Boundary quotients of the right Toeplitz algebra of the affine semigroup over the natural numbers

We consider the semigroup crossed product of the additive natural numbers by the multiplicative natural numbers. We study its Toeplitz C*-algebra generated by the right-regular representation, which we call the right Toeplitz algebra. We analyse its structure by studying three distinguished quotients. We show that the multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of the additive rationals by an action of the multiplicative rationals, and study its ideal structure. We identify the Crisp-Laca boundary quotient as the C*-algebra of the corresponding group built from rational numbers. There is a natural dynamics on the right Toeplitz algebra and all its KMS states factor through the additive boundary quotient. We describe the KMS simplex for inverse temperatures greater than one.

to have shared with Vaughan what, in retrospect, were far too few precious moments. He will be remembered and he will be missed.
Astrid. Everyone agrees that Vaughan has done wonderful mathematics and wonderful things for mathematics all over the world. But I really treasure how generous Vaughan was with his time and energy, and in particular how kind and supportive he was of me when I first moved to NZ in 2010. His influence on the NZ mathematics community has been colossal.

INTRODUCTION
We consider the semidirect product group Q ⋊ Q × + and its Toeplitz algebras. In previous work [23], we studied the usual Toeplitz algebra T (N ⋊ N × ) associated to the left-invariant partial order with positive cone N ⋊ N × . The main results there show that there is a natural dynamics on T (N ⋊ N × ) that admits a rich supply of KMS states and exhibits phase transitions like those of the Bost-Connes system [5].
The semigroup N ⋊ N × also induces a right-invariant partial order on Q ⋊ Q × + , but the corresponding Toeplitz algebra T rt (N ⋊ N × ) is known to be quite different. For example, the left Toeplitz algebra T (N ⋊ N × ) has an ideal I which contains every other proper ideal, and for which the boundary quotient T (N ⋊ N × )/I is the simple purely infinite C * -algebra C Q of Cuntz [12]. The right Toeplitz algebra T rt (N ⋊ N × ), on the other hand, has many one-dimensional representations. Curiously, though, in spite of this seemingly huge difference, the two Toeplitz algebras are known to have the same K-theory [13, §6.4].
Our goal here is to study the structure of the right Toeplitz algebra T rt (N ⋊ N × ). As suggested in [13], we view it as the left Toeplitz algebra of the opposite semigroup (N ⋊ N × ) op , because we can then use the general results about left-invariant quasi-lattice ordered groups from [28] and [21]. We then avoid a proliferation of ops by realising (N ⋊ N × ) op as the isomorphic semidirect product N × ⋉ N. This notation leads us to use similar notation for crossed-product C * -algebras, and we discuss some of the ramifications in §4.
The Toeplitz algebra T (N × ⋉ N) is generated by an isometric representation V of N × and a single isometry S which generates a representation of N. We use these generators to give a presentation of T (N × ⋉ N) (Proposition 3.3). Then we focus on three quotients of T (N × ⋉ N): the multiplicative boundary quotient ∂ mult T (N × ⋉ N) in which the Vs are unitary, the additive boundary quotient ∂ add T (N × ⋉ N) in which S is unitary, and the Crisp-Laca boundary quotient ∂T (N × ⋉ N) in which they all are.
Our first main result is a structure theorem for ∂ mult T (N × ⋉ N). For the proof, we realise ∂ mult T (N × ⋉ N) as a crossed product of the usual Toeplitz algebra T (Q + ) by an action of the group Q × + (Proposition 5.4), and analyse the structure of this crossed product C * -algebra. The structure of the Toeplitz algebra T (Q + ) is described by results of Douglas [15]. We identify the commutator ideal in the Toeplitz algebra as a corner in a crossed product of a commutative C * -algebra, and then combine the actions to get an action of Q × + ⋉ Q on the commutative algebra, which we can study using the well-established (i.e., now relatively old-fashioned!) theory of transformation group algebras. The final result (Theorem 6.10) describes a composition series for ∂ mult T (N × ⋉ N) with a large commutative quotient, and two other simple subquotients.
Then, following [23], we consider a dynamics σ : R → Aut T (N × ⋉ N) such that σ t fixes S and σ t (V a ) = a it V a , and aim to describe the KMS states of (T (N × ⋉ N), σ). We have been successful for inverse temperatures β > 1, by applying general results from [20] about KMS states on semidirect products in [23,Theorem 7.1]; they are parametrised by probability measures on T (see Theorem 8.1 below). At β = 1, the arguments break down: we can see by example (using various combinations of point masses on T) that there are many KMS 1 states, but we do not know how to parametrise them.
In Appendix A we discuss how our results intersect with those of [19] and [1], and, in particular, realise the additive boundary quotient as a Nica-Toeplitz-Pimsner algebra analogous to the ones studied in [19].

QUASI-LATTICE ORDERED GROUPS
Suppose that G is a group and P is a submonoid which generates G and satisfies P ∩ P −1 = {e}. Then there is a partial order ≤ l on G such that g ≤ l h ⇐⇒ g −1 h ∈ P; the subscript in ≤ l is there to remind us that the partial order is invariant under left multiplication by elements of G. Following Nica [28], the pair (G, P) is quasi-lattice ordered if (QL) for all n ≥ 1 and all x 1 , . . . , x n in G with a common upper bound in P also have a least common upper bound in P. It follows from [10,Lemma 7] that (QL) is equivalent to (QL1) every element of G with an upper bound in P has a least upper bound in P. The subset of elements of G which have upper bounds in P (that is, to which (QL1) applies) is PP −1 = pq −1 : p, q ∈ P .
An isometric representation of P is a homomorphism V of P into the semigroup of isometries in a C * -algebra, and V is Nica covariant if the range projections satisfy V p V * p V q V * q = V p∨q V * p∨q for all p, q ∈ P. A key example of a Nica-covariant representation is the Toeplitz representation T : P → B(ℓ 2 (P)), which is characterised in terms of the usual basis {ǫ p : p ∈ P} by T q ǫ p = ǫ qp . The Toeplitz algebra T (P) is the C * -subalgebra of B(ℓ 2 (P)) generated by {T p : p ∈ P}. The pair (G, P) also has a right-invariant partial order such that g ≤ r h ⇐⇒ hg −1 ∈ P. Then there is a homomorphism W : P → B(ℓ 2 (P)) characterised by W q ǫ p = ǫ pq −1 if q ≤ r p and 0 otherwise.
Each W q is a coisometry: W * q : ǫ p → ǫ pq is an isometry. We define the right Toeplitz algebra T rt (P) to be the C * -subalgebra C * (W p : p ∈ P) = C * (W * p : p ∈ P) of B(ℓ 2 (P)). There is a dual notion of right quasi-lattice order, and a corresponding notion of Nica covariant co-representations in which the initial projections satisfy W * p W p W * q W q = W * p∨q W * p∨q for all p, q. The Toeplitz co-representation W is Nica covariant.
We can study the right Toeplitz algebra T rt (P) by applying existing theory to the opposite semigroup (G op , P op ), which is the set G op = {g op : g ∈ G} with g op h op = (hg) op . Indeed, the map g → g op is an anti-isomorphism which maps P to P op and satisfies g op ≤ l h op ⇐⇒ (g op ) −1 h op ∈ P ⇐⇒ (hg −1 ) op ⇐⇒ hg −1 ∈ P ⇐⇒ g ≤ r h.
Then the unitary U : ℓ 2 (P) → ℓ 2 (P op ) characterised by Uǫ p = ǫ p op satisfies UW * q U * ǫ p op = UW * q ǫ p = Uǫ pq = ǫ q op p op = T q op ǫ p op . Thus conjugation by U gives an anti-isomorphism of T rt (P) = C * (W q ) onto T (P op ) = C * (T q ), which is the object we study in this paper.

THE OPPOSITE OF THE AFFINE SEMIGROUP AND ITS TOEPLITZ ALGEBRA
We use N for the natural numbers which for us include 0. We write N × for the multiplicative semigroup {1, 2, 3, . . . } of N. We write Q for the rational numbers, Q + for the additive semigroup of non-negative rational numbers and Q × + for the multiplicative group of positive rational numbers.
We begin by checking that the established theory applies to T (N × ⋉ N). We recall from [10, Definition 26] that (G, P) is lattice ordered every element of G has a least upper bound in P. By [10, Lemma 27], (G, P is lattice ordered if and only if G = PP −1 and (G, P) is quasi-lattice ordered.
Proof of Proposition 3.1. We take elements (a, r) and (b, s) of Q × + ⋉ Q. Then, by analogy with the discussion at the foot of page 648 in [23], we have: We now aim to verify that (a, r) has a least upper bound in N × ⋉ N. We write a = c −1 d and r = c −1 k with d, c ∈ N × and k ∈ Z, and with common denominator c ∈ N × as small as possible. Then from (3.1) we have This says, first, that for n ≥ cr = k, (d, n) is an upper bound for (a, r). It also suggests that with n := max(0, k) (remember that k is negative if r is), (d, n) is a candidate for the least upper bound.
To confirm that (d, n) is a least upper bound, suppose that (b, p) ∈ N × ⋉ N and (a, r) ≤ (b, p). Then (3.1) implies that f := a −1 b ∈ N × and l := p − a −1 br ∈ N. But then f satisfies a = bf −1 ∈ N × and r = (p − l)f −1 ∈ Z. Thus "c as small as possible" implies that there exists e ∈ N × such that f = ce. Now we have Note that n is either 0 or k. Since p ∈ N, we trivially have p − (d −1 b)0 ∈ N. So we suppose that n = k. Then Thus we have (d, n) ≤ (b, p), as required. Proposition 3.1 allows us to apply the existing results about left-invariant quasi-lattice ordered groups to (Q × + ⋉ Q, N × ⋉ N). In particular, since Q × + ⋉ Q is a semidirect product of abelian groups, it is amenable, and hence (Q × + ⋉ Q, N × ⋉ N) is amenable as a quasi-lattice ordered group (see [28, §4.5]). Thus we can apply [28, §4.2] (or [21,Corollary 3.8]) to (Q × + ⋉ Q, N × ⋉ N). This gives: The left-regular representation of N × ⋉ N on ℓ 2 (N × ⋉ N) is characterised in terms of its action on point masses by T (a,m) ε (b,n) = ε (ab,bm+n) .
We now give a presentation of the Toeplitz algebra of N × ⋉ N.
Let C * (s, v) be the universal C*-algebra generated by elements s and {v a : a ∈ N × } satisfying the relations (T0)-(T4). Then there is an isomorphism π of C * (s, v) onto the Toeplitz algebra T (N × ⋉ N) such that π(s) = S and π(v a ) = V a .
Proof. The relation (T0) says that the generators are isometries. Relation (T1) holds because (1,1) = V a S a , and (T3) because (a, 0)(b, 0) = (ab, 0). Relation (T3) follows from Nica-covariance for the pair (a, 0) and (b, 0), and we claim that (T4) follows from Nica covariance for (1, 1) and (a, 0). Indeed, we have (1, 1) ∨ (a, 0) = (a, a), and T (a,a) = V a S a . Thus Nica covariance gives Now suppose S and {V a : a ∈ N × } are elements of a C * -algebra A satisfying (T0)-(T4), and define L : N × ⋉ N → A by L (a,m) = V a S m . Then from (T1) we have Thus L is a homomorphism of N × ⋉ N into the semigroup of isometries in A. We next check that L is Nica covariant. In the following calculations, we write a = gcd(a, b)a ′ and similarly for b. Then we have on one hand, , and on the other . These last two expressions are the same because (for example) lcm(a, b) = gcd(a, b)a ′ b ′ = ab ′ , S k S * k S l S * l = S k∨l S * (k∨l) , and lcm(a, b)(a −1 m ∨ b −1 n) = b ′ m ∨ a ′ n by left invariance of the partial order. Thus L is Nica covariant.
Since the elements s and {v a : a ∈ N × } satisfy the relations (T0)-(T4), the previous paragraph gives us a Nica-covariant representation L : N × ⋉N → C * (s, v) such that L (a,m) = v a s m for all (a, m). Now Corollary 3.2 gives a homomorphism π L : T (N × ⋉ N) → C * (v, s) such that π L (T (a,m) ) = v a s m . On the other hand, the first paragraph of the proof shows that S and the V a satisfy the relations in C * (v, s), and hence the universal property of C * (v, s) gives a homomorphism of C * (s, v) into T (N × ⋉ N) which is an inverse for π L .
Relations (T0)-(T4) imply that . We write q mult , q add and q CL for the quotient maps of T (N × ⋉ N) onto ∂ mult T (N × ⋉N), ∂ add T (N × ⋉N) and ∂T (N × ⋉N), respectively. It is possible to verify directly that this is indeed the boundary quotient from [11], but we compute it explicitly below in Proposition 3.6.
Remark 3.5. Let (G, P) be a quasi-lattice ordered group. In the literature, for example in [11] and [7], the additive, multiplicative and boundary quotients of T (P) are defined using the canonical isomorphism of T (P) onto a crossed product C(X) ⋊ G of a partial action of G on a compact space X. The quotients correspond to distinguished closed and invariant subsets of X. Proposition 3.6. The Crisp-Laca boundary quotient ∂T (N × ⋉ N) is isomorphic to the group C * -algebra C * (Q × + ⋉ Q). Proof. We will show that ∂T (N × ⋉N) has the universal property of the group C * -algebra C * (Q × + ⋉Q). .
We think that mental checks suffice for (T1)-(T3) and (T5), but (T4) is more interesting: 0 if x < a, and e a −1 (x−a) = e a −1 x−1 if x ≥ a because a −1 x < 1 ⇐⇒ x < a. Thus we have a representation π mult of ∂ mult T (N × ⋉ N) on ℓ 2 (Q + ). Proposition 3.8. Resume the notation of Proposition 3.3. Then the images of S and {V a : a ∈ N × } in ∂ add T (N × ⋉ N) are isometries satisfying: The C * -algebra ∂ add T (N × ⋉ N) is universal for C * -algebras generated by isometries s and {v a : a ∈ N × } satisfying the (lower-case analogues of) these relations.
Proof. The relation (T0) in Proposition 3.3 just says that the elements are isometries, and (A1) follows from (T1). The relations (T2) and (T3) say precisely that V is a Nica-covariant representation, and hence give (A2). The relation (A3) holds because we modded out the ideal generated by To see that ∂ add T (N × ⋉ N) is universal for these relations, we suppose we have R, W a satisfying them. Since (A3) implies that R is unitary, we have and hence the relation (T4) is also satisfied in C * (R, W). Thus Proposition 3.3 gives a homomorphism of T (N × ⋉ N) into C * (R, W). Since R is unitary, this homomorphism factors through a homomorphism of ∂ add T (N × ⋉ N). So (A1) is satisfied. The usual argument shows that V is Nica covariant, and S is unitary because we allow m to be a negative integer.

BACKWARDS CROSSED PRODUCTS
Our choice of the conventions N × ⋉ N and Q × + ⋉ Q for semidirect products, which we use because it allows us to apply results about the usual left-invariant partial order on (G, P), leads us to adopt slightly unorthodox conventions for crossed products of C * -algebras. Suppose that G is a group (in this paper, always discrete) and α : G → Aut A is an action of G on a C * -algebra A. The crossed product is then the C * -algebra generated by a universal covariant representation (i A , i G ) of the system (A, G, α). As usual, there is up to isomorphism exactly one such crossed product. Here we denote it by G ⋉ α A, suppress the representation i A of A, write u = i G , and view the crossed product as G ⋉ α A := span{u g a : g ∈ G, a ∈ A}.
We now suppose that we have a right action (q, n) → n · q of one group Q by automorphisms of another group N, and Q ⋉ N is the semidirect product with (p, m)(q, n) := (pq, (m · q)n). The unitary representations of Q ⋉ N are given by pairs of representations U : Q → U(H) and V : N → U(H) satisfying U m V p = V p U m·p . The following lemma for iterated crossed products is well-known, for example as Proposition 3.11 in [33], but we need to know the formula for the induced action with our conventions. Lemma 4.1. Suppose that α : Q ⋉ N → Aut A is an action of the semidirect product Q ⋉ N by automorphisms of a C * -algebra A. Then there is an action β of Q on the crossed product N ⋉ α| N A such that β p (u n a) = u n·p −1 α (p,e N ) (a). We write u, v and w for the unitary representations of N in Proof. For fixed p ∈ Q, we define π : A → N ⋉ α| N A by π(a) = i A (α (p,e) (a)) and U : N → U(N ⋉ α| N A) by U n = u n·p −1 = i N (n · p −1 ). Then for all n ∈ N, a ∈ A we have π(α (e,n) (a)) = i A (α (p,e) (α (e,n) (a))) = i(α (p,n) (a)) = i A (α (e,n·p −1 ) (α (p,e) (a))) = u n·p −1 i A (α (p,e) (a))u * n·p −1 = U n π(a)U * n , and the universal property of the crossed product gives a homomorphism β p := π ⋉ U from N ⋉ α| N A to itself. A straightforward calculation shows that β p • β q = β pq , which implies that each β p is an automorphism, and that β is an action of Q on N ⋉ A, as claimed.
The map U : n → w (e,n) is a unitary representation of N in (Q ⋉ N) ⋉ α A, and the pair (i A , U) is a covariant representation of (A, N, α| N ). Thus there is a homomorphism i N⋉A : , and this gives a homomorphism back from (Q ⋉ N) ⋉ α A which is an inverse for π. In this section we show that ∂ mult T (N × ⋉ N) and ∂T (N ⋊ N × ) are crossed products of an action of the group Q × + by automorphisms of the Toeplitz algebra T (Q + ) and the group algebra C * (Q), respectively.

THE MULTIPLICATIVE
We start by considering the Toeplitz algebra T (Q + ). Since the group (Q, Q + ) is totally ordered, every isometric representation of Q + is Nica covariant, and it follows from [21, Theorem 3.7] (for example) that T (Q + ) is universal for isometric representations of Q + . Indeed we have the following slightly stronger result: The statement is a particular case of [15, Theorem 1].

Proposition 5.2. There is an isometric representation
We first check that W r is well-defined by (5.1): is an isometry, and Proof. Since S and V a are isometries, we have , it suffices to assume that (c, d) = (ak, bk) (otherwise swap the two elements). Then we compute using the relation V k S k = SV k : Proof of Proposition 5.2. Lemma 5.3 implies that that there is a well-defined function W : The representation π mult of Example 3.7 has π mult (q mult (S b S * b ))e x = 0 for x < b. Therefore q mult (S b S * b ) = 1. Now (5.4) implies that W r W * r = 1. In particular this implies that W is a semigroup of nonunitary isometries, and hence by Lemma 5.1 the induced homomorphism π W is injective.
Proof. The relation (T5) in the multiplicative boundary quotient implies that a → q mult (V * a ) is a homomorphism of the semigroup N × into the unitary group U ∂ mult T (N × ⋉ N) . Thus it extends uniquely to a unitary representation U of the enveloping group Q × + . Let a, b ∈ N × . The adjoint of relation (T4) implies that Since the maps a → U a π W (T b )U * a and a → π W (α a (T b )) agree on N × and they are both homomorphisms, they also agree on Q × Thus (U, π W ) is a covariant representation.
Since the range of U ⋉ π W is a C * -algebra containing all the generators q mult (S) and q mult (V a ), U ⋉ π W is surjective. We know from Proposition 5.2 that π W is injective. It follows from the presentation (T1)-(T5) that there is a continuous action β : Thus it follows from the dual-invariant uniqueness theorem 1 that U ⋉ π W is faithful.
We can use the isomorphism of Proposition 5.4 to study the structure of the multiplicative boundary quotient by analysing the crossed product Q × + ⋉ α T (Q + ), and we will do this in the next section. However, we can already get quite a bit more information using Douglas' work on the Toeplitz algebras of dense subgroups of R [15]. The canonical representation u of Q + in the group algebra C * (Q) induces a homomorphism π u : T (Q + ) → C * (Q). Douglas proved that the kernel of π u is the commutator ideal C(Q + ) of T (Q + ), and that the ideal C(Q + ) is simple (see the Corollary on [15, page 147]). Thus there is an exact sequence For a ∈ Q × + , the automorphisms α a map commutators to commutators, and so the ideal C(Q + ) is invariant for the action α defined at (5.5). Thus it follows from [33,Proposition 3.19], for example, that there is an exact sequence ) is the C * -algebra of a transformation-group, we can compute its ideal structure by studying the action of Q × + on Q, and we do this in the next section.
We finish this section by identifying the image of the ideal Q × + ⋉ α C(Q + ) in the multiplicative boundary quotient and the image of the quotient by Q × + ⋉ α C(Q + ) as the boundary quotient.
Proof. By Proposition 5.4, there is a homomorphism U : It is straightforward to check that (T5) and (T6) imply that each W r is unitary: for example, if To check that W is a representation, fix r, s ∈ Q. If either both r, s ≥ 0 or both r, s ≤ 0, then it is straightforward to check that W r+s = W r W s . So take r = a −1 b > 0, s = c −1 d < 0 where a, b, c ∈ N × , and d ∈ −N × . Write a = a ′ gcd(a, c) and c = c ′ gcd(a, c) where a ′ , c ′ ∈ N × , and calculate as at (5.3) to get that and distinguish two cases. First, suppose that r + s ≥ 0. Then bc ′ + a ′ d ≥ 0 and using q CL (SS * ) = 1 we get that Second, suppose that r + s < 0. Then bc ′ + a ′ d < 0 and again To see ( U, π W ) is covariant for (Q × + , C * (Q), α), let a ∈ Q × + and r ∈ Q. We have π W (α a (u r )) = π W (u ar ) = W ar . Using the covariance of (U, π W ) we have This gives (3).
For (4) we start by proving that the right-hand square of the diagram (5.7) commutes. It suffices to check this for the generators of T (Q + ) and Q × + . Let r ∈ Q + . Then the upper route is T r → u r → W r and the lower route is T r → W r → W r . Next let a ∈ Q × + . Then the upper route is a → a → U a and the lower route is a → U a → U a . It follows that the right-hand square commutes.
Next we show that U⋉π W is an isomorphism. To do this, we view Q × be the representation such that π L = π L • q CL . We claim that π L is an inverse for U ⋉ π W . We check on generators. For (a, m) ∈ N × ⋉ N we have and the other inclusion follows from a similar argument using the inverses of U ⋉ π W and U ⋉ W. Thus U ⋉ π W Q × + ⋉ α C(Q + ) = ker q CL , and it is now clear the left-hand square commutes as well. 6. THE STRUCTURE OF THE CROSSED PRODUCT Q × + ⋉ α T (Q + ). In this section we study the structure of the crossed products in the short exact sequence The main theorem of this section, Theorem 6.1, describes the primitive-ideal space of Q × + ⋉ α T (Q + ). At the end of the section we pull this result across to ∂ mult T (N × ⋉ N) using the isomorphism of Proposition 5.4. To study T (Q + ) we realise it as a corner in a crossed product, which goes back at least as far as Murphy [27], and was used extensively by Phillips and Raeburn in [29], which will be our main reference. We define Then B Q (−∞, ∞] is a C * -subalgebra of ℓ ∞ (R), and consists of the functions f : R → C that are continuous at all x ∈ R \ Q, are right-continuous at r ∈ Q, have limits as x → r − and x → r + for r ∈ Q (not necessarily equal), and have limits as x → ∞ (see [29,Proposition 3.1]). We consider also the C * -subalgebra The map ǫ ∞ : f → lim x→∞ f(x) is a homomorphism, and there is an exact sequence is invariant for the action τ, and hence [33,Proposition 3.19] gives an exact sequence With P := i B Q (1 [0,∞) ), there is an isomorphism π of T (Q + ) onto the corner P Q ⋉ τ B Q (−∞, ∞] P such that π(T r ) = π Q (r)P for r ∈ Q + . (This follows, for example, by applying [29,Proposition 3.2] to the trivial cocycle σ ≡ 1.) This isomorphism carries the commutator ideal C(Q + ) onto the corner P Q ⋉ τ B Q (R) P.
We now want to study the action of Q × + on the corner that corresponds to the action α on the Toeplitz algebra. Our key observation is that there is a related action of the semidirect product . Observe that we have γ 0,r = τ r . Then Lemma 4.1 gives: There is an action such that β a (u r f) = u ra −1 γ a,0 (f). We write u, v and w for the canonical unitary representations of Q + , Q × + and Q × + ⋉ Q in Q × + ⋉ Q in the respective crossed crossed products by τ, β and γ. Then there is an isomorphism 2 such that (in the notation of §4) π(v a u r f) = w a,r f.
The action β of Q × + fixes the projection P, and hence restricts to an action on P Q⋉ τ B Q (−∞, ∞] P. When we view this corner as the Toeplitz algebra T (Q + ), the generating isometries are T r = Pu r P. Thus β is characterised by the formula β a (T r ) = T a −1 r , and is not quite the action α which we have been studying. But it is very closely related, and the crossed product is the same. More precisely, we have: ) is a crossed product for (A, G, β), and G ⋉ β A is canonically isomorphic to G ⋉ α A.
Proof. Since G is abelian, the map inv : g → g −1 is an isomorphism of G onto G. Now a calculation shows that if π : A → B(H) and U : G → U(H), then (π, U) is a covariant representation of (A, G, α) if and only if (π, U • inv) is a covariant representation of (A, G, β). So (G ⋉ α A, i A , i G • inv) is a crossed product for (A, G, β).
The isomorphism of T (Q + ) onto the corner in Q ⋉ τ B Q (−∞, ∞] restricts to an isomorphism of the commutator ideal C(Q + ) onto a corner in Q ⋉ τ B Q (R), and we can realise Q × + ⋉ β C(Q + ) as a corner in Q × + ⋉ β Q ⋉ τ B Q (R) . The automorphisms γ a,r : (a, r) ∈ Q × + ⋉ Q satisfy ǫ ∞ (γ a,r (f)) = ǫ ∞ (f), and hence induce automorphisms of the ideal B Q (R) = ker ǫ ∞ . The following technical result may be of some independent interest -certainly, we were surprised to discover it. Proposition 6.4. The C * -algebra (Q × + ⋉ Q) ⋉ γ B Q (R) is simple. We will prove this using a result of Archbold and Spielberg [3]. To apply their result, we have to prove that the action of Q × + ⋉ Q on the spectrum of B Q (R) is topologically free and minimal. We will use the following criterion for topological freeness. Lemma 6.5. Suppose that G acts on a locally compact space X, and that x ∈ X : S x = {e} is dense in X. Then the action on X is topologically free in the sense of [3, Definition 1].
Proof. Suppose that g 1 , · · · , g n ∈ G \ {e}. We want to show that that is dense in X. Suppose that U is an open subset of X. Then there exists y ∈ U such that S y = {e}. Then g i · y = y for 1 ≤ i ≤ n, and hence y ∈ U belongs to the intersection (6.1). 2 It is possibly important that the isomorphism U ⋉ π W of Proposition 5.4 does not carry the action α into β. The unitary representation U in that Proposition carries a into q mult (V * a ). The action β comes from the action γ of Q × + rather than α.
The spectrum of B Q (R) is described in Lemma 3.6 of [29]. As a set . Lemma 3.6 of [29] also describes the topology of B Q (R) ∧ in terms of convergence of sequences. Lemma 6.6. Let γ : . For x ∈ R and λ ∈ Q, the induced action of Q × + ⋉ Q + on B Q (R) ∧ is given by

This action is topologically free and minimal.
Proof. Let x ∈ R, λ ∈ Q, (a, r) ∈ Q × + ⋉ Q + and f ∈ B Q (R). Then and, similarly, This proves that the induced action is given by (6.2). We now verify that the action is topologically free. Let x ∈ R. Then (a, r) · ǫ x = ǫ x ⇐⇒ r = (a − 1)x.
The action is minimal when B Q (R) has no nontrivial (Q × + ⋉ Q + )-invariant ideals. Thus, since B Q (R) is commutative, the action is minimal if and only if the only non-empty closed (Q × + ⋉ Q + )invariant subset is B Q (R) ∧ . Thus it suffices to show that every orbit closure is B Q (R) ∧ . Let φ ∈ B Q (R) ∧ . First suppose that φ = ǫ x for some x ∈ R. Let y ∈ R. Choose sequences {s i }, {r i } ⊂ Q such that x + s i → y + and x + r i → y − . Then by Lemma 3.6 of [29], Let y ∈ R. Choose {r i } ⊂ Q such that r i → y + . For each r i and j ≥ 1, choose z i,j > y such that r i − 1 j < z i,j < r i . Then z i,j → r − i as j → ∞. For f ∈ B Q (R) the right continuity of f gives using the right continuity of f. Thus again we have (Q × + ⋉ Q + ) · φ = B Q (R) ∧ , and the action is minimal.
Proof of Proposition 6.4. Lemma 6.6 says that the action of Q × + ⋉ Q on B Q (R) ∧ satisfies the first two hypotheses of the Corollary to Theorem 2 in [3]. The other hypothesis ("regularity") is automatic because the group Q × + ⋉ Q is amenable (see the comment before the Corollary). Thus the result follows from that Corollary.
, and since the latter is isomorphic to a corner in the simple algebra we can now use Proposition 6.4 to prove following corollary.
Thus β a is the automorphism of the Toeplitz algebra T (Q + ) = C * (T p ) such that β a (T p ) = T a −1 p . We know from Corollary 6.4 that (Q × + ⋉ Q) ⋉ γ B Q (R) is simple, and hence so is Q × . Thus so is the corner associated to the projection 1 [0,∞] , which is Q × + ⋉ β C(Q + ). Since the group Q × + is abelian, the crossed product by the action α : a → β −1 a is isomorphic to We now analyse the structure of the crossed product Q × + ⋉ α C * (Q) by the action induced by the action α : Q × + → Aut T (Q + ). We use the Fourier transform F to identify C * (Q) with C( Q): by convention, we use the transform such that F(u r )(γ) = γ(r), and then the action α is given by α a (γ)(r) = γ(ar) = (a · γ)(r).
Next we realise Q as a solenoid. Let m, n ∈ N × . Then m | n implies that n = mc for some c ∈ N × , and m −1 Z ⊂ n −1 Z with inclusion map m −1 k → n −1 ck. We then view Q as the direct limit The dual group of each n −1 Z is T, and hence we can view Q as the inverse limit For z = (z n ) ∈ S we have the character γ z in Q with formula γ z (n −1 k) = z k n . Let a, b ∈ N × . The action of Q × + is given by ab −1 · γ z (n −1 k) = γ n (ab −1 n −1 k) = γ z ((nb) −1 ak) = z ak nb = γ z a ((nb) −1 k). Thus (ab −1 · z) is the sequence z ab −1 in S with (z ab −1 ) n = z a nb . The constant sequence 1 is fixed under the action of Q × + . Hence the evaluation map ǫ 1 : f → f(1) gives an exact sequence which is preserved by the action of Q × + . (We observe that the homomorphism ǫ 1 is essentially the homomorphism φ : C * (Q) → C such that φ(u r ) = 1 for all r, in the sense that φ = ǫ 1 • F. This homomorphism is used in Proposition 6.9 below, and satisfies φ • π u = ψ.) Thus it follows from [33,Proposition 3.19] that there is an exact sequence Lemma 6.8. The action of Q × + on S \ {1} is free and minimal. Proof. For freeness, we take z = (z n ) ∈ S. We first suppose that a · z = z and aim to show that z = 1. We have z a n = z n for all n, and hence z a−1 n = 1 for all n. Then 1 = z a−1 n(a−1) = z n for all n, and z = 1, as we wanted.
Next we take a, b ∈ N × such that a > b (the case a < b is similar). Then (ab −1 ) · z = z =⇒ z a bn = z n = z b bn for all n (6.4) =⇒ z a−b bn = 1 for all n =⇒ z bn = 1 for all n (because a − b ∈ N × ) =⇒ z n = z b bn = 1 for all n. Thus the action is free on S \ {1}, as claimed.
To establish minimality, we take 1 = w ∈ S and aim to prove that the orbit of w is dense in S. To see this, we take z ∈ S. Since the inverse limit is topologised as a subset of n∈N × T, and a basic open neighbourhood U of z has the form n∈F U n × n / ∈F T for some finite subset F of N × . We write N := lcm{n : n ∈ F}. Since each map z N → z n −1 N N is continuous for each n ∈ F, we can find a neighbourhood V of z N such that w N ∈ V implies w n ∈ U n for all n ∈ F. Then we replace U by V × n =N T . Now we need to find a, b ∈ N × such that ((ab −1 ) · w) N ∈ V. Since w = 1 there exists b such that w b = 1, and then w bn = 1 for all n ∈ N × . First, we suppose that there exists w bN = 1 that is not a root of unity. Then {w p b : p ∈ N} is dense in T, and we can choose a ∈ N × such that w a bN ∈ N. Then ((ab −1 ) · w) N = w a bN ∈ V, as required. The alternative is that all w cN = 1 are roots of unity. For u ∈ T, we write o(u) = n if and only if u = e 2πik/n for some k co-prime to n. We claim that {o(w cN ) : c ∈ N × } is unbounded. To see this, suppose not. Then there exists a ∈ N × such that w a cN = 1 for all c. Then we have (aN −1 ) · w = 1 for all c ∈ N × , which is impossible for w ∈ S \ {1} because the implication (6.4) implies that w = 1.
The map γ → ǫ γ is a homeomorphism of Q × + onto Prim C * (Q × + ) (strictly speaking, the homeomorphism is γ → ǫ γ • F). Thus exactness of (6.3) implies that is a homeomorphism of Q × + onto the closed subset P ∈ Prim Q × + ⋉ α C * (Q) : ker(id ⋉ φ) ⊂ P (see [31,Proposition A.27], for example). The complement of this set in Prim Q ×  [31,Definition A.19]), it is all of Prim Q × + ⋉ α C * (Q) . Proof of Theorem 6.1. The homomorphism ψ : T (Q + ) → C is equivariant for the action α of Q × + on T (Q + ) and the trivial action of Q × + on C. Thus there is a homomorphism id ⋉ψ, as asserted. The exactness of the sequence (5.6) implies that P → (id ⋉π u ) −1 (P) is a homeomorphism of Prim Q × + ⋉ α C * (Q) onto a closed subset of Prim Q × + ⋉ α T (Q + ) . Since I := ker(id ⋉π u ) = Q × + ⋉ α C(Q + ) is simple by Corollary 6.7, the only ideal in Q × + ⋉ α T (Q + ) that does not factor through id ⋉ψ is O = {0}. It is the intersection of the primitive ideals containing it, and hence must itself be primitive. The set {O} is open because the complement is closed, and dense because every primitive ideal contains O. The remaining assertions follow from Proposition 6.9.
It is now a relatively straightforward matter to use the isomorphism of Proposition 5.4 to convert Theorem 6.1 into a parallel result about ∂ mult T (N × ⋉ N). Proof. By Proposition 5.5, the isomorphism U ⋉ π W : Since each u a is unitary, t satisfies t a t * a = 1, and hence there is a homomorphism π := π t on ∂ mult T (N × ⋉ N), as claimed. The homomorphism id ⋉ψ : . Thus ker π = ker(id ⋉ψ). Now the theorem follows from the parallel statements in Theorem 6.1.

KMS STATES
We consider the dynamics σ : R → Aut T (N × ⋉ N) such that σ t (S) = S and σ t (V a ) = a it V a for a ∈ N × . We want to study the KMS states of the system (T (N × ⋉ N), σ). We begin in this section by proving some results which are valid for all β. Proof. Suppose that φ is a KMS β state of (T (N × ⋉N), σ)). Since the generator S is fixed by the action α, we have 1 = φ(S * S) = φ(SS * ). Thus φ vanishes on the element 1 − SS * . Since α t (1 − SS * ) = 1 − SS * , we deduce from [2, Lemma 6.2], for example, that φ vanishes on the ideal generated by 1 − SS * , and hence factors through a state of the quotient ∂ add T (N × ⋉ N).
The next result is an analogue of [23,Lemma 8.3]. As there, we write For our system (T (N × ⋉ N), σ), the spanning elements V a S m (V b S n ) * satisfy and hence are analytic.
The following result is essentially proved in the proof of [23,Lemma 8.3] (look in the second last paragraph of that proof). gcd(a, d).
Proof of Proposition 7.2. Suppose that π is a KMS β state of (T (N × ⋉ N), α) and V a S m (V b S n ) * is a fixed spanning element. Then the KMS condition gives Since β > 0 and V * a V a = 1, we have Together, (7.2) and (7.3) imply (7.1). Now we suppose that φ satisfies (7.1), and aim to prove that φ is a KMS β state. Since the elements V a S m (V b S n ) * are analytic and span a dense subalgera, it suffices to prove that gcd(a, d) as in Lemma 7.3 above. On one hand, we have Thus (7.1) implies that On the other hand, we have Next, we observe that so the Kronecker delta functions in (7.4) and (7.5) are the same. Thus φ(xy) = 0 = φ(xy) when ca ′ = bd ′ . When ca ′ = bd ′ , we have By Lemma 7.3 we have a ′ = b ′ . Therefore the coefficient on the right-hand side is which is the same as the coefficient in the formula (7.4). Finally we have (using again that a ′ = b ′ and c ′ = d ′ ) We deduce that when as required.

KMS STATES FOR LARGE INVERSE TEMPERATURES
We consider now KMS β states of (T (N × ⋉ N), σ) for β > 1. We begin by stating our main result. (1) For each state ϕ on the Toeplitz algebra T = C * (S) there is a unique ground state ω ϕ of T (N × ⋉ N) such that (8.1) ω ϕ (V a S m S * n V * b ) = δ a,b δ a,1 ϕ(S m S * n ) for a, b ∈ N × and m, n ∈ N. The map ϕ → ω ϕ is an affine w*-homeomorphism of the set of states of T onto the ground states of (T (N × ⋉ N), σ).
(2) For each probability measure µ on the circle and each β > 1 there is a unique KMS β state of T (N × ⋉ N) such that The map µ → ψ µ,β is an affine w*-homeomorphism of the simplex of probability measures on T onto the simplex of KMS β states of (T (N × ⋉ N), σ).
Part (2) of this theorem is very similar to part (3) of [23,Theorem 7.1], which is about KMS β states of T (N ⋊ N × ) for β ∈ (2, ∞]. Our proof relies on general results from [20] about KMS states on semigroup crossed products. So we begin by writing T (N × ⋉ N) as a semigroup crossed product. The next proposition describes the underlying algebra C of this crossed product, and the following one describes the crossed-product decomposition.

Proposition 8.2.
There is an action θ of Q * + by automorphisms of T (N × ⋉ N) such that θ χ (S) = S and θ χ (V a ) = χ(a)V a for χ ∈ Q * + and a ∈ N × . The fixed-point algebra of this action is (8.3) C := T (N × ⋉ N) θ = span{V a S m S * n V * a : a ∈ N × , m, n ∈ N}, and the projection V a V * a is central in C for each a ∈ N × . Proof. If s and {v a : a ∈ N × } are universal generators for T (N × ⋉ N), then so are s and {χ(a)v a : a ∈ N × } for each χ ∈ Q * + , and hence the universal property gives an automorphism θ χ . The universal property also implies that χ → θ χ is a group homomorphism, and because the automorphisms are all isometric, it suffices to check continuity on the spanning family (3.2), and this is straightforward. The fixed-point algebra of θ is the range of the associated conditional expectation E θ on T (N × ⋉ N), and hence (8.3) follows from (3.2). Now we suppose that a ∈ N × , and aim to prove that V a V * a is central in T (N × ⋉ N) θ . Let b ∈ N × and m, n ∈ N. We write a = a ′ gcd(a, b), b = b ′ gcd(a, b) with gcd(a ′ , b ′ ) = 1 and ab ′ = ba ′ = lcm(a, b). Using first (T3) and then the adjoints of (T1) and (T4), we find that . Similarly, using (T1) and (T4) directly gives a is central as claimed. The next proposition describes T (N × ⋉ N) as a semigroup crossed product to which Theorem 12 of [20] applies. Proposition 8.3. Let C be the subalgebra of T (N × ⋉N) defined in (8.3). For a ∈ N × , we define α a : C → C by α a (X) = V a XV * a . (1) Then {α a : a ∈ N × } is a semigroup of endomorphisms of C, we have α a (C) = V a V * a CV a V * a , and α a has a left inverse given by θ(Y) = V * a YV a . (2) There is a canonical isomorphism which matches up the two copies of C, and for a ∈ N × takes V a ∈ T (N × ⋉ N) to the generating isometry w a ∈ C ⋊ α N × . Proof. For a, b ∈ N × and m, n ∈ Z we have Thus α a maps C to C and α a (C) ⊂ V a V * a CV a V * a . It follows from (8.4) that α a (C) = V a V * a CV a V * a . Since V is a semigroup of isometries, {α a } is a semigroup of endomorphisms of C with left inverses as described.
To prove the existence of the isomorphism (8.5), we observe that the inclusion ι : C ⊂ T (N × ⋉ N) and the semigroup of isometries {V a : a ∈ N × } form a covariant pair for the semigroup dynamical system (C, N × , α). Thus we have a homomorphism ι ⋊ V : C ⋊ α N × → T (N × ⋉ N). This proves in particular that the image of C in C ⋊ α N × is faithful and that w a maps to V a .
To obtain the inverse of ι ⋊ V, we show that the relations (T0)-(T4) hold for the isometry S ∈ C and the canonical semigroup of isometries w a viewed as elements of C ⋊ α N × . Relations (T0) and (T2) reflect that S and w are isometric representations. For (T3), let a, b ∈ N × . First notice that covariance implies that w a w * a = α a (1) = V a V * a . Thus (a,b) . Now suppose that gcd(a, b) = 1, so that lcm(a, b) = ab. Then To prove that (T1) holds in the crossed product, we recall that V * a SV a = S a in the C*-algebra C because (T4) holds in T (N × ⋉ N). Thus w * a Sw a = w * a (w a w * a Sw a w * a )w a = w * a (V a V * a SV a V * a )w a = w * a V a S a V * a w a = w * a α a (S a )w a = w * a w a S a w * a w a = S a . Now Sw a = Sw a w * a w a = Sα a (1)w a = S(V a V * a )w a = V a S a V * a w a using (T1) for (S, V) = V a V * a Sw a using (T4) for (S, V) = α a (1)Sw a = w a w * a Sw a = w a S a , giving (T1). A similar argument with S * in place of S shows that (T4) holds in the crossed product. Using the universal property of T (N × ⋉ N) from Proposition 3.3 shows that the assignment S → S and V a → w a gives an inverse for ι ⋊ V.
For α to respect the lattice structure in the sense of [20, Definition 3] requires that α a (C) is an ideal in C and that α a (1)α b (1) = α a∨b (1) for a, b ∈ N × . The first follows because α a (C) = V a V * a CV a V * a and V a V * a is central in C. The second follows because and because a ∨ b = lcm(a, b) in the lattice-ordered semigroup N × .
Proof of Theorem 8.1. For each p in the set P of primes, denote by N × p the set of numbers with all prime factors less than p. We now define C p := span{V a S m S * n V * a : a ∈ N × p , m, n ∈ N}. A computation like that in (8.4) gives , and since the prime factors of ab ′ = ba ′ = lcm(a, b) appear in a or b, we deduce from (8.6) that C p is a C*-subalgebra of C. Since N × q ⊂ N × p for q ≤ p, we have C q ⊂ C p for q ≤ p, and p∈P C p = span{V a S m S * n V * a : a ∈ N × , and m, n ∈ N is dense in C. For p ∈ P, we consider the projection since Proposition 8.2 says that each V q V * q is central in C, the projection Q p is also central in C p . Let a ∈ N × p \ {1}. Then a has at least one prime factor q ≤ p. Since V is Nica-covariant for N × , we have The relations (T1) and (T4) imply that SV q V * q = V q V * q S. Thus Q p SQ p = Q p S = SQ p . We claim that the map S → Q p SQ p extends to an isomorphism π QpS of T = C * (S) onto Q p C p Q p . To see this, we observe that and hence Q p S = Q p SQ p is an isometry in the corner Q p C p Q p . On the other hand, since Q p ε 1,0 = ε 1,0 and (Q p S) * ε 1,0 = 0, Q p S is a proper isometry in C p . Thus Coburn's theorem gives the required isomorphism π QpS . In the previous paragraph we saw that Q p S generates Q p C p Q p , and hence the range of π QpS is all of Q p C p Q p .
Let ϕ be a state of T . For each prime p, we get a state ϕ • π −1 QpS to a positive functional ω p on C p by setting ω p (c) = ϕ • π −1 QpS (Q p cQ p ) for c ∈ C p . Since ω p (1) = 1, ω p is a state (by [4, II.6.2.5], for example). Then for a ∈ N × p we have QpS (Q p S m S * n Q p ) if a = 1 = δ a,1 ϕ(S m S * n ).
For q ≤ p, we have N × q ⊂ N × p , and hence (8.7) implies that for a ∈ N × q we have ω p (V a S m S * n V * a ) = δ a,1 (ϕ(S m S * n ) = ω q (V a S m S * n V * a ). Thus ω p | Cq = ω q , and we can define ω ∞ : p C p → C by ω ∞ (c) = ω p (c) for c ∈ C p . Since each ω p is a linear functional with norm 1, ω ∞ also has norm 1, and extends to a linear functional on the closure C; since each ω p is positive, so is ω ∞ . Since 1 ∈ C and ω p (1) = for all p, we have ω ∞ (1) = 1. Thus ω ∞ is a state.
To complete the proof of part (1), it remains to show that ω ϕ is a ground state for every ϕ and that every ground state arises this way. From Proposition 8.3 we know that the semigroup {α a } respects the lattice structure, and that the dynamics on T (N × ⋉ N) ∼ = C ⋊ α N × has the form σ N of [20,Theorem 12] for the function N : N × → (0, ∞) defined by N(a) = a for a ∈ N × . Thus the hypothesis of [20,Theorem 12] are satisfied, and part (2) of that theorem says that a state ω of (T (N × ⋉ N), σ) is a ground state if and only if it factors through the conditional expectation E θ and its restriction to the fixed point algebra C vanishes on all the corners α a (C) for a = 1, or equivalently if and only if ω(V a S m S * n V * b ) = δ a,b δ a,1 ω(S m S * n ) for all a, b ∈ N × and m, n ∈ N, which is (8.1). Thus ω is a ground state if and only if it is equal to ω ϕ for ϕ := ω| T . We have now proved part (1).
For part (2), fix β > 1. We seek to apply [20,Theorem 20] to T (N × ⋉ N) ∼ = C ⋊ α N × with function N(a) = a. The critical inverse temperature is 1. As we noted above, the hypotheses of [20,Theorem 12] are satisfied, and these are also hypotheses of [20,Theorem 20]. We also need to know that the isometries {V a : a ∈ N × } generate a faithful copy of T (N × ) in T (N × ⋉ N) and that their ranges V a V * a = α a (1) generate a faithful copy of the diagonal subalgebra B N × ; this follows from [21,Theorem 3.7]. Thus [20,Theorem 20] gives an affine weak* homeomorphism T β from the space of tracial ground states of C ⋊ α N × ∼ = T (N × ⋉ N) onto the KMS β states of C ⋊ α N × . The normalising factor ζ N (β) in that theorem is here the Riemann zeta function. Thus the formula for T β is where γ is the left inverse of α.
By part (1), the tracial ground states of T (N × ⋉ N) are parametrised by traces on the Toeplitz algebra T = C * (S). If φ is a trace on T , then φ factors through a state of of the quotient C(T) by the relation 1 − SS * = 0, and hence φ is given by a probability measure on the circle 3 .
It remains to establish (8.2). The formula for T β ω, when applied to the tracial ground state ω = ω τµ arising from a probability measure µ on T, becomes But ω τµ (V * d V a S m S * n V * b V d ) = δ a,b δ a ′ ,1 ω τµ (S m S * n ) by (8.1). So the only nonzero terms on the righthand side of (8.8) are the ones where a ′ = 1, equivalently the ones where d is a multiple of a. 3 To see this, note (following the argument of [17,Lemma 2.2]) that the tracial property gives φ(SS * ) = φ(S * S) = φ(1) = 1. Thus with p := 1 − SS * , we have φ(p) = 0. Now for any a ∈ T we have 0 ≤ φ(pa * ap) ≤ φ(p a 2 p) = a 2 φ(p) = 0, and since the positive elements span the ideal generated by p, it follows that φ vanishes on pT p. But now for any a, b ∈ T , we have φ(apb) = φ((ap)(pb)) = φ((pb)(ap)) = 0, and φ vanishes on the ideal generated by p.

Thus
Remark 8.4. This proof is quite different from the one for N ⋊ N × in [23], which uses a Hilbertspace representation. We can also realise the KMS β states of T (N × ⋉ N) spatially, and we outline the construction. For a probability measure µ on T, we consider the Hilbert space L 2 (T, µ), and define H µ := ℓ 2 (N × , L 2 (T, µ)). For f ∈ L 2 (T, µ) and c ∈ N × , we define fe d : c → δ c,d f.
gives a specific formula for a state, and we can deduce from Proposition 7.2 that it is a KMS state. The details are messy, but easier than the ones in [23, §9], and we wind up with the formula (8.2), and hence we are describing the same KMS states. However, we were not able to adapt the arguments of [23, §10] to prove that every KMS β state has the form ψ β,µ . So the extra information that we get from using the results in [20] is the surjectivity of the parametrisation in [20, Theorem 20].

KMS STATES WITH INVERSE TEMPERATURE 1
For each probability measure µ on T, compactness of the state space implies that there is a sequence β n → 1+ such that {ψ βn,µ } converges weak* to a state ψ 1,µ . It follows on general grounds [6, Proposition 5.3.23] that the limit ψ 1,µ is a KMS 1 state of (T (N × ⋉ N), σ) (or one can apply Proposition 7.2). We naturally ask whether it is the only KMS 1 state, as in [23] for the usual left Toeplitz algebra T (N ⋊ N × ).
We answer this by looking at some specific measures where we can compute the values of the state explicitly. We leave the detailed study of critical and subcritical equilibria to future work. So we have ψ β,µ V a S m S * n V * b = δ a,b δ m,n ζ(β) −1 a −β c∈N × c −β = δ a,b δ m,n a −β .
In the limit as β → 1 + , we find ψ 1,µ V a S m S * n V * b = δ a,b δ m,n a −1 . Thus we have Thus we have We now take a = b, so that the delta function disappears. If m − n is even, then as in the previous example, If m − n is odd, then we note that To sum up, we have

APPENDIX A. CONCLUDING REMARKS
A.1. Finitely many primes. We now consider a finite set E of prime numbers, and the set N E of integers n ∈ N whose prime factors all belong to E. Then N E is an (additive) subgroup of N, and the set N × E = N E \ {0} is a (multiplicative) subgroup of N × which is isomorphic via prime factorisation to N E ∼ = N |E| . Since the case d < ∞ is explicitly allowed in [20, §4], we can run the argument of §8 for the semidirect product N × E ⋉ N E . However, because there are now only finitely many primes in play, the Dirichlet function a∈N × E a −β converges for all β > 0, with sum Hence for all β > 0, there is a bijection µ → ψ µ,β of the set P(T) of probability measures onto the simplex of KMS β states of (T (N × E ⋉ N E ), σ) such that A.2. A crossed-product realisation of the boundary quotient. The commutation relations in Proposition 3.8 make the additive boundary quotient ∂ add T (N × ⋉ N) look a little like a crossed product of the singly generated C * -algebra C * (S) by an endomorphic action α of N × satisfying α a (S) = S a . However, it is not one of the familiar "semigroup crossed products" studied in [32], [21], [22], [16] or [18], for example; it is more like the "semicrossed products" introduced recently in [14] and [19]. Here we make this connection precise. We view ∂ add T (N × ⋉ N) as the universal C * -algebra generated by elements satisfying the relations of Proposition 3.8. As usual, we use lower case (s, v a ) for the generators of ∂ add T (N × ⋉ N) to emphasise that they have a universal property.
To write ∂ add T (N × ⋉ N) as a crossed product, we need to identify the coefficient algebra C * (s). Relation (A3) says that s is unitary, and implies that C * (s) is commutative and isomorphic to C(σ(s)). Since s is unitary, we have σ(s) ⊂ T, and we claim that it is all of T. To see this, we use the representation (S, V) on ℓ 2 (N × ⋊ Z) of Example 3.9. An elementary calculation shows that, for each z ∈ T, the point mass e 1,0 in ℓ 2 (N × ⋊ Z) is not in the range of S − z1, and hence z ∈ σ(S). Since σ(S) ⊂ σ(s) ⊂ T, we deduce that σ(s) is also all of T. Thus (C * (s), s) is canonically isomorphic to (C(T), ι), where ι is the function ι : z → z.
Since C(T) is the universal algebra generated by the unitary ι, and each ι a is unitary, there are endomorphisms {α a : a ∈ N × } of C(T) characterised by α a (ι) = ι a . It follows by approximation that α a is given on more general functions f ∈ C(T) by α a (f)(z) = f(z a ).
We now consider a Nica-Toeplitz-Pimsner algebra N T (C(T), N × , α) which is similar to the ones studied by Kakariadis in [19]. Our N T (C(T), N × , α) is universal for pairs consisting of a unitary operator S on a Hilbert space H and a Nica-covariant representation V of N × on H such that (A.2) π(f)V a = V a π(α a (f)) for f ∈ C(T) and a ∈ N × .
Remark A.1. Our N T (C(T), N × , α) is similar to, but different from, that of [19, Definition 2.1]. First, the unitary operator S ∈ U(H) gives a unital representation π S of the coefficient algebra A = C(T), and since we can recover S via the formula S := π S (ι), these unitaries S are in oneto-one correspondence with the representations π described in [19, Definition 2.1(i)]. Second, our semigroup N × is not finitely generated whereas the semigroup Z n + in [19] (which we would denote by N n ) is finitely generated. If we fix n primes Proof. Proposition 3.8 says that ∂ add and N T (C(T), N × , α) have equivalent presentations, and hence are canonically isomorphic algebras.
For the second assertion, we need to check the extra relations imposed in [19, §2.3]. View N × as N P . Since each z → z a is surjective, each endomorphism α p is injective, and the ideals I p in [19] are all A = C(T). So the extra relations are p∈S (1 − V p V * p ) = 0 for all finite subsets S of P.
These are equivalent to 1 − V p V * p = 0 for all p ∈ P, and hence say merely that each V a is unitary. Thus the isomorphism π S × V induces an isomorphism of the Crisp-Laca boundary quotient ∂TT (N × ⋉ N) onto the quotient N O(C(T), N × , α) of N T (C(T), N × , α).
Remark A.3. It follows from Proposition A.2 and Proposition 3.6 that the Cuntz-Nica-Pimsner algebra N O(C(T), N × , α) is isomorphic to the group algebra C * (Q × + ⋉ Q). Indeed, the algebra N O(C(T), N × , α) is generated by a unitary element S and a unitary representation U, and the proof of §5 shows how such S and U combine to give a unitary representation W of Q + (and hence also of Q) alongside the representation U of Q × + coming from V. The pair (U, W) now give a unitary representation of Q × + ⋉Q, and hence a homomorphism of C * (Q × + ⋉Q) into N O(C(T), N × , α) which is the required isomorphism.
A.3. Relations with work of Kakariadis on KMS states. In [19, §4], Kakariadis has computed the KMS states on crossed products of the the form N T (A, N n , α) for endomorphic actions α of the additive semigroup N n (denoted there by Z n + ) on unital C * -algebras A. Our results overlap with his, and especially in the case of finitely many primes, as discussed in §A.1, and A = C(T).
To see this, we suppose again that E is a finite set of primes, and use the notation of §A.1. Then the map M : N E → N × E defined by M(x) = p∈E p xp is an isomorphism. We define τ : N E → End C(T) in terms of the action α : N × E → End C(T) of Proposition A.2 by τ = α • M. We then define λ ∈ [0, ∞) E by λ p = ln p, and use it to define an action σ λ of R on N T (C(T), N E , τ), as in [19, §4.1]. (We have simplified his notation a little by noting that λ p β > 0 for all p, and hence there is no need to put hypotheses on the vector β.) The simplex of tracial states on C(T) is the simplex of P(T) of probability measures on T. Hence [19,Theorem 4.4] says that, for every β > 0, there is an affine homeomorphism µ → φ µ,β of P(T) onto the simplex of KMS β states of N T (C(T), N E , τ) such that φ µ,β (V x fV * y ) = δ x,y e − x,λ β for all x, y ∈ N E and f ∈ C(T).
To transfer this into our notation, we note that Similarly, we have e −βλp = p −β . Next we take f(z) = z m and note that τ w (f)(z) = α M(w) (f)(z) = f(z M(w) ) = z M(w)m .
Thus we have When we pull this back using the isomorphism M : N E → N × E to a state on N T (C(T), N × E , α) (effectively replacing x, y by a = M(x), b = M(y)), we recover the state ψ µ,β of §A.1.
A.4. LCM semigroups. Afsar, Brownlowe, Larsen and Stammeier have recently proved a very general result [1,Theorem 4.3] about the KMS states on the C * -algebras of right LCM semigroups which unifies the main results of [23], [7], [24], [25] and [9]. So it is natural to ask whether their result applies also to the semigroup N × ⋉ N. The answer is that it does not seem to. To see why, we need to view N × ⋉ N as a right LCM semigroup, and see that it is not "admissible" in the sense of [1, Definition 3.1].
The semigroup S := N × ⋉ N is certainly a right LCM semigroup, because it is the positive cone in the quasi-lattice ordered group (Q × + ⋉ Q, N × ⋉ N). Indeed, in the original development [8, §3-4] of the theory of right LCM semigroup algebras, the authors used the paper [21] on quasi-lattice ordered groups as a template. So we can try to apply the result in [1] to S = N × ⋉ N. (As a trivial point of notation, we continue to write s ∨ t < ∞ to mean sS ∩ tS = ∅.) Since S is lattice ordered, the "core subsemigroup" S c := s ∈ S : s ∨ t < ∞ for all t ∈ S is all of S. Thus there are no elements of S\S c , hence trivially no "core irreducible" elements. In the notation of [1], we have S 1 ci = {1} and S 1 ci S c = S, which is (A1) in [1, Definition 3.1]. The second condition (A2) holds vacuously. But (A3), which postulates the existence of a homomorphism N : S → N × with certain properties, seems to be a problem.
Suppose that N : N × ⋉ N → N × is a homomorphism. Then we take a ∈ N × with a > 1, and compute N(a, a) two ways. On the one hand, we have Thus for c ∈ N × belonging to the range of N, the set N −1 (c) contains all (a, m) with N(a, 0) = c, and is in particular infinite. But here, since S c = S, the equivalence relation ∼ used in [1] satisfies s ∼ t ⇐⇒ s ∨ t < ∞, and hence we have s ∼ t for all s, t in our lattice-ordered semigroup S. Thus |N −1 (c)/ ∼ | = 1, and N cannot satisfy condition (A3)(a) in [