The Dixmier property and tracial states for C*-algebras

It is shown that a unital C*-algebra A has the Dixmier property if and only if it is weakly central and satisfies certain tracial conditions. This generalises the Haagerup-Zsido theorem for simple C*-algebras. We also study a uniform version of the Dixmier property, as satisfied for example by von Neumann algebras and the reduced C*-algebras of Powers groups, but not by all C*-algebras with the Dixmier property, and we obtain necessary and sufficient conditions for a simple unital C*-algebra with unique tracial state to have this uniform property. We give further examples of C*-algebras with the uniform Dixmier property, namely all C*-algebras with the Dixmier property and finite radius of comparison-by-traces. Finally, we determine the distance between two Dixmier sets, in an arbitrary unital C*-algebra, by a formula involving tracial data and algebraic numerical ranges.

In [37], Haagerup and Zsidó established a definitive result about the Dixmier property for simple C * -algebras: a simple unital C * -algebra has the Dixmier property if and only if it has at most one tracial state. For non-simple C * -algebras, the Dixmier property imposes serious restrictions on the ideal structure: if a C * -algebra has the Dixmier property, then it is weakly central ([8, p. 275]), see Definition 1.3. One of our main results is a complete generalisation of Haagerup and Zsidó's, showing that the Dixmier property is equivalent to this ideal space restriction together with tracial conditions: Theorem 1.1 (Theorem 2.6). Let A be a unital C * -algebra. Then A has the Dixmier property if and only if all of the following hold.
(i) A is weakly central, (ii) every simple quotient of A has at most one tracial state, and (iii) every extreme tracial state of A factors through some simple quotient.
A characterisation of the singleton Dixmier property is an immediate consequence of this result (Corollary 2.8): it corresponds to the case that in (ii), every simple quotient has exactly one tracial state. We also take the opportunity to remove a separability condition from a result in [3]: a postliminal C * -algebra A has the (singleton) Dixmier property if and only if Z(A/J) = (Z(A) + J)/J for every proper closed ideal J of A (Theorem 2.12).
The case of trivial centre in Theorem 1.1 is already an interesting generalisation of the Haagerup-Zsidó theorem: a unital C * -algebra A has the Dixmier property with centre Z(A) = C1 if and only if A has a unique maximal ideal J, A has at most one tracial state and J has no tracial states (Corollary 2.10). This result has a crucial application in Section 3 (see below).
In Section 3, we consider a strengthening of the Dixmier property, called the uniform Dixmier property, in which the number of unitaries used to approximately average an element depends only on the tolerance (and not the particular element). This is closely related to the uniform strong Dixmier property studied in [28,Section 7.2], as well as the uniform averaging properties recently considered in [65,Section 5] and [66,Section 6]. Many of the classical examples of C * -algebras with the Dixmier property turn out to have the uniform Dixmier property, including von Neumann algebras and C * r (F 2 ) (see Remark 3.4). Adding to this, we show that any C * -algebra with the Dixmier property and with finite radius of comparison-by-traces has the uniform Dixmier property (Corollary 3.22). We use Corollary 2.10 to characterise, in terms of two distinct uniformity conditions, when a tracial unital C * -algebra with the Dixmier property and trivial centre has the uniform Dixmier property (Theorem 3.24). Finally, following a suggestion by the referee, we find explicit constants for the uniform  Furthermore, if A has no tracial states then condition (a) is vacuously satisfied.
Note that, in order to verify condition (b) in Theorem 1.2, it suffices to check simple quotients (that is, quotients of A by maximal ideals). Theorem 1.1 is proven using the Katětov-Tong insertion theorem (see Theorem 2.5 below) to produce candidate central elements corresponding to any given self-adjoint element a ∈ A, and then using Theorem 1.2 to verify that these candidates are indeed in the respective Dixmier set.
In Section 4, motivated by Theorem 1.2, we show that for elements a and b in an arbitrary unital C * -algebra A, the distance between the Dixmier sets D A (a) and D A (b) can be read off from tracial data and the algebraic numerical ranges of a and b in quotients of A (Theorem 4.12). This result extends Theorem 1.2 in several ways: first by considering the Dixmier sets of a pair of elements a and b (rather than one of them being zero), second by providing a distance formula between these sets (rather than focusing on the case that this distance is zero), and third by allowing the elements a and b to be non-self-adjoint. We also show that, in certain cases, the distance between D A (a) and D A (b) is attained (Proposition 4.10). In this section, we obtain elements in Z(A) by using Michael's selection theorem, rather than the Katětov-Tong theorem (cf. [58], [90]) {a ∈ A : τ (a * a) = 0} is a closed (two-sided) ideal of A and is easily seen to coincide with the kernel of the Gelfand-Naimark-Segal (GNS) representation π τ (and with the right kernel). We shall refer to this ideal as the trace-kernel ideal for τ . When C is a commutative C * -algebra (generally, arising as the centre of another C * -algebra A) and N ⊆ C is a maximal ideal, define φ N ∈ P (C) to be the (unique) pure state satisfying   [60,61]) A C * -algebra A is said to be weakly central if Ψ (as just described) is injective.
When A is weakly central, Ψ is a homeomorphism since its domain is compact and its range is Hausdorff. Misonou used the Dixmier property to show that every von Neumann algebra is weakly central ( [61,Theorem 3]). As observed in [8, p. 275], the same method shows that every unital C * -algebra with the Dixmier property is weakly central. Although weak centrality does not imply the Dixmier property (consider any unital simple C * -algebra with more than one tracial state), Magajna has given a characterisation of weak centrality in terms of a more general kind of averaging involving elementary completely positive mappings ( [58]).
A Glimm ideal of a unital C * -algebra A is an ideal N A (= AN A) generated by a maximal ideal N of Z(A) (see [31,Section 4]); note that N A is already closed by the Banach module factorisation theorem (a fact that does not require N to be maximal).
Let A be a C * -algebra with centre Z(A). A centre-valued trace on A is a positive, linear contraction R : A → Z(A) such that R(z) = z (z ∈ Z(A)) and R(ab) = R(ba) (a, b ∈ A). The equivalence of (i) and (ii) in the next result, together with the description of the centre-valued trace R, is essentially well-known and easy to see. It underlies Dixmier's approach to the trace in a finite von Neumann algebra [23,24,48]. A detailed proof is given in [3, 5.1.3] using the same methods as in the case of a von Neumann algebra (see, for example, [24,Corollaire III.8.4]). (The equivalence with (iii) is probably also well-known, although we were unable to find a reference.) Since this holds for all φ ∈ P (Z(A)), we obtain that z 1 = z 2 as required for (i).
Since the Dixmier property passes to quotients ([5, p. 544]), it is immediate from Proposition 1.4 (iii) that the singleton Dixmier property passes to quotients of unital C *algebras. More generally, the singleton Dixmier property passes to ideals and quotients of arbitrary C * -algebras [3, Proposition 5.1.11].
The next theorem will not be applied until Section 3, but we include it here as it may be of independent interest (cf. [2,Theorem 4.3]). In [64, Lemma 2.1 (i)], it is shown that a limit of sums of self-adjoint commutators in a quotient can be lifted (as below), but at a cost of in the norm. Theorem 1.6 shows that this cost can be avoided. The proof uses a technique from Loring and Shulman's [57]; the result almost follows from [57,Theorem 3.2], except that they work with polynomials (in non-commuting variables) whereas we need to work with a series (of commutators). Here [A, A] means the span of commutators in A, i.e., the span of elements of the form [a, b] = ab − ba, where a, b ∈ A. Recall also that a quasicentral approximate unit of an ideal J of A is an approximate unit (u λ ) for J which is approximately central in A.
We will need the following in the proof of this theorem.
Lemma 1.5. Let A be a C * -algebra, J a closed ideal of A and (u λ ) a quasicentral approximate unit of J. Let 0 < δ < 1 and a ∈ A. Then Proof. This is a special case of [57, Theorem 2.3].
Theorem 1.6. Let A be a C * -algebra, let J be a closed ideal of A and letā ∈ A/J be a self-adjoint element in [A/J, A/J]. Then there exists a self-adjoint lift a ∈ [A, A] ofā such that a = ā .
Proof. We may assume without loss of generality that ā = 1. The strategy of the proof is as follows: We will construct a sequence (a (n) ) ∞ n=1 of self-adjoints lifts ofā such that a (n) ∈ [A, A] for all n, a (n) → 1, and the sequence (a (n) ) ∞ n=1 is Cauchy. This is sufficient to prove the theorem, for then lim n a (n) is the desired lift.
Since ∞ n=1 δ n < ∞, the final item ensures that the sequence is Cauchy, and so upon finding such a (n) , we are done. Let us start with a self-adjoint lift a (1) ∈ [A, A] ofā such that a (1) < 1 + 1 . This can be done by [ for some x (1) i ∈ A, where the series is norm convergent. Now fix n ≥ 1, and suppose that we have defined a self-adjoint a (n) that is a lift ofā, such that a (n) < 1 + n , and such that a (n) has the form Let (u λ ) be a quasicentral approximate unit of J, and define Define a (n+1) as in (1.2) using the new elements x (n+1) i (the new series also converges since only finitely many terms were changed). It is clear that a (n+1) is a self-adjoint lift ofā and that a (n+1) ∈ [A, A]. Presently, the element a (n+1) depends on λ. We will choose λ suitably. We have .
Exploiting the approximate centrality of (u λ ) (using [103, Proposition 1.8] to get that (1 − δ n u λ ) 1 2 is approximately central), we can choose λ large enough such that We have Using (1.3), we find that the norm of the image of kn i ] in the quotient A/J is less than ā + n+1 /3 = 1 + n+1 /3. So, by Lemma 1.5, we can choose λ large enough such Then, for such choices of λ we have a (n+1) < 1 + n+1 . Now consider a (n+1) − a (n) : Again using approximate centrality of (u λ ), we may possibly increase λ to get Thus, a (n+1) − a (n) ≤ 4δ n , as required.
We recall from [17, Proposition 2.7] that, for a ∈ A, a ∈ [A, A] if and only if τ (a) = 0 for all tracial states of A. Thus Theorem 1.6 can clearly be rephrased in terms of tracial states of A/J and of A instead of commutators.

Tracial characterisations of the Dixmier property and the singleton Dixmier property
We begin with a few straightforward (and probably well-known) facts. The next result can be proved by using a quasicentral approximate unit or the GNS representation, or the invariance of the extension under unitary conjugation. A proof using an arbitrary approximate unit is given in [84,Lemma 3.1].
Lemma 2.2. Let J be a nonzero closed ideal of a C * -algebra A and let τ ∈ T (J). Then the unique extension of τ to a state of A (see [68, 3.1.6]) is a tracial state. Lemma 2.3. Let J be a proper closed ideal of a unital C * -algebra A. Then for any τ ∈ ∂ e T (A/J), τ • q J ∈ ∂ e T (A). Lemma 2.4. Let A be a unital C * -algebra and suppose that τ ∈ ∂ e T (A). Then τ | Z(A) is a pure state on Z(A).
Proof. Let z ∈ Z(A) be a positive contraction. Then the function τ z : A → C given by τ z (a) := τ (za) is a tracial functional on A which clearly satisfies τ z ≤ τ . Since τ is an extreme tracial state, it follows that τ z is a scalar multiple of τ , and so In particular, this shows that τ | Z(A) is multiplicative, and therefore a pure state.
We will also need the following.
Theorem 2.5 (Katětov-Tong insertion theorem). Let X be a normal space and Y a closed subspace. Let f : X → R be upper semicontinuous, g : Y → R be continuous, and h : X → R be lower semicontinuous, satisfying Then there existsg : X → R continuous such thatg| Y = g and Proof. We reduce this to the standard form of the Katětov-Tong insertion theorem (see [50] or [99]), which is the case that Y = ∅. Define f 1 , h 1 : X → R by Using that f is upper semicontinuous, that Y is closed, and that f ≤ g on Y , it follows that f 1 is upper semicontinuous. Likewise, h 1 is lower semicontinuous. It is also clear that f 1 ≤ h 1 . Therefore by the standard form of the Katětov-Tong insertion theorem, there exists a continuous functiong : X → R such that The definitions of f 1 and h 1 ensure that (2.1) holds.
Here is our first main theorem, characterising the Dixmier property in terms of other conditions that are more readily verified, namely weak centrality and tracial conditions. Theorem 2.6. Let A be a unital C * -algebra. The following are equivalent.
(i) A has the Dixmier property Thus G is injective, and it is surjective by (iii)(b). We will show that Y is closed in Max(A) (and hence compact) and that the bijection G is continuous for the weak * -topology on the Hausdorff space ∂ e T (A) (and hence G is a homeomorphism).
Let M belong to the closure of Y in Max(A) and let (M i ) be an arbitrary net in Y that is convergent to M . Since T (A) is weak * -compact, there exist τ ∈ T (A) and a subnet ( It follows from the Cauchy-Schwartz inequality for states that τ annihilates the Glimm ideal  Finally, we have seen above that when A has the Dixmier property, ∂ e T (A) is homeomorphic to the compact set Y and so the Choquet simplex T (A) is a Bauer simplex (possibly empty).
Suppose that A is a unital C * -algebra with the Dixmier property and that θ : Z(A) → C(Max(A)) is the canonical * -isomorphism induced by the Gelfand transform for Z(A) and the homeomorphism Ψ : Max(A) → Max(Z(A)). Let a = a * ∈ A, let f and h be the associated spectral functions on Max(A) and let g be the associated function on the closed subset Y of Max(A) (see the proof of Theorem 2.6). Then it follows from Theorem 1.2 that Thus D A (a) ∩ Z(A) is closed under the operations of max and min (regarding self-adjoint elements of Z(A) as continuous functions on Max(A)). Furthermore, if z 1 , z 2 , z 3 are selfadjoint elements of Z(A) such that z 1 ≤ z 2 ≤ z 3 and z 1 , z 3 ∈ D A (a) then z 3 ∈ D A (a).
In the case where A is a properly infinite von Neumann algebra (and hence for a general von Neumann algebra), Ringrose has shown that D A (a) ∩ Z(A) is an order interval in the self-adjoint part of Z(A) and has given a formula for the end-points in terms of spectral theory (see [77, Corollary 2.3, Theorem 3.3 and Remark 3.5]). The next result gives a different spectral description for the end-points.
Corollary 2.7. Let A be a properly infinite von Neumann algebra and let a = a * ∈ A. Then, with the notation above, the spectral functions f and h are continuous on Max(A), [39,Proposition 1]. It follows that the functions f and h are continuous on Max(A). Since A has no tracial states, the subset Y of Max(A) is empty. The result now follows from the discussion above.
We now show how Theorem 2.6 leads to necessary and sufficient conditions for the singleton Dixmier property. Proof. The equivalence of (i), (ii) and (iii) follows from Theorem 2.6 and Proposition 1.4. It is also clear that (iv) implies (v) (note that r e maps extreme tracial states into P (Z(A)) by Lemma 2.4).
(i)⇒(iv): Suppose that A has the singleton Dixmier property. Then (iv)(a) holds by Proposition 1.4. For (iv)(b), we proceed as in the well-known case of a finite von Neumann algebra (cf. [24,Proposition III.5.3]). For the surjectivity of r, we observe that if φ ∈ S(Z(A)) then φ • R ∈ T (A), where R : A → Z(A) is the unique centre-valued trace of A, and (φ • R)| Z(A) = φ. The injectivity of r follows from the facts that A has the Dixmier property and tracial states are constant on Dixmier sets. Since r is a weak * -continuous bijection from the compact space T (A) to the Hausdorff space S(Z(A)), it is a homeomorphism. Then Since r e is injective, Since r e is injective, Let K be a proper closed ideal of A and suppose that A/K has the singleton Dixmier property. Let P be a primitive ideal of A containing K and let M be a maximal ideal of A containing P . Since A/K has the singleton Dixmier property, it follows from Proposition 1.4 that T ((A/K)/(M/K)) is non-empty and hence M ∈ Y . On the other hand, P ∩ Z(A) is a prime ideal of Z(A) and hence It follows that P ⊇ N A = J. Since this holds for all P ∈ Prim(A/K), we obtain that K ⊇ J.
Conversely, suppose that K ⊇ J. Since A has the Dixmier property, so does A/K. Let M be a maximal ideal of A that contains K. Since M ∩ Z(A) ⊇ J ∩ Z(A) ⊇ N and Ψ(Y ) is closed in Max(Z(A)), we obtain that M ∈ Y . Thus T ((A/K)/(M/K)) is non-empty and so A/K has the singleton Dixmier property by Proposition 1.4.
The uniqueness of J is immediate from its stated property.
We highlight the special case of Theorem 2.6 in which Z(A) is trivial, which generalises results from [37]. This case plays a crucial role in our investigation of the uniform Dixmier property for C * -algebras with trivial centre, in Section 3.2.
Corollary 2.10. Suppose that A is a unital C * -algebra. The following conditions are equivalent.
(i) Z(A) = C1 and A has the Dixmier property.
(ii) A has a unique maximal ideal J, A has at most one tracial state and J has no tracial states. (iii) A has a unique maximal ideal J, A/J has at most one tracial state and J has no tracial states. When these hold, A has the singleton Dixmier property exactly when it has a tracial state τ , and in this case, If A has the Dixmier property and no tracial states then Proof. Suppose that (i) holds. By Theorem 2.6 ((i) ⇒ (ii)), A is weakly central and hence, since Z(A) = C1, A has a unique maximal ideal J. Since J ∩ Z(A) = {0}, A has at most one tracial state by Theorem 2.6(ii)(a) and if A does have a tracial state then it annihilates J by Theorem 2.6(ii)(b). By Lemma 2.2, J has no tracial states. Thus (ii) holds. Conversely, suppose that (ii) holds. Then Z(A) = C1 (by Lemma 2.1) and A is weakly central. If A has a tracial state then it must annihilate J since J has no tracial states. Thus (i) holds by Theorem 2.6((ii) ⇒ (i)).
(ii)⇔(iii) is immediate. The statement concerning the singleton Dixmier property follows from Corollary 2.8 (i)⇔(ii), and the final statement follows from Theorem 1.2.
An example of a non-simple C * -algebra with a unique maximal ideal, with the Dixmier property but not the singleton Dixmier property is the "Cuntz-Toeplitz algebra" A := C * (S 1 , . . . , S n ) where 2 ≤ n < ∞ and S 1 , . . . , S n are isometries on an infinite dimensional Hilbert space with mutually orthogonal range projections having sum less than 1 (cf. [7,Theorem 11]). Corollary 2.10 above motivates the following question. Is there an example of a unital C * -algebra A containing a unique maximal ideal J such that A has a unique tracial state and A/J has no tracial states? A non-separable example is the multiplier algebra M (J) where J is a non-unital hereditary subalgebra of a UHF algebra; here, J is simple and has a unique trace, and by [25, Theorem 3.1 and its proof], M (J)/J is simple and infinite. Thus J is the unique maximal ideal of M (J), and the extension of the trace on J is the unique trace on M (J).
For a separable nuclear example, one may utilise a construction of Kirchberg [52] as pointed out by Ozawa at the end of [67, Section 3]. Thus J and A are C * -subalgebras of the CAR algebra M 2 ∞ such that J is hereditary in M 2 ∞ and is an ideal in A such that A/J ∼ = O ∞ . Since M 2 ∞ is simple and has a faithful, unique tracial state, J also has both of these properties (note that any tracial state of J can be extended to a bounded tracial functional on M 2 ∞ ). Suppose that A has a maximal ideal M distinct from J. Then M ∩ J = {0} and so contradicting the fact that A has a faithful tracial state induced from M 2 ∞ . It follows from a theorem of Cuntz and Pedersen [17,Theorem 2.9], as in the proof of [62,Theorem 14], that A has a faithful, unique tracial state. Even though A/J satisfies a strong form of the Dixmier property [7, Theorem 8], A itself does not have the Dixmier property because its tracial state does not vanish on J.
This example also shows that, in Corollary 2.8, the condition (v)(a) does not follow from condition (v)(b). On the other hand, to see that condition (v)(a) does not imply condition (v)(b) in Corollary 2.8, consider any simple unital C * -algebra with more than one tracial state.
The following concerns the Dixmier property for non-unital C * -algebras; a non-unital C * -algebra A is said to have the (singleton) Dixmier property if the unitisation A + C1 has the same property. A C * -algebra A (with or without an identity) is said to have the centre-quotient property if Z(A/J) = (Z(A) + J)/J for every proper closed ideal J of A. Vesterstrøm showed that, for unital A, the centre-quotient property is equivalent to weak centrality ([100, Theorems 1 and 2]). Dixmier observed that the centre-quotient property is a simple consequence of the Dixmier property in a von Neumann algebra ([24, p.259, Ex. 7]). Similarly, it is easily seen that if a C * -algebra has the Dixmier property then it also has the centre-quotient property ([3, 2.2.2]). The next result was obtained in [3, 4.3.1, 5.1.9] under the additional assumption that either A is separable or there is a finite bound on the covering dimension of compact Hausdorff subsets of the spectrumÂ. The method was very different from that used below. Theorem 2.12. Let A be a postliminal C * -algebra. The following conditions are equivalent.
(i) A has the centre-quotient property.
(ii) A has the singleton Dixmier property.
(iii) A has the Dixmier property.
Proof. (i)⇒(ii): Suppose first of all that A is a unital postliminal C * -algebra with the centre-quotient property. Then A is weakly central ( [100]). Furthermore, A automatically satisfies conditions (iii)(a) and (iii)(b) of Corollary 2.8. For (iii)(a), recall that a simple, unital C * -algebra of type I is * -isomorphic to M n for some n ∈ N. For (iii)(b), note that if τ ∈ ∂ e T (A) then π τ (A) is a finite factor of type I (see [22, 6.8.7 and 6.8.6]) and so ker π τ is maximal. By Corollary 2.8, A has the singleton Dixmier property.
Secondly, suppose that A is a non-unital postliminal C * -algebra with the centre-quotient property. Then it is easily seen that A + C1 has the centre-quotient property (note that if J is a closed ideal of A + C1 then either J ⊆ A or else (A + C1)/J is canonically * -isomorphic to A/(A ∩ J). Thus A + C1 is a unital postliminal C * -algebra with the centre-quotient property and so has the singleton Dixmier property by the first part of the proof.
(iii)⇒(i): For the convenience of the reader, we give the details in the case where A is a non-unital C * -algebra with the Dixmier property. The unital case is even easier (and could alternatively be obtained via weak centrality and [100]). Let J be a closed ideal of A and let q : Corollary 2.13. Let A be a postliminal C * -algebra such that every irreducible representation of A is infinite dimensional. Then A has the singleton Dixmier property.
Proof. As in the proof of [3, 4.3.2], the use of a composition series with liminal quotients shows easily that the centre of A is {0}. Since the same applies to any nonzero quotient of A, it follows that A has the centre-quotient property and hence the singleton Dixmier property.

The uniform Dixmier property
In this section, we introduce and study the following uniform version of the Dixmier property (cf. [66, Section 6] and [65, Section 5]). Definition 3.1. A unital C * -algebra A has the uniform Dixmier property if for every > 0 there exists n ∈ N such that for all a ∈ A, there exist unitaries u 1 , . . . , u n ∈ U(A) such that Theorem 3.2. Let A be a unital C * -algebra. The following are equivalent: (i) A has the uniform Dixmier property.
(ii) There exist m ∈ N and 0 < γ < 1 such that for every self-adjoint a ∈ A we have that for some z ∈ Z(A) and u 1 , . . . , u m ∈ U(A). (iii) There exist m ∈ N and 0 < γ < 1 such that for every self-adjoint a ∈ A we have that

1)
for some z ∈ Z(A), some u 1 , . . . , u m ∈ U(A), and some t 1 , . . . , t m ∈ [0, 1] such that (iv) There exists a function Φ : A → Z(A) such that for every > 0 there exists n ∈ N such that for all a ∈ A we have that for some unitaries u 1 , . . . , u n ∈ U(A).
Proof. This proof uses known ideas from the theory of the Dixmier property and of sequence algebras, and is included for completeness. The implications (i)⇒(ii)⇒(iii) and (iv)⇒(i) are clear. Let us prove that (iii)⇒(iv). Given an arbitrary element a ∈ A, we can decompose a as b + ic where b, c are self-adjoint and b , c ≤ a . By a standard argument of successive averaging (cf. the proofs of [48, Lemmas 8.3.2 and 8.3 In this way, we extend (3.1) to all a ∈ A at the expense of changing (m, γ) for (m 2k , 2γ k ) (where k is chosen so that 2γ k < 1). Henceforth, let us instead assume, without loss of generality, that the constants (m, γ) are such that (3.1) is valid for all a ∈ A.
Let a ∈ A. Then there exists for some unitaries u 1 , . . . , u m ∈ A and scalars t 1 , . . . , t m ∈ [0, 1] such that m i=1 t i = 1. Set Applying the same argument to a 1 − z 1 we find z 2 ∈ Z(A), and a convex combination of m unitary conjugates of where a 2 is a convex combination of m unitary conjugates of a 1 (whence, also a convex combination of m 2 unitary conjugates of a). Then Continuing this process ad infinitum we find a k ∈ D(a) and z k ∈ Z(A) for k = 1, 2, . . . such that a k is a convex combination of m unitary conjugates of a k−1 and Recall that a k is a convex combination of m k unitary conjugates of a. Notice also that the rightmost side tends to 0 as k → ∞. This shows that for each > 0 there exists n such that a − Φ(a) ≤ a for all a ∈ A, where a is a convex combination of n unitary conjugates of a. It remains to show that this convex combination may be chosen to be an average (for a larger n). Let > 0. Pick n ∈ N such that for any a ∈ A we have for some v 1 , . . . , v n ∈ U(A) and λ 1 , . . . , λ n ≥ 0 such that n i=1 λ i = 1. Now let N ≥ 2n/ . We can find non-negative rational numbers of the form (To find such µ i , first set p 1 to be the greatest integer such that p 1 N ≤ λ 1 ; then having picked p 1 , . . . , p i−1 , pick p i to be the greatest integer such that p 1 +···+p i N ≤ λ 1 + · · · + λ i .) Let u 1 , . . . , u N be given by listing each unitary v i a total of p i times, so that Thus, N is as desired.
for all a ∈ A. Hence, the uniform Dixmier property implies the Dixmier property. Moreover, if A has the singleton Dixmier property, then Φ must be the centre-valued trace. That Theorem 3.2 (ii) implies the Dixmier property has been used many times before to show the Dixmier property (e.g., [24,72]).
We will find it useful to keep track of the constants (m, γ) such that Theorem 3.2 (ii) is satisfied. If there exist m ∈ N and 0 < γ < 1 such that for every self-adjoint a ∈ A we have that for some z ∈ Z(A) and some u 1 , . . . , u m ∈ U(A), then we say that A has the uniform Dixmier property with constants (m, γ). In particular, all finite dimensional C * -algebras have the uniform Dixmier property with constants m = 2 and γ = 3/4. For C * r (F 2 ), Powers's original argument that C * r (F 2 ) is simple can be used to derive explicit constants, and more generally, C * r (G) has the uniform Dixmier property for any Powers group G as defined in [20, p. 244]. See Example 3.8.
There have been significant recent advances in the understanding of when C * r (G) has the properties of simplicity and of unique trace (for a discrete group G) [15,36,49,51,56]; in particular, if C * r (G) is simple, then it also has a unique trace. Therefore, simplicity and the Dixmier property coincide for C * r (G); it turns out that, in fact, the Dixmier property is witnessed using only group unitaries to do the averaging ( [36,Theorem 4.5] or [51,Theorem 5.3]). However, it is not clear when C * r (G) has the uniform Dixmier property. Question 3.5. Is there a discrete group G for which C * r (G) has the Dixmier property (i.e., is simple), but not the uniform Dixmier property? Is the uniform Dixmier property for C * r (G) the same as being able to average uniformly using group unitaries? In Corollary 3.11 below we show that all AF C * -algebras with the Dixmier property have the uniform Dixmier property. In Section 3.2 we show that all C * -algebras with the Dixmier property and finite radius of comparison-by-traces have the uniform Dixmier property. More examples, and explicit constants, are discussed in Section 3.3.
(i) If A is a unital C * -algebra with the uniform Dixmier property with constants (m, γ), then all of the quotients of A have the uniform Dixmier property, also with constants (m, γ).
. are unital C * -algebras with the uniform Dixmier property with constants (m, γ), then ∞ n=1 A n has the uniform Dixmier property, also with constants (m, γ). Proof. This is straightforward.
(i): For every self-adjoint a ∈ A/I we can find a self-adjoint liftã ∈ A with the same norm. Then there exist unitaries u 1 , . . . , u m ∈ U(A) such that (3.2) holds forã. Passing to the quotient A/I we get the same for a.
(ii): Let a = (a n ) n ∈ n A n be self-adjoint. For each n we may find m unitaries u 1,n , . . . , u m,n ∈ U(A n ) and z n ∈ Z(A) such that m i=1 1 m u i,n a n u * i,n − z n ≤ γ a n .
Let u i = (u i,n ) n for i = 1, . . . , m and define z : as desired.
It will be convenient in the proof of Proposition 3.7 below to use the following notation from [5]: For a unital C * -algebra A and a subgroup V of U(A), Av(A, V) is the set of all mappings (called averaging operators) α : A → A which can be defined by an equation of the form Elementary properties of such mappings α are described in [5, 2.2].
k=1 be an increasing sequence of C * -subalgebras of A whose union is dense in A, all containing the unit. Suppose that A k has the singleton Dixmier property for all k. The following are equivalent: (i) A has the Dixmier property.
(ii) The limit lim k→∞ R k (a) exists for all a ∈ ∞ k=1 A k , where R k denotes the centrevalued trace on A k for all k.
(iii) A has the singleton Dixmier property and Note that an inductive limit of C * -algebras with the singleton Dixmier property need not have the Dixmier property (e.g., there exist simple, unital AF algebras with more than one tracial state).
Proof. Glimm's argument for UHF algebras [30,Lemma 3.1] shows that k≥1 U(A k ) is norm-dense in U(A) (in brief, if a n → u then a n (a * n a n ) −1/2 → u). Since multiplication is jointly continuous for the norm-topology on A, it follows that, for all a ∈ A, We shall use this repeatedly.
(i)⇒(iii): Let us first show that A has the singleton Dixmier property. Suppose that Enlarging n if necessary, we can find b ∈ A n such that a − b < /4. Notice then that It follows that z 1 − z 2 ≤ and hence that z 1 = z 2 , as required.
Since R n is constant on Dixmier sets in A n , R n (α(a)) = R n (a), and since R(a) ∈ Z(A), Suppose that A has the singleton Dixmier property. Let R : A → Z(A) denote its centre-valued trace. If A also has the uniform Dixmier property then by Theorem 3.2 (iv) (applied to a − R(a)) and Remark 3.3, there exist M ∈ N and 0 < Υ < 1 such that for every self-adjoint a ∈ A we have that for some u 1 , . . . , u M ∈ U(A). We will find it necessary to keep track of these constants in the theorem below, so we will say in this case that A has the uniform singleton Dixmier property with constants (M, Υ).
Since R is contractive, tracial and fixes elements of (ii): This is immediate since we always have a − R(a) ≤ 2 a . Proof. The uniform singleton Dixmier property, and indeed the constants (M, Υ), pass to quotients (by the same proof as for Theorem 3.6 (i), using Theorem 1.6 in place of lifting self-adjoint elements to self-adjoint elements); thus, we may reduce to the case that the connecting maps of the inductive limit are inclusions. So let us assume that the C * -algebras (A k ) ∞ k=1 form an increasing sequence of subalgebras of A whose union is dense in A. We denote the centre-valued trace on A k by R k . By Proposition 3.7, A has the singleton Dixmier property. We denote its centre-valued trace by R. Let a ∈ A be a self-adjoint contraction with R(a) = 0. Let > 0. Find a self-adjoint contraction b ∈ A k , for k large enough, such that a − b < . Find n > k such that R n (b) − R(b) < (its existence is guaranteed by Proposition 3.7). Thus, Since A n has the uniform singleton Dixmier property with constants (M, Υ), we have that for some unitaries u 1 , . . . , u M ∈ U(A n ). Hence, Thus, A has the uniform singleton Dixmier property with constants (M, Υ(1 + 2 ) + 3 ) for any sufficiently small > 0. Necessary and sufficient conditions for a unital AF C * -algebra to have the Dixmier property have been given in [8,Theorem 6.6]. The example in [8,Example 6.7] shows how these conditions can be verified by using a Bratteli diagram.
In the following, we let ω be a free ultrafilter on N and denote by A ω the ultrapower of A under ω. Generally, many of the arguments used with sequence algebras n A n / n A n also work with A ω (and more generally, ultrapowers ω A n ); for example we could have used ultraproducts in Theorem 3.6 instead of sequence algebras. However, A ω has some advantages in terms of its size. For example, if A is simple and purely infinite then A ω is simple [81, Proposition 6.2.6], whereas n A/ n A has a maximal ideal corresponding to each free ultrafilter. Likewise, if A has a unique trace then A ω has a unique distinguished trace (which is potentially unique -see Theorem 3.24), whereas n A/ n A has a (distinguished) trace corresponding to each free ultrafilter. For more about ultrapowers, see [54,Section 3]. The following theorem is a standard application of ultraproducts. Proof. (i)⇒(ii): Let m ∈ N and 0 < γ < 1 be such that A has the uniform Dixmier property. By Theorem 3.6 (i), ∞ (A) has the uniform Dixmier property (with the same constants), and then by Theorem 3.6 (ii), so does the quotient A ω . Moreover, since Z( ∞ (A)) = ∞ (Z(A)), and ∞ (A) has the centre-quotient property (since it has the Dixmier property), ∞ (Z(A)) is mapped onto the centre of A ω by the quotient map. Thus, Z(A) ω = Z(A ω ). 1 (ii)⇒(i): Suppose that (ii) holds and, for a contradiction, that (i) does not. Using Theorem 3.2 (iii)⇒(i), we have that condition (iii) of Theorem 3.2 does not hold, and in particular it does not hold for γ = 1/2. Thus, for each n ≥ 1 there exists a n ∈ A such that a n = 1 and for all u 1 , . . . , u n ∈ U(A) and t 1 , . .
Let a ∈ A ω be the element represented by the sequence (a n ) n . Since A ω has the Dixmier property, there exist u 1 , . . . , Since Z(A) ω = Z(A ω ), we can lift z to a bounded sequence (z n ) n from Z(A). We may also lift each u i to a sequence (u i,n ) n from U(A) (either by using [5,Proposition 2.5] in the initial choice of the elements u i or the fact that unitaries from A ω always lift to a sequence of unitaries). We have lim n→ω k i=1 t i u i,n a n u * i,n − z n < 1 2 .
In particular, for some n ≥ k we must have which gives a contradiction.
Remark 3.13. The argument in the previous proof shows, more generally, that for a class C of C * -algebras, the following are equivalent: (i) There exist constants (m, γ) such that every algebra A in C has the uniform Dixmier property with constants (m, γ) (ii) For every sequence (A n ) ∞ n=1 from C, ω A n has the Dixmier property and Z( ω A n ) = ω Z(A n ). For a C * -algebra A, the condition Z(A ω ) = Z(A) ω is related to norms of inner derivations, as follows. Firstly, recall that the triangle inequality shows that ad(a) ≤ 2d(a, Z(A)), where ad(a) is the inner derivation of A induced by a ∈ A (that is, ad(a)(x) := xa − ax). Proposition 3.14. Let A be a C * -algebra. The following are equivalent: By scaling, we may assume that d(b n , Z(A)) = 1 for all n ≥ 1. Then, for each n ≥ 1, there exists z n ∈ Z(A) such that b n − z n < 2. Let c n := b n − z n (n ≥ 1) and let c ∈ A ω correspond to the bounded sequence (c n ) n . Note that d(c n , Z(A)) = d(b n , Z(A)) = 1 (n ≥ 1) and ad(c n ) = ad(b n ) → 0 as n → ∞. For any bounded sequence (a n ) n in A, lim n→ω a n c n − c n a n = 0 and so c ∈ Z(A ω ). On the other hand, for any bounded sequence (y n ) n in Z(A), lim n→ω c n − y n ≥ 1 and so c / ∈ Z(A) ω .
2n and there exists a n ∈ A such that a n ≤ 1 and Then, for all n ≥ 1, b n − z n ≤ K(A) b n a n − a n b n + 1 n .
Recalling that b ∈ Z(A ω ), we obtain that lim n→ω b n a n − a n b n = 0 and hence that lim n→ω b n − z n = 0. Since z n ≤ 2 b n + 1 2n , (z n ) n is a bounded sequence and so b ∈ Z(A) ω .
It is easily seen that the method of proof of Proposition 3.14 also shows that K(A) < ∞ if and only if the centre of ∞ (A)/c 0 (A) is the canonical image of ∞ (Z(A))/c 0 (Z(A)).
Bearing in mind that the Dixmier property is a necessary condition for the uniform Dixmier property, we record the following simple corollary of the results in this section.
for all τ ∈ T (A) and some r > r, then a is Cuntz below b. (Recall that a is said to be Cuntz below b if d n bd * n → a for some sequence (d n ) in ∞ k=1 M k (A).) The radius of comparison-by-traces of A is the minimum r such that A has radius of comparison-by-traces at most r. If no such r exists then we say that A has infinite radius of comparison-by-traces.
In [14] the radius of comparison of A is defined as above, except that in (3.5) τ ranges through all 2-quasitraces of A normalised at the unit. We use the name "radius of comparisonby-traces" to emphasise that the comparison of a and b in (3.5) is done only on tracial states. Clearly, the radius of comparison-by-traces dominates the radius of comparison. If the C * -algebra A is exact, then by [35] its bounded 2-quasitraces are traces so the two numbers agree.  Suppose that a and b satisfy (3.5) for all tracial states τ ∈ K and some r > r. We will prove that a is Cuntz below b (which clearly shows that A has radius of comparison-by-traces at most r). Let > 0 and choose r < r < r . We claim that there exists δ > 0 such that (3.6) Indeed, let g ∈ C 0 ((0, a ]) + be such that g (t) = 1 for t ≥ . Then The function τ → τ (g (a)) + r is continuous on T (A) while τ → d τ ((b − 1 n ) + ) is lower semicontinuous for all n. Since for all τ ∈ K and K is compact, there exists n such that g (a)) + r for all τ ∈ K, thus yielding the desired δ. Decreasing δ if necessary, let us also assume that (b − δ) + is full. Letting τ range through T (A i ) ⊆ K in (3.6), and using that A i has radius of comparison-by-traces at most r, we obtain that (a i − ) + is Cuntz below (b i − δ) + for all i. Hence, using [80, Proposition 2.4], we obtain Since is arbitrary, we get that a is Cuntz below b, as desired.
(ii): Here we follow closely arguments from [63]. We first establish two claims. for all n = 1, 2, . . . and all τ ∈ K. By the proof of (i), a ⊕n is Cuntz below b ⊕n ⊕ 1 r for all n ∈ N. Now let τ ∈ T (A). Then nd τ (a) ≤ nd τ (b) + r . Letting n → ∞ we get that d τ (a) ≤ d τ (b), proving our claim. Claim 2 : If a, b ∈ A + , with b full, are such that τ (a) ≤ τ (b) for all τ ∈ K then τ (a) ≤ τ (b) for all τ ∈ T (A). Let us prove this. Let > 0. Since for all positive elements c ∈ A and all σ ∈ T (A) (see for example [27,Proposition 4.2]), one can construct positive elements a n , b n (in matrix algebras over A), and find natural numbers r n , s n such that lim n→∞ 1 r n d σ (a n ) = σ((1 − )a) and lim n→∞ 1 for all tracial states σ, with both sequences increasing. Since b is full, we have that for all τ ∈ K. Using lower semi-continuity and the compactness of K, as in part (i), we obtain n ∈ N such that τ ((1 − )a) ≤ 1 sn d τ (b n ) for all τ ∈ K. Hence, 1 for all m and for all sufficiently large n. By the first claim applied to the positive elements a ⊕sn m and b ⊕rm for any σ ∈ T (A). Taking the limit as m → ∞, we obtain σ((1 − )a) ≤ σ(b). Letting → 0 proves the claim.
Let us now show that K = T (A). By the Hahn-Banach theorem, it suffices to show that for all self-adjoint a ∈ A, if τ (a) = 0 for all τ ∈ K then τ (a) = 0 for all τ ∈ T (A). If a is a self-adjoint such that τ (a) = 0 for all τ ∈ K then τ (a + t1) = τ (t1) for all τ ∈ K. Moreover, for t > a both a + t1 and t1 are positive and full. It follows by Claim 2 that τ (a + t1) = τ (t1) for all τ ∈ T (A) and t > a , which yields the desired result.
In the proof of the next result we make use of Theorem 4.4, proven in Section 4 below, and whose proof is independent from the results of this section. Theorem 4.4 is an extension of Theorem 1.2 from the introduction to non-self-adjoint elements. Theorem 3.20. Let r ∈ [0, ∞). There exists M ∈ N such that if A is a unital C * -algebra with radius of comparison-by-traces at most r and a ∈ A is such that 0 ∈ D A (a), then for some unitaries u 1 , . . . , u M ∈ A.
Proof. Suppose, for the sake of contradiction, that there exist unital C * -algebras A 1 , A 2 , . . . with radius of comparison-by-traces at most r, and contractions a n ∈ A n such that 0 ∈ D A (a n ) for all n, but any average of n unitary conjugates of a n has norm greater than 1/2. Let A := ∞ n=1 A n and a := (a n ) n ∈ A. We will show that 0 ∈ D A (a) relying on Theorem 4.4 from Section 4. To show that 0 ∈ D A (a), it suffices to check conditions (a) and (b) of Theorem 4.4. Notice that τ (a) = τ (a n ) = 0 for all τ ∈ T (A n ) and all n. It follows by Theorem 3.19 (ii) that τ (a) = 0 for all τ ∈ T (A), i.e., condition (a) holds. In order to show that a satisfies condition (b), we prove that it satisfies the equivalent form (b"), stated right before the proof of Theorem 4.4. Let t , t > 0 be such t > t and let w ∈ C. Since 0 ∈ D An (a n ), we have, by condition (b") applied to a n , that (Re(wa n ) + t) + is a full element of A n (i.e., it generates A n as a closed two-sided ideal). For all τ ∈ T (A n ) we have where we have used that d τ (c) ≥ τ (c)/ c for any c ≥ 0 in the first inequality. Choose N ≥ (2 + r)(|w| + t)/t. Then d τ ((Re(wa n ) + t) ⊕N + ) ≥ 2 + r. Since A n has radius of comparison-by-traces at most r, the above (including fullness of (Re(wa n ) + t) + ) implies that 1 ∈ A n is Cuntz below (Re(wa n ) + t) ⊕N + . Thus, there exists a partial isometry v n ∈ M N (A n ) such that 1 = v * n v n and v n v * n ≤ C · (Re(wa n ) + t ) ⊕N + , where C > 0 depends on t and t but not on n. Then, setting v := (v n ) n ∈ M N (A), we get 1 = v * v and vv * ≤ C · (Re(wa n ) + t ) ⊕N + . Hence, (Re(wa) + t ) + is full for all t > 0 and w ∈ C. This proves condition (b"). It follows that 0 ∈ D A (a). Thus, there is a finite convex combination of unitary conjugates of a whose norm is less than 1 2 . Enlarging the number of terms if necessary, we may assume that this convex combination is an average (see the proof of Theorem 3.2 (iii)⇒(iv)). So, there exist M ∈ N and unitaries u 1 , . . . , u M ∈ A such that We arrive at a contradiction by projecting onto A M .
Remark 3.21. For the case of self-adjoint elements (which is all that is needed in the next corollary), Theorem 3.20 can be proven using Theorem 1.2 in place of Theorem 4.4.

Corollary 3.22.
Let r ∈ [0, ∞). Then there exist constants (m, γ) such that every unital C * -algebra with the Dixmier property and with radius of comparison-by-traces at most r has the uniform Dixmier property with constants (m, γ). In particular, every simple unital C * -algebra with at most one tracial state and radius of comparison-by-traces at most r has the uniform Dixmier property with constants (m, γ).
Proof. Let M ∈ N be as in Theorem 3.20. Suppose that A is a unital C * -algebra with the Dixmier property and radius of comparison-by-traces at most r. Now let a ∈ A be a self-adjoint element and choose z ∈ D A (a) ∩ Z(A). Then 0 ∈ D A (a − z), and so  Let us explain why these examples have finite radius of comparison-by-traces (which is the same as finite radius of comparison, since they are exact). For n ∈ N, Villadsen's algebra A with stable rank n + 1 is constructed, according to [102,Section 3], as A = lim − → A i where A i = p i (C(X i ) ⊗ K)p i , with X i a certain space of dimension n(1 + 2 · 1! + 4 · 2! + · · · + 2i · i!) and p i a certain projection of constant rank (i + 1)!. We compute By [98,Theorem 5.1], it follows that A i has radius of comparison at most 2n. Hence by [14,Proposition 3.2.4], the radius of comparison of A is at most 2n.
3.2. C * -algebras with trivial centre. If A is a unital C * -algebra with trivial centre, then by Corollary 2.10 A has the Dixmier property if and only if we have one of the following four cases: (1) A is simple and has no tracial states, (2) A is simple and has a unique tracial state, (3) A has no tracial states and a unique non-zero maximal ideal, (4) A has a unique tracial state and its trace-kernel ideal is the unique nonzero maximal ideal of A.
Cases (2) and (4) have the singleton Dixmier property while cases (1) and (3) do not. Now, since A is unital and has the Dixmier property, K(A) ≤ 1 < ∞ and so Z(A ω ) = Z(A) ω = C1 by Proposition 3.14. (That C ω = C is because every bounded sequence of complex numbers has a unique limit under ω, i.e., the map taking (x n ) ∞ n=1 ∈ n C to lim n→ω x n induces an isomorphism C ω → C.) Thus, by Corollary 3.15, in order for A to have the uniform Dixmier property A ω must also fall in one of the four cases above. In Theorem 3.24 below we take this analysis further to obtain explicit conditions for having the uniform Dixmier property when A falls in cases (2) and (4) above.
Suppose that A is in either case (2) or (4). Let τ denote the unique tracial state of A. Then τ induces a canonical tracial state τ ω on A ω , by τ ω (a) := lim n→ω τ (a n ), whenever a is represented by the sequence (a n ) n . Let J := {a ∈ A ω : τ ω (a * a) = 0}, the trace-kernel ideal for τ ω . Using the Kaplansky density theorem, one can see that A ω /J is isomorphic to the tracial von Neumann ultrapower of π τ (A) , where π τ is the GNS representation associated to τ ([54, Theorem 3.3]). In particular, this quotient is a finite factor, and is therefore simple, so that J is a maximal ideal.
In the next result, conditions (i) and (iii) are both expressed purely in terms of the C * -algebra A. However, in order to show that these conditions are equivalent, we introduce A ω so that we can apply Corollary 2.10 and Corollary 3.15.
Theorem 3.24. Let A be a C * -algebra with the Dixmier property, trivial centre, and unique tracial state τ . The following are equivalent: (i) A has the uniform Dixmier property.
(ii) τ ω is the unique tracial state on A ω and the trace-kernel ideal J is the unique maximal ideal of A ω . (iii) Both of the following hold: (a) there exists m ∈ N such that if a ∈ A is a self-adjoint contraction satisfying τ (a) = 0 then there exist contractions x i ax * i = 1.
Proof. Recall that since A is unital and has the Dixmier property, K(A) ≤ 1 < ∞ and so Z(A ω ) = Z(A) ω = C1 by Proposition 3.14.
(ii)⇔(iii): We will first show that (a) is equivalent to τ ω being the unique tracial state on A ω , then that (b) is equivalent to J being the unique maximal ideal of A ω .
For Suppose that τ ω is the unique tracial state on A ω and, for a contradiction, that (a) doesn't hold. Then for each n ∈ N there exists a self-adjoint contraction a n ∈ A such that τ (a n ) = 0 and for all tuples (x 1 , . . . , x n ) of contractions in A.
Since the sequence (a n ) n is bounded, it defines a self-adjoint element a ∈ A ω , and this element clearly satisfies τ ω (a) = 0. Since τ ω is the unique tracial state, it follows (as mentioned above) that there exist m ∈ N and y 1 , . . . , y m ∈ A ω such that By increasing m if necessary, we may assume that all of the elements y i are contractions. Lifting each y i to a sequence (x i,n ) n of contractions in A, we have for ω-almost all n ∈ N, In particular, for some n ≥ m, we obtain a contradiction to (3.7). This proves that if A ω has a unique tracial state then (a) holds. Now suppose that (a) holds, which provides a number m. If a ∈ A ω is a self-adjoint contraction satisfying τ ω (a) = 0, then we may lift a to a sequence (a n ) ∞ n=1 of self-adjoint elements satisfying τ (a n ) = 0 and a n ≤ a for all n. (To achieve this, we first lift a to any bounded sequence of self-adjoint elements, then correct the tracial state on each element by adding an appropriate scalar, and finally scale to obtain a n ≤ a .) By applying (a) to each a n , we can arrive at elements x 1 , . . . , x m ∈ A ω such that In other words, this shows that A ω satisfies (a), with τ ω in place of τ . Next, by iterating, we see that if a ∈ A ω is a self-adjoint contraction and satisfies τ ω (a) = 0, then for any k ∈ N , there exist mk contractions x 1 , . . . , x mk ∈ A ω such that It follows that a ∈ (A ω ) 0 . By R-linearity, a is self-adjoint and τ ω (a) = 0} and hence τ ω is the unique tracial state of A ω . Now, suppose that J is the unique maximal ideal of A ω and let us prove that (b) holds. Suppose for a contradiction that (b) doesn't hold. Then there exists > 0 and, for each n ∈ N, a contraction a n ∈ A + such that τ (a n ) > yet n i=1 x i a n x * i = 1 (3.8) for all contractions x 1 , . . . , x n ∈ A.
Define a ∈ A ω by the sequence (a n ) n , so that τ ω (a) ≥ . Since J is the unique maximal ideal of A ω , the ideal generated by a is A ω . Hence, there exists y 1 , . . . , y m ∈ A ω such that and by increasing m if necessary we may assume that all of the elements y i are contractions. Lift each y i to a sequence (y i,k ) k of contractions. Then, for ω-almost all indices k, we have Pick k ≥ 2m such that this holds. Set so that the spectrum of b is contained in [1/2, 3/2]. Therefore, (2b) −1/2 y i,k is a contraction, and in contradiction to (3.8).
Finally assume that (b) holds, and we'll prove that J is the unique maximal ideal of A ω . Let I be an ideal of A ω , such that I ⊆ J. Therefore, I contains a positive contraction a ∈ J, so that r := τ ω (a) > 0. Using := r/2, we get some n ∈ N from (b).
We may lift a to a sequence (a k ) k of positive contractions such that τ (a k ) > r/2 for each k. Then for each k there exist n contractions x 1,k , . . . , x n,k ∈ A such that Letting x i ∈ A ω be the element represented by the sequence (x i,k ) k , we have and therefore I = A ω . This shows that J is the unique maximal ideal of A ω .
Under the hypotheses of Theorem 3.24, it is unclear whether there is any relation between the conditions that τ ω is the unique tracial state on A ω (equivalently, condition (a)) and that J is the unique maximal ideal of A ω (equivalently, condition (b)). In [79,Theorem 1.4], LR showed that there is a simple unital (and nuclear, in fact AH) C * -algebra A with unique tracial state, which doesn't satisfy (iii)(a) in Theorem 3.24 (i.e., A ω doesn't have a unique tracial state). Since A has the Dixmier property by [37], this shows that the Dixmier property is strictly weaker than the uniform Dixmier property.
Let us briefly discuss the cases when A is unital, has the Dixmier property, trivial centre, and no tracial states (i.e., cases (1) and (3) from the beginning of this section). If A is simple and purely infinite, then A ω is also simple and purely infinite ([81, Proposition 6.2.6]), whence has the Dixmier property, and so A has the uniform Dixmier property. In the cases that A is not simple and purely infinite, we have little to say about whether A has the uniform Dixmier property. In such cases, A ω has no tracial states either, but it is not simple, for if A ω is simple and non-elementary then A must be simple and purely infinite (  Let A be a simple unital C * -algebra with no tracial states, which is not purely infinite. Then there is a bounded sequence of self-adjoint elements (a n ) ∞ n=1 with 1 ∈ D A (a n ) for all n, but 1 ∈ D Aω ((a n ) n ). (However, it is conceivable that D Aω ((a n ) n ) meets Z(A ω ) in another point, so this does not show that A does not have the uniform Dixmier property.) To see this, first, since A ω is non-simple, there exists a positive element a ∈ A ω of norm 2 that is not full. Lift a to a bounded sequence (a n ) n of positive elements. Since a = 2 and a is not invertible, for ω-almost all n, the convex hull of the spectrum of a n contains 1. Modifying a n for n in an ω-null set, we can arrange that the convex hull of the spectrum of a n contains 1 for all n. Since A is simple, it follows that 1 ∈ D A (a n ) for all n. However, if 1 ∈ D Aω (A) then with I := Ideal(a), 1 ∈ D Aω/I (q I (a)) = D Aω/I (0), which is a contradiction.

Explicit constants.
Suppose that A is a unital C * -algebra that has the Dixmier property as well as one of the following properties: (1) finite nuclear dimension, or (2) finite radius of comparison by traces. Then A has the uniform Dixmier property for suitable constants (m, γ) (i.e., (3.2) holds). For finite nuclear dimension, this follows from [65,Theorem 5.6]. For finite radius of comparison, this is Corollary 3.22 obtained above. These results are proven by contradiction, with repeated use of the Hahn-Banach Theorem, thereby not yielding explicit values for the constants (m, γ). In fact, we do not know explicit values for (m, γ) holding globally in either one of these two cases. (On the other hand, explicit constants may be extracted from the methods used in [87] and [66], for simple C * -algebras with real rank zero, strict comparison by traces, and a unique tracial state.) In this section, prompted by an interesting question posed by the referee, we find explicit values for the constants (m, γ) for a variety of C * -algebras with the uniform Dixmier property. When the C * -algebras have the singleton Dixmier property, we also estimate the constants (M, Υ) (i.e, for which (3.4) holds).
Let A be a C * -algebra. Let h ∈ A be a self-adjoint element and let [l(h), r(h)] be the smallest interval containing the spectrum of h, i.e., the numerical range of h. Set ω(h) := r(h) − l(h) and note that ω(h) ≤ 2 h .
We first consider uniform Dixmier property constants for von Neumann algebras (slightly improving the constants that can be extracted from Dixmier's original argument [24, Lemma 1 of §III.5.1]). Let W be a von Neumann algebra and h a self-adjoint element of W . Let e ∈ W be a central projection. In the next lemma ω e (h) denotes ω(eh) in the von Neumann algebra eW .
Proof. It is shown in [65, Proposition 3.2] that given two self-adjoint elements h 1 , h 2 ∈ W with finite spectrum, it is possible to find projections P 1 , . . . , P N adding up to 1, a unitary u ∈ W , and self-adjoint central elements λ 1 , µ 1 , . . . , λ N , µ N ∈ Z(W ) with finite spectrum such that (Note: [65, Proposition 3.2] is stated for positive elements but it is easily extended to self-adjoint elements by adding a scalar.) Let us apply this result to the self-adjoint elements h and −h. We then get Let i, j ∈ S with i ≤ j. Then Thus, ω e k (h) ≤ ω e k (h)/2 for all k, as desired.
Proof. Let 0 < < 1/2 and let 0 = g = g * ∈ W . By the spectral theorem, there is a self-adjoint element h ∈ W with finite spectrum such that g − h < g and h ≤ g . Apply Lemma 3.27 to h to obtain a unitary u ∈ W and central projections e 1 , . . . , e n as in the statement of that lemma, and then define the central element z := n k=1 α k e k , where α k is the midpoint of the spectrum of e k (h + uhu * )/2 in e k W (that is, the midpoint of the interval [l(e k (h + uhu * )/2), r(e k (h + uhu * )/2)]). Then we see that Since the e k are orthogonal central projections, it follows that (h + uhu * )/2 − z ≤ 1 2 h . Then (g + ugu * )/2 − z ≤ h /2 + g ≤ (1/2 + ) g .
Proof. It is well-known that C(X, M n ) has the singleton Dixmier property: for example, the Dixmier property holds by [5, Proposition 2.10] and the singleton Dixmier property is then a consequence of the fact that every simple quotient has a tracial state (see Proposition 1.4). We prove (ii) first, because the argument is more similar to the previous proof.
(ii): Let h ∈ C(X, M n ) be a self-adjoint element. By [94,Theorem 10], h is approximately unitarily equivalent to a diagonal self-adjoint h = diag(λ 1 , . . . , λ n ), where the eigenvalue functions λ 1 , . . . , λ n ∈ C(X, R) are arranged in decreasing order: λ 1 ≥ λ 2 ≥ · · · ≥ λ n . Note that a self-adjoint element a in a unital C * -algebra satisfies (3.2) if and only if every unitary conjugate of a does so (with the same central element z). Hence, by an approximation argument similar to that in the proof of Theorem 3.28, it suffices to establish (3.2) with m = 2 and γ = 1/2 for diagonal self-adjoint elements of the form above. So assume that h is diagonal with decreasing eigenvalue functions. Let u ∈ M n be the permutation unitary such that uhu * = diag(λ n , . . . , λ 1 ). Seth : The same estimates used in the proof of Lemma 3.27 show that ω(h) ≤ ω(h)/2. It follows that h − λ 1 + λ n 2 · 1 n ≤ 1 2 h .
As observed above, this shows that C(X, M n ) has the uniform Dixmier property with m = 2 and every γ ∈ (1/2, 1). The constants M = 4 and Υ ∈ (1/2, 1) are then derived from the constants (m, γ) as in the proof of Theorem 3.28.
(i): Let > 0 be given. Let f ∈ C(X, M n ) be a self-adjoint contraction. For each x ∈ X, by Proposition 3.29, we may find λ x ∈ R and a unitary u x ∈ M n such that Evidently, we may assume λ x ∈ [−1, 1]. By continuity, we may then find a neighbourhood W x of x such that From the open cover {W x : x ∈ X} of X, using compactness and the fact that X has dimension d, we may find a finite refinement of the form {W . . , d}, since all unitaries in M n are homotopic to the identity, we may produce a unitary u (i) ∈ C(X, M n ) such that as follows. We may find disjoint open sets V ) * ) has distance at most (d + 1)/(d + 2) + to the centre. Note that, by a partition-of-unity argument, the distance fromf to the centre is equal to the supremum over all x ∈ X of the distance fromf (x) to Z(M n ) = C1 n (see [89,Theorem 2.3] for a more general result). For x ∈ X, pick i 0 , j such that x ∈ W (i 0 ) j . Without loss of generality, i 0 = 0. Then A similar argument is used to get uniform singleton Dixmier property constants. Here we may replace f with f − R(f ), so that f (x) has trace 0 for all x ∈ X. Then we use the same argument as above, with the uniform singleton Dixmier property constants (4, 1 2 ) from Proposition 3.29, and with λ x = 0 for all x (and thereby λ (i) j = 0 for all i, j), to get M = 3(d + 1) + 1 = 3d + 4 and (for any sufficiently small > 0) Υ = 1 2 4 3d + 4 + 3d 3d + 4 + = 3d + 2 3d + 4 + .
Consider the following property of unital C * -algebras A: (P): There exist M ∈ N and 0 < Υ < 1 such that if h ∈ A is a self-adjoint such that τ (h) = 0 for all τ ∈ T (A) then for some unitaries u 1 , . . . , u M . Note that if A has the property (P) for some (M, Υ) then it also has (P) for (M k , Υ k ) (k = 2, 3, . . .).
Suppose that A has the Dixmier property and has the property (P) for some (M, Υ). For h = h * ∈ A and z 1 , z 2 ∈ D A (h) ∩ Z(A), we have τ (z 1 − z 2 ) = 0 for all τ ∈ T (A) and hence 0 ∈ D A (z 1 − z 2 ) = {z 1 − z 2 }. Thus z 1 = z 2 . An elementary argument with real and imaginary parts shows that A has the singleton Dixmier property. It then follows from (P) that A has the uniform singleton Dixmier property with the same constants (M, Υ) (as introduced in (3.4)).
Conversely, suppose that A has the uniform singleton Dixmier property with constants (M, Υ). If h = h * ∈ A vanishes on all tracial states of A then h also vanishes on the centre-valued trace of A. Thus A has the property (P) with the same (M, Υ). But (P) may hold much more generally: if every quotient of A has a bounded trace and A has either finite nuclear dimension or finite radius of comparison by traces then A has (P) for some (M, Υ) ([65, Theorem 5.6] for the former case, Theorem 3.20 in the latter).
In the following results, it will occasionally be convenient to write a ≈ b to mean a − b < .
Theorem 3.31. Let A be a unital C * -algebra with decomposition rank one and stable rank one. Then A has (P) with constants (M, Υ) for M = 15 and every Υ ∈ (11/15, 1). In particular, if A also has the Dixmier property, then it has the uniform singleton Dixmier property with these constants.
Proof. Let > 0 be given. Let us factorise the diagonal embedding ι : Suppose that, for every n ∈ N, M n has the uniform Dixmier property with constants (m, γ) (we shall describe suitable values m > 1 and γ towards the end of the proof). Then any product of finite-dimensional C * -algebras has the uniform Dixmier property with constants (m, γ) (Theorem 3.6(ii)), hence the Dixmier property and hence the singleton Dixmier property by Proposition 1.4 (note that the product of the centre-valued traces is a centrevalued trace on the product). Hence N 0 and N 1 have the uniform Dixmier property with constants (m, γ) (Theorem 3.6) and the singleton Dixmier property (see the remark after Proposition 1.4). By Lemma 3.9, N 0 and N 1 have the uniform singleton Dixmier property with constants (m, 2γ) (and for this we require 2γ < 1). Hence, as noted above, N 0 and N 1 have property (P) with constants M = m > 1 and Υ = 2γ.
Let h ∈ A be a self-adjoint contraction that is zero on every tracial state. Then the same is true for ψ i (h) for i = 0, 1. So for i = 0, 1 there exist unitaries u i,1 , . . . , u i,M −1 ∈ N i such that so that x i,j depends on n, and as n → ∞. Since A has stable rank one, every element in A ∞ has a unitary polar decomposition, with the unitary element belonging to A ∞ . (Indeed, since A has stable rank one every element of A ∞ lifts to an element (a k ) ∈ ∞ k=1 A such that a k is invertible for all k; further, such an (a k ) has polar decomposition in ∞ k=1 A.) So x i,j = U i,j |x i,j | for some unitary U i,j ∈ A ∞ . Then, remembering that x i,j depends on n, as n → ∞. From (3.10), for n sufficiently large we get for i = 0, 1. We choose n so that in addition, we will estimate its norm. We manipulate the sum on the right side: Let e 0 := φ 0 (1) and e 1 := φ 1 (1). Then h 0 ≤ e 0 , h 1 ≤ e 1 , and e 0 + e 1 = 1. Next from (3.11) and the fact that e 0 +e 1 = 1, it follows that [U i,j , e 1−i ] < for i = 0, 1 and j = 1, . . . , M −1. Hence, U 1,j h 0 U * 1,j + U 0,j h 1 U * 0,j ≤ U 1,j e 0 U * 1,j + U 0,j e 1 U * 0,j ≈ 2 e 0 + e 1 = 1, which implies that U 1,j h 0 U * 1,j + U 0,j h 1 U * 0,j ≤ (1 + 2 )1. A similar argument shows that U 1,j h 0 U * 1,j + U 0,j h 1 U * 0,j ≥ −(1 + 2 )1. The norm of the right side of (3.12) is at most 2M (Υ + ) + (M − 1)(1 + 2 ) from which we obtain that Thus A has property (P) with constants M = 2M −1 and Υ = (2M Υ +M −1)/(2M −1)+2 (provided that this value of Υ is less than 1). It follows from Proposition 3.29 that, for every n, the C * -algebra M n has the uniform Dixmier property with constants m = 2 3 = 8 and γ = (1/2) 3 = 1/8. By the discussion above, we may take M = 8 and Υ = 1/4 and hence obtain that A has (P) with constants (M, Υ) for M = 15 and every Υ ∈ (11/15, 1).
Our next goal is to prove Theorem 3.33 below. But first we need a lemma and some preliminaries.
Given a self-adjoint element h, let us say that the spectrum of h has gaps of size at most δ if every closed subinterval of [l(h), r(h)] of length δ intersects the spectrum of h.
Hence b −1 c is invertible and so c is invertible, which contradicts that 1/3 is in the spectrum ofk. A similar argument shows that the spectrum ofh intersects (2/3 − /2, 2/3 + /2). It follows that the spectrum ofh has gaps of size at most 1/3 + .
(ii): Choose points l(h) = t 0 < t 1 < . . . < t n = r(h) in the spectrum of h such that t i+1 −t i ≤ δ for all i. Perturb h by functional calculus using an increasing continuous function close to the identity function that takes the constant value t i in a small neighborhood of each t i , so that the new h satisfies h − h < , has spectrum contained in [l(h), r(h)] and has the property that there exist pairwise orthogonal non-zero positive elements e 0 , e 1 , . . . , e n such that h e i = t i e i . For each i = 0, 1, . . . , n, choose x i ∈ e i Ae i such that x 2 i = 0 and x * i x i (and hence x i x * i ) has spectrum equal to [0,1]. This is possible since e i Ae i is simple and non-elementary. Now let We claim thath and x are as desired. It follows from the pairwise orthogonality of the e i that x 2 = 0, that Let us show thath has spectrum [l(h) − δ/2, r(h) + δ/2]. Note that all of the elements h , lie in a commutative C * -subalgebra C containing the unit 1 of A. Evaluating the right-hand side of the expression forh on the points of the spectrum of C where x * i x i is supported, all other terms except for h vanish, while h takes the constant value t i . Since the spectrum of x * i x i is [0, 1], we obtain the interval [t i − δ/2, t i ] in the spectrum of h. Evaluating on the points where x i x * i is supported, we obtain the interval [t i , t i + δ/2] in the spectrum ofh. Doing this for all i, we obtain the interval [l(h) − δ/2, r(h) + δ/2] in the spectrum ofh. Evaluating on any other point in the spectrum of C, we obtain a value in the spectrum of h which is contained in [l(h), r(h)]. Thus the spectrum ofh is as required.
Let A be simple, unital, non-elementary, with stable rank one and with strict comparison by traces. Using Cuntz semigroup classification results, one can prove the existence of a nuclear C * -subalgebra B ⊆ A with rather special properties. [64,Theorem 4.1] spells out the properties of B that we need: (i) B ∼ = C ⊗ W , where C is a simple AF C * -algebra and W is the Jacelon-Razak algebra.
(ii) Every tracial state τ on B extends uniquely to a tracial state on A.
(iii) Every non-invertible self-adjoint element h in A with connected spectrum is approximately unitarily equivalent to a self-adjoint element in B.  [43] and [64] involves using these properties to reduce the proof of certain properties of self-adjoint elements in A to the case of self-adjoint elements in B. We we will use the same technique to prove the following theorem. In this theorem, the initial hypotheses on A ensure that T (A) is non-empty (surely, if we had T (A) = ∅ then the strict comparison-by-traces property and the simplicity of A would imply that A is purely infinite, contradicting that A has stable rank one.) Thus the later additional assumption that A has the Dixmier property is equivalent to assuming that A has a unique tracial state [37].
Theorem 3.33. Let A be a simple, unital, non-elementary C * -algebra with stable rank one and strict comparison by traces. Then A has (P) with constants M = 3 · 7 3 and Υ = 0.86. Suppose, in addition, that A has the Dixmier property. Then A has the uniform singleton Dixmier property with these constants.
Proof. Suppose that the unitisation B + C1 of B has (P) with constants (M , Υ ). We show how to obtain constants for A. Suppose that h ∈ A is a self-adjoint element that is zero on every trace and such that h = 1. Let > 0. From Lemma 3.32 (i), we obtain h 1 = (1/3)h + (2/3)uhu * whose spectrum has gaps at most 2/3 + . Applying Lemma 3.32 (ii) to h 1 with δ = 2/3 + , we obtainh 1 , x ∈ A, as described, such that (3.13) Notice, for later use, that since the positive elements x * x and xx * are orthogonal Also, by our choice of δ, the spectrum ofh 1 is exactly the interval From this and h 1 ≤ 1, we see that h 1 ≤ 4/3 + /2. Moreover,h 1 is non-invertible. Indeed, its spectrum contains the closed interval [l(h 1 ), r(h 1 )] and the latter contains 0 since h 1 is zero on all traces and T (A) = ∅ (as argued before this theorem). By property (iii) of the C * -subalgebra B above, there is a self-adjoint element b ∈ B which is approximately unitarily equivalent toh 1 . Notice, from (3.13), that sup{|τ (h 1 )| : τ ∈ T (A)} < . Applying this average on both sides of (3.13), we find an average of 3M unitary conjugates of the original element h with norm at most Υ ( 4 3 Since can be chosen arbitrarily small, we find that, provided 4 3 Υ + 1 3 < 1, A has (P) with constants M = 3M and every Υ ∈ ( 4 3 Υ + 1 3 , 1). (3.14) Finally, let us find suitable constants for the unitisation B + C1 of B. Since B has decomposition rank 1 and stable rank one, B + C1 has the same properties and so has (P) with constants M = 15 and arbitrary Υ ∈ (11/15, 1) by Theorem 3.31. Therefore, it also has (P) with constants M = 15 3 and arbitrary Υ ∈ ((11/15) 3 , 1). Putting the latter constants into the formula (3.14), we get that A has (P) with constants M = 3 · 15 3 and Υ = 0.86.

The distance between Dixmier sets
In this section, we derive results about the distance between Dixmier sets D A (a) and D A (b). Along the way, we obtain a description of D A (a) ∩ Z(A) for C * -algebras with the Dixmier property and we point out some cases in which the distance between D A (a) and D A (b) is attained. Here by the distance between two subsets D 1 , D 2 of a C * -algebra, we mean Lemma 4.1. Let A be a unital C * -algebra and let a, b ∈ A. The distance between the sets D A (a) and D A (b) is equal to the distance between the sets D A * * (a) and D A * * (b). D A (b)). Let us prove the opposite inequality. Let > 0 be given. Let a ∈ D A * * (a) and b ∈ D A * * (b) be such that a − b < r + . Approximating a and b by averages of unitary conjugates there exists some N and unitaries u 1 , . . . , u N , v 1 , . . . , v N ∈ U(A * * ) such that The weak * -closure of this convex set (in A * * ) contains an element of norm less than r + (namely, the element appearing in (4.1)), so by the Hahn-Banach theorem, S must also contain an element of norm less than r + . (Otherwise, the Hahn-Banach theorem ensures the existence of a functional λ ∈ A * such that Re(λ(x)) < r + for all x < r + and Re(λ(s)) ≥ r + for all s ∈ S; but then λ ≤ 1 and Re(λ(s)) ≥ r + for all s in the weak * -closure of S, which is a contradiction.) However, note that S ⊆ D A (a) − D A (b), so this shows that d(D A (a), D A (b)) ≤ r + . Since is arbitrary, we are done.
Given a unital C * -algebra A and a ∈ A, let W A (a) := {ρ(a) : ρ ∈ S(A)}, the algebraic numerical range of a. Since the state space S(A) is weak * -compact and convex, W A (a) is a compact convex subset of C.
Lemma 4.2. Let A be a unital C * -algebra and let a, b ∈ A. The following are true: Proof. Thus, d(W A (a), W A (b)) ≤ a − b . But if α, β ∈ Av(A, U(A)) are averaging operators then W A (α(a)) ⊆ W A (a) and W A (β(b)) ⊆ W A (b). So Passing to the infimum over all α, β ∈ Av(A, U(A)) we get that as desired. Lemma 4.1 allows us to reduce the calculation of the distance between Dixmier sets to the case that the ambient C * -algebra is a von Neumann algebra. To deal with the von Neumann algebra case we rely on the following theorem of Halpern and Strȃtilȃ-Zsidó:  Condition (b) need only be checked on all the quotients by maximal ideals of A. A reformulation of (b) is (b') on every nonzero quotient there exists a state that vanishes on a; i.e., 0 ∈ W A/I (q I (a)) for all closed ideals I of A. To see this, suppose that ρ(a) = 0 for some ρ ∈ S(A). For all w ∈ C, ρ(wa) = 0 and so ρ(Re(wa)) = 0. Hence Re(wa) is not invertible and negative. Conversely, suppose that 0 / ∈ W A (a). Then by convexity of W A (a), for a suitable w ∈ C and > 0, Re(ρ(wa)) ≤ − < 0 for all states ρ, i.e., ρ(Re(wa)) ≤ − < 0 for all states ρ. But this implies that Re(wa) is negative and invertible. This equivalence holds similarly in every nonzero quotient. Notice that if every nonzero quotient of A has a tracial state then (b') follows from (a).
Another reformulation of (b) is the following: (b") (Re(wa) + t) + is a full element (i.e., generates A as a closed two-sided ideal) for all t > 0 and all w ∈ C. To see that this is equivalent to Theorem 4.4 (b), notice first that Re(wa) ≤ −t1 in the quotient by the closed two-sided ideal generated by (Re(wa) + t) + , where a → a is the quotient map for this ideal. So, assuming (b), this quotient must be {0}, i.e., (Re(wa) + t) + is full for all t > 0. On the other hand, if Re(wa) ≤ −t1 in the quotient by some ideal I, then clearly (Re(wa) + t1) + ∈ I. So, if t > 0, and assuming (b"), we get that I = A.
Proof of Theorem 4.4. Since traces are constant on Dixmier sets, if 0 ∈ D A (a) then τ (a) = 0 for all τ ∈ T (A), i.e., (a) holds. Also, if 0 ∈ D A (a) then 0 ∈ D A (wa) for any w ∈ C (indeed, any central element) and this prevents Re(wa) from being invertible and negative. The same holds for quotients since q I (D A (a)) ⊆ D A/I (q I (a)). Thus, (b) holds as well.
Suppose now that a ∈ A is such that (a) and (b) hold. If A ⊆ B (where B is a C * -algebra with the same unit as A) then (a) and (b) also hold in B. This is clear for condition (a), since traces of B restrict to traces of A. This is also clear for condition (b"), which is equivalent to (b). It follows that a satisfies (a) and (b) in the von Neumann algebra A * * . Let A * * f ⊕ A * * pi be the decomposition of A * * into a finite and a properly infinite von Neumann algebra and let a = a f + a pi be the corresponding decomposition of a. From condition (a) we get that R(a f ) = 0, where R denotes the center-valued trace, which in turn implies that 0 ∈ D A * * f (a f ). On the other hand, from condition (b) we get that for every maximal ideal I of A * * pi there exists a state on A * * pi /I that vanishes on q I (a pi ). By Theorem 4.3, 0 ∈ D A * * pi (a pi ). Since we may extend unitary elements in A * * f (respectively A * * pi ) by adding the unit of A * * pi (respectively A * * f ), we conclude that 0 ∈ D A * * (a). By Lemma 4.1, 0 ∈ D A (a).
The next result extends the discussion in Section 2 (after Theorem 2.6) concerning the form of the sets D A (a) ∩ Z(A) in a unital C * -algebra A which has the Dixmier property. Here, the element a ∈ A is not required to be self-adjoint. Our next goal is to extend Theorem 4.4 to a distance formula between the sets D A (a) and D A (b) (Theorem 4.12 below). Note that one cannot reduce the calculation of this distance to the case that one element is 0 by looking at the distance between D A (b − a) and 0, since  . Let (I λ ) λ be a collection of closed ideals of a C * -algebra A and let I be a closed ideal of A such that λ I λ ⊆ I. Then every state of A which vanishes on I is a weak * -limit of convex combinations of states vanishing on the I λ 's.
Recall that for topological spaces X and Y , a set-valued function φ : X → {subsets of Y } is defined to be lower semicontinuous if for every open set U of Y , the set is open in X. Later in this section we will use the Michael selection theorem ([59, Theorem 3.1']).
The next lemma is implicit in a strategy mentioned in [58]. Let c λ ∈ λ denote the point of tangency. Let us pass to a subnet M λ → M such that the c λ → c, and let be the tangent line at c. Since the sets Φ a (M λ ) are uniformly bounded (they are all inside the ball B a (0)), there exists λ 0 such that the sets Φ a (M λ ) for λ ≥ λ 0 are all separated from the ball B /2 (w) by a single line 0 parallel to (and tangent to the circle {z : |z − w| = /2}). But, since λ ≥λ 0 M λ ⊆ M , we have by the previous lemma that any state of A which vanishes on M is a weak * -limit of convex combinations of states vanishing on the M λ 's. In particular, w (= ρ(a) for some state ρ of A which vanishes on M ) is a limit of convex combinations of elements in λ ≥λ 0 Φ a (M λ ). This contradicts that we can separate λ ≥λ 0 Φ a (M λ ) from B /2 (w) by the line 0 . Let us describe more specifically how to obtain λ 0 . The line 0 is parallel to but closer to w. We may therefore choose λ 0 such that all the points {c λ : λ ≥ λ 0 } lie on the same side of 0 and such that the lines λ and 0 intersect outside of the ball B a (0) for all λ ≥ λ 0 . Then λ 0 is as desired.
Define a set-valued function F a on Max(A) as follows: The values of F a are compact convex subsets of C. Since M → W A/M (q M (a)) is lower semicontinuous by Lemma 4.7, Y is closed, F a | Y is continuous, and τ M (a) ∈ W A/M (q M (a)) for M ∈ Y , the set-valued function F a is lower semicontinuous.
The following proposition is trivial in the case of the singleton Dixmier property. It is again lower semicontinuous and takes values in the closed intervals of R. Hence, it has a continuous selection giving rise to a central element z b ∈ D A (b) such that z a − z b ≤ r. Since A is weakly central and has no tracial states, it has the Dixmier property by Theorem 2.6 (this can also be seen from the fact that A is * -isomorphic to the tensor product of O 2 with an abelian C * -algebra). We now extend the distance formula from Proposition 4.10 to arbitrary C * -algebras. The following result gives a formula for the distance between the Dixmier sets of two elements of an arbitrary unital C * -algebra. A similar result is [65,Theorem 4.3], which gives a formula for the distance between one self-adjoint element and the Dixmier set of another; these results say the same thing in the case that both elements are self-adjoint and one is central.