∈G} a_{g}g, then a^{∗} = ∑_{g∈G} \overline{a}_{g}g^{−1}, where the bar indicates complex conjugation.

Date: Tue Nov 20 12:41:28 EST 2007.

2000 Mathematics Subject Classification. Primary: 16S34; Secondary: 20C07, 22D25, 46L99.

Key words and phrases. Atiyah conjecture, group von Neumann algebra.

A ring R is called regular, or sometimes von Neumann regular, if for every x ∈ R, there exists an idempotent e ∈ R with xR = eR [5, Theorem 1.1]. It is called finite, or directly finite, if xy = 1 implies yx = 1 for all x, y ∈ R. Finally a ∗-regular ring R is a regular ring with an involution ^{∗} with the property that x ∈ R and x^{∗}x = 0 implies x = 0. In a ∗-regular ring, given x ∈ R, there is a unique projection e such that xR = eR; so e = e^{∗} = e^{2}.

Let U(G) denote the algebra of unbounded operators on ℓ^{2}(G) affiliated to N (G) [10, §8]. Then the involution on N (G) extends to an involution on U(G), and U(G) is a finite ∗-regular algebra. Also if M is a right N (G)-module, then dim_{N (G)} M = dim_{N (G)} M ⊗_{N (G)} U(G); in particular dim_{N (G)} eU(G) = tr(e).

For any subring R of the ring S, we let D(R, S) denote the division closure of R in S; that is the smallest subring of S containing R that is closed under taking inverses. In the case G is a group and K is a subfield of ℂ, we shall set D(KG) = D(KG, U(G)). For any group G, let lcm(G) indicate the least common multiple of the orders of the finite subgroups of G, and adopt the convention that lcm(G) = ∞ if the orders of the finite subgroups of G are unbounded. One version of the strong Atiyah conjecture states that if G is a group with lcm(G) < ∞, then the L^{2}-Betti numbers of every closed manifold with fundamental group G lie in the abelian group \frac{1}{lcm(G)}ℤ. This is equivalent to the conjecture that if n ∈ ℕ, A ∈ M_{n}(ℚ G) and α: ℓ^{2}(G)^{n} → ℓ^{2}(G)^{n} is the map induced by left multiplication by A, then lcm(G) dim_{N (G)} ker α ∈ ℤ [9, Lemma 2.2]. In this paper, we shall consider more generally the case when the coefficient ring is a subfield of ℂ.

Definition 1.1. Let G be a group with lcm(G) < ∞, and let K be a subfield of ℂ. We say that the strong Atiyah conjecture holds for G over K if

lcm(G) dim_{N (G)} ker α ∈ ℤ for all α ∈ M_{n}(KG).

This is equivalent to the conjecture that if M is a finitely presented KG-module, then lcm(G) dim_{N (G)} M ⊗_{KG} N (G) ∈ ℤ [10, Lemma 10.7]. Obviously if G satisfies the strong Atiyah conjecture over ℂ, then G satisfies the strong Atiyah conjecture over K for all subfields K of ℂ. The strong Atiyah conjecture over ℂ is known for large classes of groups; for example [7, Theorem 1.5] tells us that it is true if G has a normal free subgroup F such that G/F is an elementary amenable group. If K is the algebraic closure of ℚ in ℂ, it is known for even larger classes of groups, for example [4, Theorem 1.4] for groups which are residually torsion-free elementary amenable. The following result is well known; see for example [12, Lemma 3].

Proposition 1.2. Let G be a torsion-free group (i.e. lcm(G) = 1) and let K be a subfield of ℂ. Then G satisfies the strong Atiyah conjecture over K if and only if D(KG) is a skew field.

The purpose of this paper is to generalize Proposition 1.2. We will denote the finite conjugate subgroup of the group G by Δ(G), and the torsion subgroup of Δ(G) by Δ^{+}(G) (this is a subgroup, compare [11, Lemma 19.3]. We shall prove

Theorem 1.3. Let G be a group with d := lcm(G) < ∞ and Δ^{+}(G) = 1, and let K be a subfield of ℂ that is closed under complex conjugation. Then G satisfies the strong Atiyah conjecture over K if and only if D(KG) is a d × d matrix ring over a skew field.

It seems plausible that if K is a subfield of ℂ which is closed under complex conjugation and G is a group with lcm(G) < ∞ which satisfies the Atiyah conjecture over K, then D(KG) is a semisimple Artinian ring. However we cannot prove this, though we are able to prove a slightly weaker result, and to state this we require the following definition.

Definition 1.4. Let R be a subring of the ring S. The extended division closure, E(R, S), of R in S is the smallest subring of S containing R with the properties (a) If x ∈ E(R, S) and x^{−1} ∈ S, then x ∈ E(R, S).

(b) If x ∈ E(R, S) and xS = eS where e is a central idempotent of S, then e ∈ E(R, S).

Obviously E(R, S) ⊇ D(R, S). Note that if {R_{i}} is a collection of subrings of S satisfying 1.4(a) and 1.4(b) above, then ⋂_{i} R_{i} is also a subring of S satisfying 1.4(a) and 1.4(b), consequently E(R, S) is a well defined subring of S containing R. Also if G is a group and K is a subfield of ℂ, then we write E(KG) for E(KG, U(G)). Observe that, if G is torsion-free and if the strong Atiyah conjecture holds for G over K, then D(KG) is a division ring, hence xD(KG) = D(KG) for every 0 ≠ x ∈ D(KG) and consequently E(KG) = D(KG) in this case. We are tempted to conjecture that this is always the case. We hope to show in a later paper that this should follow from a suitable version of the Atiyah conjecture.

We shall prove

Theorem 1.5. Let G be a group with lcm(G) < ∞, and let K be a subfield of ℂ that is closed under complex conjugation. Suppose that G satisfies the strong Atiyah conjecture over K. Then E(KG) is a semisimple Artinian ring.

Thus in particular if K is a subfield of ℂ that is closed under complex conjugation and G is a group with lcm(G) < ∞ which satisfies the strong Atiyah conjecture over K, then KG can be embedded in a semisimple Artinian ring. Theorem 1.5 follows immediately from the more general Theorem 2.6 in Section 2.

In Section 3 we will show, somewhat unrelated to the rest of the paper, that KG can be embedded in a least subring of U(G) that is ∗-regular.

2. Proofs

Let R be a subring of the ring S and let C = {e ∈ S | e is a central idempotent of S and eS = rS for some r ∈ R}. Then we define

C(R, S) = ∑_{e∈C} eR,

a subring of S. In the case S = U(G), we write C(R) for C(R, U(G)). For each ordinal α, define E_{α}(R, S) as follows:

• E_{0}(R, S) = R;

• E_{α+1}(R, S) = D(C(E_{α}(R, S), S), S);
• E_{α}(R, S) = ⋃_{β<α} E_{β}(R, S) if α is a limit ordinal.

Then E(R, S) = ⋃_{α} E_{α}(R, S). Also in the case R = KG where G is a group and K is a subfield of ℂ, we shall write E_{α}(KG) for E_{α}(KG, U(G)). If A ⊆ ℝ, then ⟨A⟩ will indicate the additive subgroup of ℝ generated by A.

Lemma 2.1. Let G be a group, let R be a subring of U(G), let n ∈ ℕ, and let x ∈ R. Suppose that xU(G) = eU(G) where e is a central idempotent of U(G). Then ⟨dim_{N (G)} βU(G)^{n} | β ∈ M_{n}(R)⟩ = ⟨dim_{N (G)} αU(G)^{n} | α ∈ M_{n}(R + eR)⟩.

Proof. Set E = eI_{n}, the diagonal matrix in M_{n}(R + eR) that has e’s on the main diagonal and zeros elsewhere. Then E is a central idempotent in M_{n}(U(G)). Obviously

⟨dim_{N (G)} βU(G)^{n} | β ∈ M_{n}(R)⟩ ⊆ ⟨dim_{N (G)} αU(G)^{n} | α ∈ M_{n}(R + eR)⟩,

so we need to prove the reverse inclusion. Let α ∈ M_{n}(R+eR) and write α = β+Eγ where β, γ ∈ M_{n}(R). Then we have

dim_{N (G)} αU(G)^{n} = dim_{N (G)}(β + γ)EU(G)^{n} + dim_{N (G)} β(1 − E)U(G)^{n}.

Since dim_{N (G)} β(1 − E)U(G)^{n} = dim_{N (G)} βU(G)^{n} − dim_{N (G)} βEU(G)^{n}, it suffices to prove that

dim_{N (G)} EβU(G)^{n} ∈ ⟨dim_{N (G)} δU(G)^{n} | δ ∈ M_{n}(R)⟩

for all β ∈ M_{n}(R). But EβU(G)^{n} = β(xI_{n})U(G)^{n} and the result follows. □

Lemma 2.1 immediately gives the following corollary.

Corollary 2.2. Let G be a group, let R be a subring of U(G), and let n ∈ ℕ. Then ⟨dim_{N (G)} αU(G) | α ∈ M_{n}(R)⟩ = ⟨dim_{N (G)} αU(G) | α ∈ M_{n}(C(R))⟩.

Proof. Let e_{1}, …, e_{m} be central idempotents of U(G) such that for each i, there exists α_{i} ∈ R with e_{i}U(G) = α_{i}U(G). Then by induction on m, Lemma 2.1 tells us that the result is true if α ∈ M_{n}(R + e_{1}R + ⋯ + e_{m}R). Since M_{n}(C(R)) is the union of M_{n}(R + e_{1}R + ⋯ + e_{m}R), the result is proven. □

Lemma 2.3. Let R be a subring of the ring S, let n ∈ ℕ, and let A ∈ M_{n}(D(R, S)). Then there exist 0 ≤ m ∈ ℤ and X, Y ∈ GL_{m+n}(S) such that X diag(A, I_{m})Y ∈ M_{m+n}(R).

Proof. This follows from [3, Proposition 7.1.3 and Exercise 7.1.4] and [8, Proposition 3.4]. □

Lemma 2.4. Let G be a group and let K be a subfield of ℂ. Then ⟨dim_{N (G)} xU(G) | x ∈ M_{n}(KG), n ∈ ℕ⟩ = ⟨dim_{N (G)} xU(G) | x ∈ M_{n}(E(KG)), n ∈ ℕ⟩.

Proof. Obviously

⟨dim_{N (G)} xU(G) | x ∈ M_{n}(KG), n ∈ ℕ⟩ ⊆ ⟨dim_{N (G)} xU(G) | x ∈ M_{n}(E(KG)), n ∈ ℕ⟩.

We shall prove the reverse inclusion by transfinite induction. So let n ∈ ℕ and x ∈ M_{n}(E(KG)). Then we may choose the least ordinal α such that x ∈ M_{n}(E_{α}(KG)). Clearly α is not a limit ordinal, and the result is true if α = 0, so we may write α = β +1 for some ordinal β and assume that the result is true for all y ∈ M_{n}(E_{β}(KG)). By Corollary 2.2 the result is true for all y ∈ M_{n}(C(E_{β}(KG))) and now the result follows from Lemma 2.3. □

The following result from [6] will be crucial for our work here. Because of this, and because we use a slightly different formulation, we state it here.

Lemma 2.5. [6, Lemma 2] Let G be a group, let n ∈ ℕ, and let α_{1}, …, α_{n} ∈ U(G). Then (∑_{j=1}^{n} α_{j}α_{j}^{∗})U(G) ⊇ α_{1}U(G).

Proof. By induction on n and [6, Lemma 2], we see that (∑_{j=1}^{n} α_{j}α_{j}^{∗})U(G) ⊇ α_{1}α_{1}^{∗}U(G). The result now follows by applying [6, Lemma 2] in the case β = 0. □

Theorem 2.6. Let G be a group and let K be a subfield of ℂ which is closed under complex conjugation. Suppose there is an ℓ ∈ ℕ such that ℓ dim_{N (G)} αU(G)^{n} ∈ ℤ for all α ∈ M_{n}(KG) and for all n ∈ ℕ. Then E(KG) is a semisimple Artinian ring.

Proof. First observe that Lemma 2.4 tells us that

(2.7)  ℓ dim_{N (G)} αU(G) ∈ ℤ for all α ∈ E(KG).

Next note that the hypothesis tells us that E(KG) has at most ℓ primitive central idempotents. Indeed if e_{1}, …, e_{ℓ+1} are (nonzero distinct) primitive central idempotents, then e_{i}e_{j} = 0 for i ≠ j and we see that the sum ⨁_{i=1}^{ℓ+1} e_{i}U(G) is direct. But

dim_{N (G)} ⨁_{i=1}^{ℓ+1} e_{i}U(G) = ∑_{i=1}^{ℓ+1} dim_{N (G)} e_{i}U(G) ≥ (ℓ + 1)/ℓ > 1

by (2.7), and we have a contradiction. Thus E(KG) has n primitive central idempotents e_{1}, …, e_{n} for some n ∈ ℕ, n ≤ l. For each i, 1 ≤ i ≤ n, choose 0 ≠ α_{i} ∈ e_{i}E(KG) such that dim_{N (G)} α_{i}U(G) is minimal.

Fix m ∈ {1, 2, …, n}. Since ℓ dim_{N (G)} αU(G) ∈ ℤ for all α ∈ E(KG) by (2.7), we may choose g_{1}, …, g_{r} ∈ G with dim_{N (G)}(∑_{i=1}^{r} g_{i}α_{m}α_{m}^{∗}g_{i}^{−1})U(G) maximal. Note that if g_{r+1} ∈ G, then

(∑_{i=1}^{r+1} g_{i}α_{m}α_{m}^{∗}g_{i}^{−1})U(G) ⊇ ∑_{i=1}^{r} g_{i}α_{m}U(G) ⊇ (∑_{i=1}^{r} g_{i}α_{m}α_{m}^{∗}g_{i}^{−1})U(G)

by Lemma 2.5, hence

dim_{N (G)}(∑_{i=1}^{r+1} g_{i}α_{m}α_{m}^{∗}g_{i}^{−1})U(G) ≥ dim_{N (G)}(∑_{i=1}^{r} g_{i}α_{m}α_{m}^{∗}g_{i}^{−1})U(G)

and by maximality of dim_{N (G)}(∑_{i=1}^{r} g_{i}α_{m}α_{m}^{∗}g_{i}^{−1})U(G), we see that

dim_{N (G)}(∑_{i=1}^{r+1} g_{i}α_{m}α_{m}^{∗}g_{i}^{−1})U(G) = dim_{N (G)}(∑_{i=1}^{r} g_{i}α_{m}α_{m}^{∗}g_{i}^{−1})U(G).

It follows that

(∑_{i=1}^{r+1} g_{i}α_{m}α_{m}^{∗}g_{i}^{−1})U(G) = (∑_{i=1}^{r} g_{i}α_{m}α_{m}^{∗}g_{i}^{−1})U(G)

and we deduce from Lemma 2.5 that gα_{m}U(G) ⊆ (∑_{i=1}^{r} g_{i}α_{m}α_{m}^{∗}g_{i}^{−1})U(G) for all g ∈ G. Let f ∈ U(G) be the unique projection such that

fU(G) = ∑_{i=1}^{r} g_{i}α_{m}α_{m}^{∗}g_{i}^{−1}U(G).

Then gfU(G) = ∑ gg_{i}α_{m}α_{m}^{∗}g_{i}^{−1}U(G) ⊆ ∑ gg_{i}α_{m}U(G) ⊆ fU(G) for all g ∈ G, thus gfU(G) = fU(G) and we deduce that gfg^{−1}U(G) = fU(G) for all g ∈ G. Also gfg^{−1} is also a projection, thus gfg^{−1} = f for all g ∈ G and we conclude that f is a central projection in E(KG). Since f ≠ 0, fU(G) ⊆ e_{m}U(G) and e_{m} is primitive, we conclude that f = e_{m} and consequently ∑_{i=1}^{r} g_{i}α_{m}U(G) = e_{m}U(G). By omitting some of the terms in this sum if necessary, we may assume that

(2.8)  ∑_{1≤i≤r, i≠s} g_{i}α_{m}U(G) ≠ e_{m}U(G)

for all s such that 1 ≤ s ≤ r. We make the following observation:

(2.9)  If 0 ≠ x ∈ g_{s}α_{m}E(KG), then xU(G) = g_{s}α_{m}U(G),

where 1 ≤ s ≤ r. This is because 0 ≠ xU(G) ⊆ g_{s}α_{m}U(G) and by minimality of dim_{N (G)} α_{m}U(G), we see that dim_{N (G)} xU(G) = dim_{N (G)} g_{s}α_{m}U(G) and consequently xU(G) = g_{s}α_{m}U(G).

We claim that e_{m}E(KG) = ⨁_{i=1}^{r} g_{i}α_{m}E(KG). Set σ = (∑_{i=1}^{r} g_{i}α_{m}α_{m}^{∗}g_{i}^{−1}). Since σU(G) = e_{m}U(G), we see that

(σ + (1 − e_{m}))U(G) ⊇ σU(G) + (1 − e_{m})U(G) = e_{m}U(G) + (1 − e_{m})U(G) = U(G).

Therefore, σ + 1 − e_{m} is invertible in U(G) and hence σ + 1 − e_{m} is invertible in E(KG). Thus

e_{m}σE(KG) = e_{m}(σ + 1 − e_{m})E(KG) = e_{m}E(KG).

Moreover, σE(KG) ⊆ e_{m}E(KG) and therefore e_{m}σE(KG) = σE(KG), hence

e_{m}E(KG) = σE(KG) = ∑_{i=1}^{r} g_{i}α_{m}E(KG).

If this sum is not direct, then for some s with 1 ≤ s ≤ r, we have g_{s}α_{m}E(KG) ∩ ∑_{i≠s} g_{i}α_{m}E(KG) ≠ 0, and without loss of generality we may assume that s = 1.

So let 0 ≠ x ∈ g_{1}α_{m}E(KG) ∩ ∑_{i=2}^{r} g_{i}α_{m}E(KG). Then 0 ≠ xU(G) ⊆ g_{1}α_{m}U(G) and (2.9) shows that xU(G) = g_{1}α_{m}U(G). It follows that g_{1}α_{m}U(G) ⊆ ∑_{i=2}^{r} g_{i}α_{m}U(G), consequently

∑_{i=2}^{r} g_{i}α_{m}U(G) = e_{m}U(G),

which contradicts (2.8) and our claim is established.

Now we show that g_{1}α_{m}E(KG) is an irreducible E(KG)-module. Suppose 0 ≠ x ∈ g_{1}α_{m}E(KG). Then xU(G) = g_{1}α_{m}U(G) by (2.9) and using Lemma 2.5, we see as before that xx^{∗} + ∑_{i=2}^{r} g_{i}α_{i}α_{i}^{∗}g_{i}^{−1} + 1 − e_{m} is a unit in U(G) and hence is also a unit in E(KG). This proves that xE(KG) = g_{1}α_{m}E(KG) and we deduce that E(KG) is a finite direct sum of irreducible E(KG)-modules. It follows that E(KG) is a semisimple Artinian ring. □

Proposition 2.10. Let G be a group with Δ(G) finite and let K be a subfield of ℂ with K = \overline{K} which contains all |Δ(G)|-th roots of unity, e.g. K = ℂ or K is the algebraic closure of ℚ in ℂ. Then E(KG) = D(KG).

Proof. If e is a central idempotent in U(G), then e ∈ N (Δ(G)), in particular e ∈ ℂG, and by our assumption on K even e ∈ KG. The result follows. □

The following result is well known, but we include a proof.

Lemma 2.11. Let G be a group, let e be a projection in N (G), and let α ∈ N (G). Then tr(eαα^{∗}e) ≤ tr(αα^{∗}).

Proof. Since tr(xy) = tr(yx) for all x, y ∈ N (G), we see that tr(eαα^{∗}(1 − e)) = tr((1 − e)αα^{∗}e) = 0. Therefore tr(αα^{∗}) = tr(eαα^{∗}e) + tr((1 − e)αα^{∗}(1 − e)). Since tr((1 − e)αα^{∗}(1 − e)) ≥ 0, the result follows. □

Lemma 2.12. Let G be a group, and let (α_{n}) be a sequence in N (G) converging strongly to α. Suppose that ker α = 0. Then dim_{N (G)}(ker α_{n}) converges to 0.

Proof. By the principle of uniform boundedness, ∥α_{n}∥ is bounded. Also by multiplying everything by a unitary operator if necessary, we may assume that α is positive. Then α_{n} − α converges strongly to 0 and (α_{n} − α)^{∗} is bounded, hence (α_{n} − α)^{∗}(α_{n} − α) converges strongly to 0 and in particular lim_{n→∞} tr((α_{n} − α)^{∗}(α_{n} − α)) = 0. Let e_{n} ∈ N (G) denote the projection of ℓ^{2}(G) onto ker α_{n}. Then e_{n}α_{n}^{∗} = α_{n}e_{n} = 0 and using Lemma 2.11, we obtain

tr((α_{n} − α)^{∗}(α_{n} − α)) ≥ tr(e_{n}(α_{n} − α)^{∗}(α_{n} − α)e_{n})
= tr(e_{n}α^{∗}αe_{n}) ≥ 0.

Thus lim_{n→∞} tr(e_{n}α^{∗}αe_{n}) = 0. Suppose by way of contradiction that lim_{n→∞} dim_{N (G)}(ker α_{n}) ≠ 0. Then by taking a subsequence if necessary, we may assume that dim_{N (G)}(ker α_{n}) > ϵ for some ϵ > 0, for all n ∈ ℕ. By considering the spectral family associated to α^{∗}α [10, Definition 1.68], there is a closed α^{∗}α-invariant N (G)-submodule X of ℓ^{2}(G) and a δ > 0 such that dim_{N (G)}(X) > 1 − ϵ/2 and α^{∗}α > δ on X. Because dim_{N (G)}(X) > 1 − ϵ/2 and dim_{N (G)}(ker α_{n}) > ϵ, we find that dim_{N (G)}(X ∩ker α_{n}) > ϵ/2 (use [10, Theorem 6.7]). Let f_{n} denote the projection of ℓ^{2}(G) onto X ∩ker α_{n}, so tr f_{n} > ϵ/2. Since α^{∗}α > δ on X ∩ker α_{n}, f_{n}α^{∗}αf_{n} ≥ δf_{n}, and because of positivity of tr we see that tr(f_{n}α^{∗}αf_{n}) ≥ tr(δf_{n}) > δϵ/2. Therefore tr(e_{n}α^{∗}αe_{n}) > ϵδ/2 by Lemma 2.11, which shows that tr(e_{n}α^{∗}αe_{n}) does not converge to 0, and the result follows. □

Proposition 2.13. Let G be a group with Δ^{+}(G) = 1 and let K be a subfield of ℂ that is closed under complex conjugation. Assume that lcm(G) = d ∈ ℕ and that G satisfies the strong Atiyah conjecture over K. Then D(KG) is a d × d matrix ring over a skew field.

Let p be a prime, let q be the largest power of p that divides d, and let H ≤ G with |H| = q (so H is a “Sylow” p-subgroup of G). Set e = \frac{1}{q} ∑_{h∈H} h, a projection in ℚH. We shall use the center valued von Neumann dimension dim^{u}, as defined in [10, Definition 9.12]. Since Δ^{+}(G) = 1, we see that dim^{u}(eU(G)) = 1/q and dim^{u}((1 − e)U(G)) = (q − 1)/q. Therefore by [10, Theorem 9.13(1)],

(1 − e)U(G) ≅ eU(G)^{q−1}

and we deduce that there exist orthogonal projections e = e_{1}, e_{2}, …, e_{q} ∈ U(G) (so e_{i}e_{j} = 0 for i ≠ j) such that ∑_{i=1}^{q} e_{i} = 1 and e_{i}U(G) ≅ eU(G) for all i. By [2, Exercise 13.15A, p. 76], there exist similarities (that is self adjoint unitaries) u_{i} ∈ U(G) with u_{1} = 1 such that e_{i} = u_{i}eu_{i}. There is a countable subgroup F of G such that u_{i} ∈ N (F) for all i. By the Kaplansky density theorem [1, Corollary, p. 8] for each i (1 ≤ i ≤ q) there exists a sequence u_{ij} ∈ KF such that u_{ij} → u_{i} as j → ∞ in the strong operator topology in N (F) with u_{1j} = 1 for all j. Set v_{j} = ∑_{i=1}^{q} u_{ij}eu_{ij}. Then v_{j} → ∑_{i=1}^{q} e_{i} = 1 strongly, hence for 1 ≤ i ≤ q,

lim_{j→∞} dim_{N (F)}(v_{j}U(F)) = lim_{j→∞} dim_{N (F)}(u_{ij}U(F)) = 1

by Lemma 2.12. Now dim_{N (F)}(xU(F)) = dim_{N (G)}(xU(G)) for all x ∈ U(F), consequently

lim_{j→∞} dim_{N (G)} v_{j}U(G) = lim_{j→∞} dim_{N (G)}(u_{ij}U(G)) = 1 for 1 ≤ i ≤ q,

and since by assumption G satisfies the strong Atiyah conjecture over K, there exists n ∈ ℕ such that dim_{N (G)} v_{j}U(G) = dim_{N (G)}(u_{ij}U(G)) = 1 for 1 ≤ i ≤ q for all j ≥ n, in particular dim_{N (G)}(v_{n}U(G)) = dim_{N (G)}(u_{in}U(G)) = 1 and we conclude that v_{n} and u_{in} (1 ≤ i ≤ q) are units in U(G). Therefore v_{n} and u_{in} (1 ≤ i ≤ q) are units in D(KG) and we deduce that ∑_{i=1}^{q} u_{in}eD(KG) = D(KG), because

D(KG) = v_{n}D(KG) = ∑_{i=1}^{q}(u_{in}eu_{in})D(KG) ⊆ ∑_{i=1}^{q} u_{in}eD(KG) ⊆ D(KG).

Since dim_{N (G)} eU(G) = 1/q, we see that ⨁_{i=1}^{q} u_{in}eU(G) = U(G), a direct sum, and we deduce that

(2.14)  ⨁_{i=1}^{q} u_{in}eD(KG) = D(KG),

also a direct sum.

Now suppose that ε is a central idempotent in C(D(KG)). We want to prove that ε = 0 or 1, so assume otherwise. Now εu_{in}eU(G) ≅ εeU(G) for all i, which implies that dim_{N (G)}(εU(G)) = q dim_{N (G)}(εeU(G)). Moreover, because of the Atiyah conjecture, d dim_{N (G)}(εeU(G)) ∈ ℤ. These two observations together imply that d dim_{N (G)}(εU(G)) ∈ qℤ. Since this is true for all primes p, it follows that dim_{N (G)} εU(G) ∈ ℤ, so 0 and 1 are the only central idempotents of C(D(KG)).

Summing up, we have shown that C(D(KG)) contains no nontrivial central idempotents. Using Theorem 2.6, we see that D(KG) is a semisimple Artinian ring with no nontrivial central idempotents. Thus D(KG) is an l × l matrix ring over a division ring for some l ∈ ℕ. In particular, D(KG) is the direct sum of l mutually isomorphic D(KG)-submodules, so if f is a primitive idempotent in D(KG), we see that dim_{N (G)}(fU(G)) = 1/l. Furthermore Lemma 2.3 (or Lemma 2.4) show that l|d. On the other hand (2.14) shows that q|l, for all primes p, so d|e and the result follows.

Proof of Theorem 1.3. If G satisfies the strong Atiyah conjecture over K, then D(KG) is a d × d matrix ring over a skew field by Proposition 2.13. Conversely suppose D(KG) is a d × d matrix ring over a skew field F. We need to show that if M is a finitely presented KG-module, then lcm(G) dim_{N (G)} M ⊗_{KG} U(G) ∈ ℤ. However

M ⊗_{KG} U(G) ≅ M ⊗_{KG} M_{d}(F) ⊗_{M_{d}(F)} ⊗U(G),

consequently (M ⊗_{KG} U(G))^{d} is a finitely generated free U(G)-module and we conclude that lcm(G) dim_{N (G)} M ⊗_{KG} U(G) ∈ ℤ as required. □

3. Embeddings in ∗-regular rings

There are other closures of group rings KG in U(G) which may be useful, especially when lcm(G) = ∞. In general the intersection of regular subrings of a von Neumann regular ring is not regular [5, Example 1.10], however we do have the following result.

Proposition 3.1. Let G be a group and let {R_{i} | i ∈ I} be a collection of ∗-regular subrings of U(G). Then ⋂_{i∈I} R_{i} is also a ∗-regular subring of U(G).

Proof. Set S = ⋂_{i∈I} R_{i}. Obviously S is a ∗-subring of U(G); we need to show that S is ∗-regular, that is given s ∈ S, there is a projection e ∈ S such that sS = eS. We note that D(R_{i}, U(G)) = R_{i} for all i. Indeed if x ∈ R_{i} and x is invertible in U(G), then xR_{i} = eR_{i} where e is a projection in R_{i}, consequently xU(G) = eU(G) and since x is invertible in U(G), we must have e = 1 and we deduce that xR_{i} = R_{i}. Similarly R_{i}x = R_{i} and thus x is invertible in R_{i}, so D(R_{i}, U(G)) = R_{i} as asserted. Since R_{i} is ∗-regular, for each i ∈ I, there is a projection e_{i} ∈ R_{i} such that e_{i}R_{i} = sR_{i}. We now have e_{i}U(G) = e_{j}U(G) for all i, j and we deduce that e_{i} = e_{j} for all i, j ∈ I, so there exists f ∈ S such that f = e_{i} for all i. Since fU(G) = sU(G), we see that fs = s, so s ∈ fS and hence sS ⊆ fS. Thus the result will be proven if we can show that ss^{∗}S ⊇ fS. By Lemma 2.5,

(ss^{∗} + (1 − f))U(G) ⊇ (1 − f)U(G) + sU(G) = (1 − f)U(G) + fU(G) = U(G)

and we see that ss^{∗} + 1 − f is a unit in U(G). Let t ∈ U(G) be the inverse of ss^{∗} + 1 − f, so

(3.2)  (ss^{∗} + 1 − f)t = 1.

Since D(R_{i}, U(G)) = R_{i} for all i, we deduce that t ∈ R_{i} for all i and hence t ∈ S. Moreover fs = s and f(1 − f) = 0, so if we multiply (3.2) on the left by f, we obtain ss^{∗}t = f and the result is proven. □

Thus if K is a subfield of ℂ that is closed under complex conjugation and G is any group, then there is a least subring of U(G) containing KG that is ∗-regular.

References

[1] William Arveson. An invitation to C^{∗}-algebras. Springer-Verlag, New York, 1976. Graduate Texts in Mathematics, No. 39.

[2] Sterling K. Berberian. Baer ∗-rings. Springer-Verlag, New York, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 195.

[3] P. M. Cohn. Free rings and their relations, volume 19 of London Mathematical Society Monographs. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, second edition, 1985.

[4] Jozef Dódziuk, Peter Linnell, Varghese Mathai, Thomas Schick, and Stuart Yates. Approximating L^{2}-invariants, and the Atiyah conjecture. Comm. Pure Appl. Math., 56(7):839–873, 2003.

[5] K. R. Goodearl. von Neumann regular rings. Robert E. Krieger Publishing Co. Inc., Malabar, FL, second edition, 1991.

[6] Peter A. Linnell. Embedding group algebras into finite von Neumann regular rings. To appear in the conference proceedings of the “International Conference on Modules and Comodules”, dedicated to Robert Wisbauer, to be published by Birkhauser, available from http://arxiv.org/abs/0708.2243.

[7] Peter A. Linnell. Division rings and group von Neumann algebras. Forum Math., 5(6):561– 576, 1993.

[8] Peter A. Linnell. Noncommutative localization in group rings. In Non-commutative localization in algebra and topology, volume 330 of London Math. Soc. Lecture Note Ser., pages 40–59. Cambridge Univ. Press, Cambridge, 2006.

[9] Wolfgang Lück. L^{2}-invariants of regular coverings of compact manifolds and CW-complexes. In Handbook of geometric topology, pages 735–817. North-Holland, Amsterdam, 2002.

[10] Wolfgang Lück. L^{2}-invariants: theory and applications to geometry and K-theory, volume 44 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2002.

[11] Donald S. Passman. Infinite group rings. Marcel Dekker Inc., New York, 1971. Pure and Applied Mathematics, 6.

[12] Thomas Schick. Integrality of L^{2}-Betti numbers. Math. Ann., 317(4):727–750, 2000.

[13] Thomas Schick. Erratum: “Integrality of L^{2}-Betti numbers”. Math. Ann., 322(2):421–422, 2002.

Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA

E-mail address: linnell@math.vt.edu
URL: http://www.math.vt.edu/people/plinnell/

Mathematisches Institut, Bunsenstr. 3, D-37073 Göttingen, Germany

E-mail address: schick@uni-math.gwdg.de
URL: http://www.uni-math.gwdg.de/schick/