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The bundle of KMS state spaces for flows on a unital C*-algebra

It is shown that any bundle of KMS state spaces which can occur for a flow on a unital separable C*-algebra with a trace state can also be realized by a flow on any given unital infinite-dimensional simple AF algebra with a tracial state space affinely homeomorphic to the fiber in the bundle over 0.


Introduction
The collection of KMS state spaces for a flow on a unital C * -algebra can be thought of as a bundle of simplices over the real line. This may seem like a far-fetched analogy to other more established notions of bundles because the fibers of the bundle may be empty or they may all be mutually non-isomorphic, but we will show here that it is in fact a well-behaved and useful concept. Specifically, we use it to show that for any given unital separable infinite-dimensional simple AF algebra A and for any configuration of KMS state spaces which occurs for a flow on a unital separable C * -algebra and has the property that the simplex of 0-KMS states is affinely homeomorphic to the tracial state space of A, there is a flow on A with the same configuration. In particular, it follows that for any given closed subset F of real numbers containing 0 there are flows on A whose KMS spectrum is F . This removes the lower boundedness condition which occurs in a recent work by the second author, [Th3].
Since we deal with unital AF algebras, there are always 0-KMS states present. For flows on infinite C * -algebras this is not the case, and in a joint work with Y. Sato we have shown that for any unital, nuclear, purely infinite, simple, separable C * -algebra A in the UCT class and with torsion-free K 1 group, and for any configuration of KMS state spaces which occurs for a flow on a unital separable C * -algebra without trace states, there is also a flow on A with the same configuration; see [EST]. In both cases we depend on results from the classification of simple C * -algebras.
While the work in [Th3] was based on ideas from [BEH] and [BEK1], in the present paper the underlying ideas are closer to those presented by Bratteli,Elliott and Kishimoto in [BEK2]. In particular, the idea of considering the configuration of KMS simplices as a bundle originates from [BEK2].

Proper simplex bundles
Let S be a second countable locally compact Hausdorff space and π ∶ S → R a continuous map. If the inverse image π −1 (t), equipped with the relative topology inherited from S, is homeomorphic to a compact metrizable Choquet simplex for all t ∈ R we say that (S, π) is a simplex bundle. We emphasize that π need not be surjective, and we consider therefore also the empty set as a simplex. When (S, π) is a simplex bundle we denote by A(S, π) the set of continuous functions f ∶ S → R with the property that the restriction f π −1 (t) of f to π −1 (t) is affine for all t ∈ R.
Definition 2.1. (Compare [BEK2].) A simplex bundle (S, π) is a proper simplex bundle when (1) π is proper; that is π −1 (K) is compact in S when K ⊆ R is compact, and (2) A(S, π) separate points on S; that is for all x ≠ y in S there is an f ∈ A(S, π) such that f (x) ≠ f (y).
2.1. Proper simplex bundles from flows. In this paper all C *algebras are assumed to be separable and all traces and weights on a C * -algebra are required to be non-zero, densely defined and lower semi-continuous. Let A be a C * -algebra and θ a flow on A. Let β ∈ R. A β-KMS weight for θ is a weight ω on A such that ω ○ θ t = ω for all t, and ) . (2.1) In particular, a 0-KMS weight for θ is a θ-invariant trace. A bounded β-KMS weight is called a β-KMS functional and a β-KMS state when it is of norm one. For states alternative formulations of the KMS condition can be found in [BR]. Assume that A is unital. For each β ∈ R let S θ β be the (possibly empty) set of β-KMS states for θ. Let E(A) be the state space of A, a compact convex set in the weak* topology. Set and equip S θ with the relative topology inherited from the product topology of E(A) × R. Since S θ is a closed subset of E(A) × R by Proposition 5.3.23 of [BR], it follows that S θ is a second countable locally compact Hausdorff space. Denote by π θ ∶ S θ → R the projection to the second coordinate. Since the inverse image π θ −1 (β) is homeomorphic to S θ β , which is a Choquet simplex by Theorem 5.3.30 of [BR], the pair (S θ , π θ ) is a simplex bundle. An obvious application of Proposition 5.3.23 of [BR], using the compactness of E(A), shows that π θ is proper. Note that every self-adjoint element a ∈ A gives rise to an elementâ ∈ A(S θ , π θ ) such thatâ(ω, β) = ω(a). A continuous function f ∶ R → R also gives rise to an element of A(S θ , π θ ): It follows that A(S θ , π θ ) separates the points of S θ , showing that (S θ , π θ ) is a proper simplex bundle, which we shall call the KMS bundle of the flow. In general, π θ −1 (0) is the set of θ-invariant trace states and hence non-empty if and only if A has trace states. When A is AF it is the simplex of all trace states of A.
Remark 2.2. Let (S, π) be a proper simplex bundle. Denote by A R (S, π) the subset of A(S, π) consisting of the elements that have a limit at infinity. This is a separable real Banach space (in the supremum norm) containing the constant function 1, and its state space is a metrizable compact convex set in the weak* topology. For x ∈ S, let ev x ∈ E(A R (S, π)) denote evaluation at x. For each β ∈ R, the set is a closed convex subset of E(A R (S, π)). With K = E(A R (S, π)), the system K β , β ∈ R, has the properties required in Theorem 2.1 of [BEK1]. Thus, there is a unital, simple, separable, nuclear C * -algebra A equipped with a 2π-periodic flow θ such that π −1 (β) is affinely homeomorphic to the simplex of β-KMS states for all β ∈ R. In the construction of A in [BEK1] the algebra appears to depend on (S, π) and on the many choices made in the process of its construction, but it was shown in [EST], based on the Kirchberg-Phillips classification result, that when π −1 (0) is empty one can take A to be any given separable, simple, nuclear, purely infinite C * -algebra in the UCT class and with torsion free K 1 group. It follows from the main result we describe next that when π −1 (0) is not empty one can take A to be any infinite dimensional unital simple AF algebra whose tracial state space is affinely homeomorphic to π −1 (0).

The main result and applications
Theorem 3.1. Let (S, π) be a proper simplex bundle and let A be a unital infinite-dimensional simple AF algebra with a tracial state space affinely homeomorphic to π −1 (0). There is a 2π-periodic flow on A whose KMS bundle is isomorphic to (S, π).
By definition the KMS spectrum of a flow θ on a unital C * -algebra is the set of real numbers β for which θ has a β-KMS state. When the KMS bundle of θ is isomorphic to a proper simplex bundle (S, π), the KMS spectrum of θ is the range π(S) of π.
To exhibit possible ways to work with proper simplex bundles and to illustrate how Theorem 3.1 can be applied let us use it to prove the following statement.
Corollary 3.2. Let A be a unital infinite-dimensional simple AF algebra and let F be a closed subset of real numbers containing 0.
• There is a 2π-periodic flow on A whose KMS spectrum is F and such that there is a unique β-KMS state for all β ∈ F {0}. • There is a 2π-periodic flow θ on A whose KMS spectrum is F and such that S θ β is not affinely homeomorphic to S θ β ′ when β, β ′ ∈ F {0} and β ≠ β ′ .
Proof. Given a proper simplex bundle (S, π), set S F = π −1 (F ) and denote by π F the restriction of π to π −1 (F ). The pair (S F , π F ) is again a proper simplex bundle.
For the first item, let T (A) be the tracial state space of A and fix an element ω 0 ∈ T (A). Let S be the subset of the topological product T (A) × R. Let π ∶ S → R be the canonical projection. Then (S, π) is a proper simplex bundle such that π −1 (β) = {(ω 0 , β)} when β ≠ 0 and such that π −1 (0) is a copy of T (A). The existence of the desired flow follows from Theorem 3.1 by applying it to the bundle (S F , π F ). For the second item, note that the KMS bundle of the flow described in Theorem 1.1 of [Th2] is a proper simplex bundle (S ′ , π ′ ) such that π ′−1 (0) contains only one point, π ′ (S) = R and such that π ′−1 (β) is not affinely homeomorphic to π ′−1 (β ′ ) when β ≠ β ′ . With (S, π) the bundle defined above set Define π ′′ ∶ S ′′ → R by π ′′ (x, y) = π(y). Then (S ′′ , π ′′ ) is a proper simplex bundle and an application of Theorem 3.1, this time to (S ′′ F , π ′′ F ), gives the desired flow.

Proof of the main result
4.1. Tools. The following lemma is an immediate consequence of Lemma 3.1 of [Th3] and Theorem 2.4 of [Th1].
Lemma 4.1. Let D be a C * -algebra. Denote by ρ ∈ Aut(D) an automorphism of D and let q ∈ D a projection in D which is full 1 in D ⋊ ρ Z. Letρ denote the restriction to q(D ⋊ ρ Z)q of the dual action on D ⋊ ρ Z considered as a 2π-periodic flow. Let P ∶ D ⋊ ρ Z → D denote the canonical conditional expectation. For each β ∈ R, the map τ ↦ τ ○ P q(D⋊ρZ)q is an affine homeomorphism from the set of traces τ on D that satisfy τ ○ ρ = e −β τ and τ (q) = 1, (4.1) onto the simplex of β-KMS states forρ.
In this lemma the topology on the set of traces of D with the properties (4.1) is given by pointwise convergence on elements from the corner qDq of D.
When D is an AF algebra it is well-known that the set of its traces can be identified, via the map τ ↦ τ * , with the set Hom + (K 0 (D), R) of non-zero positive homomorphisms φ ∶ K 0 (D) → R; a fact stated as Lemma 3.5 in [Th3]. In the setting of Lemma 4.1 this implies that when D is an AF algebra the KMS spectrum and the structure of the KMS states forρ can be determined directly from the pair (K 0 (D), ρ * ). To see how, we note that by Remark 3.3 in [LN] every β-KMS state τ forρ on q(D ⋊ ρ Z)q extends uniquely to a β-KMS weightτ for the dual action. Since D is the fixed point algebra for the dual action the restriction ofτ to D is a trace on D, yielding a map from the set of β-KMS states τ forρ to Hom + (K 0 (D), R). Therefore Lemma 4.1 has the following consequence.
Corollary 4.2. In the setting of Lemma 4.1 assume that D is an AF algebra. For each β ∈ R the map (4.2) is an affine homeomorphism from the set of β-KMS states forρ on q(D ⋊ γ Z)q onto the set of positive homomorphisms φ ∈ Hom + (K 0 (D), R) that satisfy Here the topology on the elements from Hom Corollary 4.2 will be complemented by the following lemma which helps to control the Elliott invariant of q(D ⋊ γ Z)q, and to ensure that it is classified by it. It follows from Lemma 3.4 of [Th3], which is based on arguments from [Sa], [MS1] and [MS2].
Lemma 4.3. Let D be a stable AF algebra such that K 0 (D) has large denominators. 2 For any order automorphism α ∈ Aut denotes the Jiang-Su algebra, [JS].
Given a proper simplex bundle (S, π) and a closed subset F ⊆ R we denote by (S F , π F ) the proper simplex bundle with S F = π −1 (F ) and π F is the restriction of π to S F . The following lemma relates A(S F , π F ) to A(S, π) and it will be a crucial tool in the following.
Lemma 4.4. Let (S, π) be a proper simplex bundle and F ⊆ R a closed subset.
(1) The map A(S, π) → A(S F , π F ) given by restriction is surjective. ( There is an element h ∈ A(S, π) such that h(y) = h F (y) for all y ∈ S F and is a compact simplex bundle in the sense of [BEK2] and it follows from Lemma 2.2 in [BEK2] that there are elements n . This element f extends h. The assertion (2) follows in a similar way on using Lemma 2.3 in [BEK2].

4.2.
Dimension groups from proper simplex bundles. Given a Choquet simplex ∆, as usual denote by Aff∆ the set of real-valued continuous and affine functions on ∆. Fix a proper simplex bundle (S, π) with π −1 (0) non-empty. Let H be a torsion free abelian group and θ ∶ H → Affπ −1 (0) a homomorphism. We assume (1) there is an element u ∈ H such that θ(u) = 1, and Then (H, H + ) is a simple dimension group; see [EHS]. It follows from (1) of Lemma 4.4 that the map r ∶ A(S, π) → Affπ −1 (0) given by restriction is surjective and we can therefore choose a linear map L ∶ Affπ −1 (0) → A(π, S) such that r ○ L = id. We arrange, as we can, that In what follows we denote the support of a real-valued function f by supp f . Let A 00 (S, π) denote the set of elements f from A(S, π) for which supp f is compact and contained in S π −1 (0). Since the topology of S is second countable we can choose a countable subgroup G 00 of A 00 (S, π) with the following density property: When f 1 ∈L(⊕ Z H) and f ± 2 ∈ Q[e −π , 1 − e −π ] the difference between ○ π is an element of A 00 (S, π), and so by enlarging G 00 we can ensure that (4.5) Let α 0 ∈ Aut A(S, π) be defined by Since α 0 (A 00 ) = A 00 and since (1 − e −π ) −1 A 00 (S, π) ⊆ A 00 (S, π), we can enlarge G 00 further to achieve that α 0 (G 00 ) = G 00 (4.6) and that Then α 0 ○L =L ○ σ, which implies that α 0 (G k ) = G k and hence Lemma 4.6. The pair (G, G + ) has the following properties. ( Proof. (1) and (3) are obvious. (2): Let f ∈ G. Then f ∈ G k + G 00 for some k ∈ N and we can write , f 0 ∈L(⊕ Z H) and g ∈ G 00 . By definition of Q[e −π , 1 − e −π ] there are n, m ∈ N and M > 0 such that (4): Since θ(H) has the Riesz interpolation property for the strict order by Lemma 3.1 in [EHS], there is h 0 ∈ θ(H) such that f i (x) < h 0 (x) < g j (x) for all i, j and all x ∈ π −1 (0). We claim that there are elements h ± ∈ Q[e −π , 1−e −π ] and K ± ∈ N such that f To find h + and K + , note that we may assume that π(S) contains arbitrarily large positive numbers; otherwise we take K + larger that sup π(S) and h + = 0. By definition of G there is N ∈ N so large that there are polynomials p 1 , p 2 , q 1 , q 2 with rational coefficients such that for all i, j and all large x. Then p i (y) < q j (y) for all large elements y of e π(S) and it follows therefore from Lemma 4.7 that there is a polynomial h ′ with rational coefficients such that p i (x) < h ′ (x) < q j (x) for all i, j and all large x. Set h + (x) = e π(x) (e π(x) − 1) −N h ′ (e π(x) ) .
Then h + ∈ Q[e −π , 1−e −π ] and if K + is large enough f i (x) < h + (x) < g j (x) for all i, j and all x ∈ π −1 ([K + , ∞)). The pair h − , K − is constructed in a similar way: Multiply each of the functions from {f 1 , f 2 , g 1 , g 2 } with the same function of the form (e −π ) N (e −π − 1) N to get them into the form x ↦ p(e −π(x) ) for p a polynomial with rational coefficients and apply Lemma 4.7.
Having h ± and K ± we use the statement (2) of Lemma 4.4 to find for all x ∈ S and all i, j. Set and note that sup H ′′ ⊆] − K − , K + [ {0}. Set K = max{K − , K + } and let δ > 0 be given, smaller than g j (x) − H ′ (x) and H ′ (x) − f i (x) for all i, j and all x ∈ π −1 ([−K, K]). By Property 4.5 there is an element g ′ ∈ G 00 such that Lemma 4.7. Let p i , q j , i, j ∈ {1, 2}, be polynomials with rational coefficients. Assume that there is a sequence {x n } in R such that lim n→∞ x n = ∞ and such that p i (x n ) < q j (x n ) for all i, j, n. It follows that there is a polynomial h with rational coefficients and a K > 0 such that for all i, j ∈ {1, 2} and all x ≥ K.
Proof. Since polynomials only have finitely many zeros there is a K ′ > 0 such that p i (x) < q j (x) for all i, j and all x ≥ K ′ . Write q j (x) = a 0,j +a 1,j x+a 2,j x 2 +⋯+a N,j x N and p i (x) = b 0,j +b 1,j x+b 2,j x 2 +⋯+b N,j x N for some N larger than the degree of any of the four given polynomials, and set and η j = (a N,j , a N −1,j , ⋯, a 0,j ) ∈ Q N +1 .
Since p i (x) < q j (x) for all large x, we have that ξ i < lex η j for all i, j with respect to the lexicographic order < lex . Since Q N +1 is totally ordered in the lexicographic order there is an element c = (c N , c N −1 , c N −2 , ⋯, c 0 ) ∈ Q N +1 such that ξ i < lex c < lex η j for all i, j. By changing c 0 by a small amount we can arrange that c ∉ {ξ 1 , ξ 2 , η 1 , η 2 }. Then the polynomial will have the desired property.
Consider the subset Γ of (⊕ Z H) ⊕ G consisting of the elements (ξ, g) ∈ (⊕ Z H) ⊕ G with the property that there is an ǫ > 0 such thatL Γ is a subgroup of (⊕ Z H) ⊕ G.
Proof. Let g ∈ G. By definition of G there is an element ξ ∈ ⊕ Z H and k ∈ N such that g(x) =L(ξ) on π −1 ( − 1 2k , 1 2k . By combining Lemma 4.8 and Lemma 4.6 above with Lemma 3.1 and Lemma 3.2 in [EHS] we conclude that (Γ, Γ + ) is a dimension group. Given an element h ∈ H we denote in the following by h (0) the ele- Lemma 4.9. (Γ, Γ + ) has large denominators; that is for all x ∈ Γ + and m ∈ N there is an element y ∈ Γ + and an n ∈ N such that my ≤ x ≤ ny.
Lemma 4.11. Let f ∈ A R (S, π) and let ǫ > 0 be given. There is an element g ∈ G such that sup x∈S f (x) − g(x) ≤ ǫ.
Proof. An initial approximation gives us an element f 1 ∈ A(S, π) which is compactly supported and a real number r ∈ R such that Let q ∈ Q such that q − r < ǫ 6 and choose an element h ∈ H such that θ(h)(y) − f 1 (y) − r < ǫ 6 for all y ∈ π −1 (0). There is a k ∈ N such that L(θ(h))( ). Since f 1 ψ + k ○ π + f 1 ψ − k ○ π is compactly supported in S π −1 (0) it follows from Property 4.5 that there is an element g ′ ∈ G 00 such that Then is an element with the desired property.
Proof. The projection p ∶ Γ → G is surjective by Lemma 4.8. Assume that (ξ, g) ∈ Γ and p(ξ, g) = g = 0. Since (Γ, Γ + ) has large denominators by Lemma 4.9 there is for each n ∈ N an element (ξ n , g n ) ∈ Γ + and a natural number k n such that n(ξ n , g n ) ≤ v ≤ k n (ξ n , g n ). Then ±(ξ, g) ≤ (ξ n , g n ) in Γ and hence ±φ(ξ, g) ≤ φ(ξ n , g n ) ≤ 1 n . It follows that φ(ξ, g) = 0 and we conclude that there is a homomorphism φ ′ ∶ G → R such that φ ′ ○ p = φ. Let g ∈ G and assume that g(x) ≥ 0 for all x ∈ S, and let n ∈ N. There is h n ∈ H + such that 0 < θ(h n ) < 1 n , and then also a natural number k ∈ N such that 0 Let ξ ∈ ⊕ n∈Z H be an element such that (ξ, g) ∈ Γ. Then (h (0) n + ξ, g ′ n + g) ∈ Γ + and hence 0 ≤ φ ′ (g ′ n + g) ≤ φ ′ (g) + 1 n . Letting n tend to infinity we find that φ ′ (g) ≥ 0, proving that φ ′ is positive on G. Let g ∈ G and n, m ∈ N satisfy g(x) < n m for all x ∈ S. Then −n < mg(x) < n for all x ∈ S and since φ ′ (1) = 1 this leads to the conclusion that φ ′ (g) ≤ n m . Combined with Lemma 4.11 it follows from the last estimate that φ ′ extends by continuity to a linear map φ ′ ∶ A R (S, π) → R such that φ ′ (f ) ≤ sup x∈S f (x) . Using a Hahn-Banach theorem we extend φ ′ in a normpreserving way to the space of all continuous real-valued functions on S with a limit at infinity. Since φ ′ (1) = 1 the extension is positive. It follows that there is a bounded Borel measure m on S such that for all f ∈ A 0 (S, π), where A 0 (S, π) denotes the space of elements in A R (S, π) that vanish at infinity. Let C c (R) denote the set of continuous real-valued compactly supported functions on R and note that C c (R) is mapped into A 0 (S, π) by the formula F ↦ F ○ π. Since φ ○ α = sφ by assumption it follows that the measure m ○ π −1 on R satisfies It follows that m ○ π −1 is concentrated at the point β = − log s and hence that m is concentrated on π −1 (β). We can therefore define a linear functional φ ′′ ∶ Affπ −1 (β) → R by wheref ∈ A 0 (S, π) is any element withf π −1 (β) = f , which exists by (1) in Lemma 4.4. If f ≥ 0 it follows from (2) of Lemma 4.4 thatf can be chosen such thatf ≥ −ǫ for any ǫ > 0 and we see therefore that φ ′′ is a positive linear functional. Since every state of Affπ −1 (β) is given by evaluation at a point in π −1 (β) it follows in this way that there is an ω ∈ π −1 (β) and a number λ ≥ 0 such that for all g ∈ A 0 (S, π). In particular, this conclusion holds for all g ∈ G ∩ A 0 (S, π). A general element f ∈ G can be write as a sum where f ± , f 0 ∈ G, f 0 has compact support and there are natural numbers n ± ∈ N such that e n−π f − ∈ A 0 (S, π) and e −n+π f + ∈ A 0 (S, π). Then and similarly, φ ′ (f + ) = λf + (ω). It follows that φ(f ) = λf (ω). Inserting f = 1 we find that λ = 1 and the proof is complete.
Additional properties 4.13. By Lemma 4.3 we can arrange that γ has the following additional properties: (A) The restriction map µ ↦ µ B is a bijection from traces µ on B ⋊ γ Z onto the γ-invariant traces on B, and the Jiang-Su algebra, [JS].
Set C = B ⋊ γ Z . It follows from Lemma 4.10 and [E1] that B is γ-simple and hence from [E2] (see also [Ki1]) that C is simple. It follows from the Pimsner-Voiculescu exact sequence, [PV], that we can identify K 0 (C), as a group, with the quotient in such a way that the map ι * ∶ K 0 (B) → K 0 (C) induced by the inclusion ι ∶ B → C becomes the quotient map Lemma 4.14. ker S 0 = (id −α)(Γ).

4.5.
Completing the proof of Theorem 3.1 via classification theory. Let (S, π) and A be as in Theorem 3.1. With H = K 0 (A) and the assumed identification of the tracial state space T (A) of A with π −1 (0) we get the homomorphism θ ∶ H → Affπ −1 (0) from the canonical map K 0 (A) → AffT (A). It follows from Theorem 4.11 in [GH] that θ(K 0 (A)) is dense in Affπ −1 (0) and that We can therefore apply the preceding with H = K 0 (A), H + = K 0 (A) + and u = [1].
Proof. Since A is AF the K 1 group of A is trivial, and by Lemma 4.16 the same is true for eCe since eCe is stably isomorphic to C. The affine homeomorphism τ → τ ′ of Lemma 4.17 is compatible with the isomorphism of ordered groups S ∶ K 0 (eCe) → K 0 (A) from Lemma βn and τ * = ω β . Then Φ −1 (ω n ) = (τ n ○P eCe , β n ) and Φ −1 (ω) = (τ ○P eCe , β). It suffices therefore to show that lim n→∞ τ n ○ P (exe) = τ ○ P (exe) for all x ∈ C. Since τ n ○ P (e) = τ ○ P (e) = 1, it suffices to check for x in a dense subset of C. If w is the canonical unitary in the multiplier algebra of C coming from the construction of C as a crossed product, it suffices to show that lim n→∞ τ n ○ P (ebw k e) = τ ○ P (ebw k e) for all k ∈ Z and all b ∈ B. Since P (ebw k e) = 0 when k ≠ 0 it suffices to consider the case k = 0; that is, it suffices to show that lim n→∞ τ n (ebe) = τ (ebe). By approximating ebe by a linear combination of projections from eBe it suffices to show that lim n→∞ τ n (p) = τ (p) when p is a projection in eBe. This holds because