Phase transitions in a number-theoretic dynamical system

Transcription

1 Phase transitions in a number-theoretic dynamical system Iain Raeburn University of Wollongong GPOTS June 2009 This talk is about joint work with Marcelo Laca.

2 In this talk, a dynamical system consists of an action α : R Aut A of the real line R on a unital C -algebra A.

3 In this talk, a dynamical system consists of an action α : R Aut A of the real line R on a unital C -algebra A. In physical models, observables of the system are represented by self-adjoint elements of A, and states of the system by positive functionals of norm 1 on A: φ(a) is the expected value of the observable a in the state φ (which is real because a = a and φ 0).

4 In this talk, a dynamical system consists of an action α : R Aut A of the real line R on a unital C -algebra A. In physical models, observables of the system are represented by self-adjoint elements of A, and states of the system by positive functionals of norm 1 on A: φ(a) is the expected value of the observable a in the state φ (which is real because a = a and φ 0). The action α represents the time evolution of the system: the observable a at time 0 moves to α t (a) at time t, or the state φ at time 0 moves to φ α t.

5 In this talk, a dynamical system consists of an action α : R Aut A of the real line R on a unital C -algebra A. In physical models, observables of the system are represented by self-adjoint elements of A, and states of the system by positive functionals of norm 1 on A: φ(a) is the expected value of the observable a in the state φ (which is real because a = a and φ 0). The action α represents the time evolution of the system: the observable a at time 0 moves to α t (a) at time t, or the state φ at time 0 moves to φ α t. In statistical physics, an important role is played by equilibrium states, which are in particular invariant under the time evolution.

6 In C -algebraic models (A, R, α), the equilibrium states are called KMS states.

7 In C -algebraic models (A, R, α), the equilibrium states are called KMS states. The set A a := {a A : t α t (a) extends to be analytic on C} of analytic elements is a dense subalgebra of A. A state φ on A is a KMS state at inverse temperature β if φ(ab) = φ(bα iβ (a)) for all a, b A a.

8 In C -algebraic models (A, R, α), the equilibrium states are called KMS states. The set A a := {a A : t α t (a) extends to be analytic on C} of analytic elements is a dense subalgebra of A. A state φ on A is a KMS state at inverse temperature β if φ(ab) = φ(bα iβ (a)) for all a, b A a. KMS states are α-invariant.

9 In C -algebraic models (A, R, α), the equilibrium states are called KMS states. The set A a := {a A : t α t (a) extends to be analytic on C} of analytic elements is a dense subalgebra of A. A state φ on A is a KMS state at inverse temperature β if φ(ab) = φ(bα iβ (a)) for all a, b A a. KMS states are α-invariant. It suffices to check the KMS β condition on a dense subspace of A a.

10 In C -algebraic models (A, R, α), the equilibrium states are called KMS states. The set A a := {a A : t α t (a) extends to be analytic on C} of analytic elements is a dense subalgebra of A. A state φ on A is a KMS state at inverse temperature β if φ(ab) = φ(bα iβ (a)) for all a, b A a. KMS states are α-invariant. It suffices to check the KMS β condition on a dense subspace of A a. The KMS β states always form a simplex, and the extremal KMS β states are factor states.

11 In C -algebraic models (A, R, α), the equilibrium states are called KMS states. The set A a := {a A : t α t (a) extends to be analytic on C} of analytic elements is a dense subalgebra of A. A state φ on A is a KMS state at inverse temperature β if φ(ab) = φ(bα iβ (a)) for all a, b A a. KMS states are α-invariant. It suffices to check the KMS β condition on a dense subspace of A a. The KMS β states always form a simplex, and the extremal KMS β states are factor states. In a physical model we expect KMS states for most β.

12 The Cuntz algebra O n is the universal algebra generated by {s j : 1 j n} satisfying sj s j = 1 = n j=1 s jsj. There is an action α : R Aut O n such that α t (s j ) = e it s j.

13 The Cuntz algebra O n is the universal algebra generated by {s j : 1 j n} satisfying sj s j = 1 = n j=1 s jsj. There is an action α : R Aut O n such that α t (s j ) = e it s j. The relation s j s j = 1 says that s j is an isometry, and then s j s j is the range projection.

14 The Cuntz algebra O n is the universal algebra generated by {s j : 1 j n} satisfying sj s j = 1 = n j=1 s jsj. There is an action α : R Aut O n such that α t (s j ) = e it s j. The relation sj s j = 1 says that s j is an isometry, and then s j sj is the range projection. Then 1 = n j=1 s jsj implies that these range projections are mutually orthogonal in particular, sj s k = 0 for j k.

15 The Cuntz algebra O n is the universal algebra generated by {s j : 1 j n} satisfying sj s j = 1 = n j=1 s jsj. There is an action α : R Aut O n such that α t (s j ) = e it s j. The relation sj s j = 1 says that s j is an isometry, and then s j sj is the range projection. Then 1 = n j=1 s jsj implies that these range projections are mutually orthogonal in particular, sj s k = 0 for j k. So every word in the s j and sk has the form and O n = span{s µ s ν}. s µ s ν := s µ1 s µ µ (s ν1 s ν ν ),

16 The Cuntz algebra O n is the universal algebra generated by {s j : 1 j n} satisfying sj s j = 1 = n j=1 s jsj. There is an action α : R Aut O n such that α t (s j ) = e it s j. The relation sj s j = 1 says that s j is an isometry, and then s j sj is the range projection. Then 1 = n j=1 s jsj implies that these range projections are mutually orthogonal in particular, sj s k = 0 for j k. So every word in the s j and sk has the form and O n = span{s µ s ν}. s µ s ν := s µ1 s µ µ (s ν1 s ν ν ), We have α t (s µ s ν) = e it( µ ν ) s µ s ν, which makes sense for t C, so the s µ s ν are analytic elements.

17 If φ is a KMS β state then φ is α-invariant, so φ(s µ s ν) = 0 unless µ = ν.

18 If φ is a KMS β state then φ is α-invariant, so φ(s µ sν) = 0 unless µ = ν. If µ = ν, then the KMS β condition gives { φ(s µ sν) = φ(sνα iβ (s µ )) = e i2β µ φ(sνs 0 if ν µ µ ) = e β µ if ν = µ.

19 If φ is a KMS β state then φ is α-invariant, so φ(s µ sν) = 0 unless µ = ν. If µ = ν, then the KMS β condition gives { φ(s µ sν) = φ(sνα iβ (s µ )) = e i2β µ φ(sνs 0 if ν µ µ ) = e β µ if ν = µ. Lemma. φ is a KMS β state iff φ(s µ s ν) = { 0 if ν µ e β µ if ν = µ.

20 If φ is a KMS β state then φ is α-invariant, so φ(s µ sν) = 0 unless µ = ν. If µ = ν, then the KMS β condition gives { φ(s µ sν) = φ(sνα iβ (s µ )) = e i2β µ φ(sνs 0 if ν µ µ ) = e β µ if ν = µ. Lemma. φ is a KMS β state iff φ(s µ s ν) = If φ is a KMS β state on O n then { 0 if ν µ e β µ if ν = µ. 1 = φ(1) = n φ(s j sj ) = j=1 n e β = ne β = β = log n. j=1

21 If φ is a KMS β state then φ is α-invariant, so φ(s µ sν) = 0 unless µ = ν. If µ = ν, then the KMS β condition gives { φ(s µ sν) = φ(sνα iβ (s µ )) = e i2β µ φ(sνs 0 if ν µ µ ) = e β µ if ν = µ. Lemma. φ is a KMS β state iff φ(s µ s ν) = If φ is a KMS β state on O n then { 0 if ν µ e β µ if ν = µ. 1 = φ(1) = n φ(s j sj ) = j=1 n e β = ne β = β = log n. j=1 So O n has at most one KMS β state, when β = log log n. Does it have one?

22 Define Φ : O n O α n by Φ(a) = 1 0 α 2πt(a) dt.

23 Define Φ : O n O α n by Φ(a) = 1 0 α 2πt(a) dt. On α = k=1 span{s µsν : µ = ν = k} = unique tracial state τ. k=1 M n k (C) carries a

24 Define Φ : O n O α n by Φ(a) = 1 0 α 2πt(a) dt. On α = k=1 span{s µsν : µ = ν = k} = unique tracial state τ. k=1 M n k (C) carries a Theorem (Olesen-Pedersen 1978). τ Φ is a KMS log n state.

25 Define Φ : O n O α n by Φ(a) = 1 0 α 2πt(a) dt. On α = k=1 span{s µsν : µ = ν = k} = unique tracial state τ. k=1 M n k (C) carries a Theorem (Olesen-Pedersen 1978). τ Φ is a KMS log n state. Laca-Exel, Laca-Neshveyev: Most of the above works equally well for T O n = C (s j : sj s j = 1 n j=1 s jsj ). For each β there is at most one KMS β state.

26 Define Φ : O n O α n by Φ(a) = 1 0 α 2πt(a) dt. On α = k=1 span{s µsν : µ = ν = k} = unique tracial state τ. k=1 M n k (C) carries a Theorem (Olesen-Pedersen 1978). τ Φ is a KMS log n state. Laca-Exel, Laca-Neshveyev: Most of the above works equally well for T O n = C (s j : sj s j = 1 n j=1 s jsj ). For each β there is at most one KMS β state. Applying a KMS β state to the relation 1 n j=1 s jsj shows that β log n.

27 Define Φ : O n O α n by Φ(a) = 1 0 α 2πt(a) dt. On α = k=1 span{s µsν : µ = ν = k} = unique tracial state τ. k=1 M n k (C) carries a Theorem (Olesen-Pedersen 1978). τ Φ is a KMS log n state. Laca-Exel, Laca-Neshveyev: Most of the above works equally well for T O n = C (s j : sj s j = 1 n j=1 s jsj ). For each β there is at most one KMS β state. Applying a KMS β state to the relation 1 n j=1 s jsj shows that β log n. Consider Σ = k 0 {1,, n}k, and S j on l 2 (Σ ) defined by S j e µ = e jµ, giving a representation π S of T O n.

28 Define Φ : O n O α n by Φ(a) = 1 0 α 2πt(a) dt. On α = k=1 span{s µsν : µ = ν = k} = unique tracial state τ. k=1 M n k (C) carries a Theorem (Olesen-Pedersen 1978). τ Φ is a KMS log n state. Laca-Exel, Laca-Neshveyev: Most of the above works equally well for T O n = C (s j : sj s j = 1 n j=1 s jsj ). For each β there is at most one KMS β state. Applying a KMS β state to the relation 1 n j=1 s jsj shows that β log n. Consider Σ = k 0 {1,, n}k, and S j on l 2 (Σ ) defined by S j e µ = e jµ, giving a representation π S of T O n. Then φ β (a) = (1 ne β ) µ Σ e β µ (π S (a)e µ e µ ) defines a KMS β state on T O n for every β > log n.

29 Any cancellative semigroup has a Toeplitz representation T on l 2 (P) such that T x e y = e xy. The Toeplitz algebra of P is T (P) := C (T x : x P) B(l 2 (P)).

30 Any cancellative semigroup has a Toeplitz representation T on l 2 (P) such that T x e y = e xy. The Toeplitz algebra of P is T (P) := C (T x : x P) B(l 2 (P)). Examples. T (N) is generated by the unilateral shift, and is the universal C -algebra generated by an isometry (Coburn 1967).

31 Any cancellative semigroup has a Toeplitz representation T on l 2 (P) such that T x e y = e xy. The Toeplitz algebra of P is T (P) := C (T x : x P) B(l 2 (P)). Examples. T (N) is generated by the unilateral shift, and is the universal C -algebra generated by an isometry (Coburn 1967). Roughly, the same is true when P is the positive cone in a totally ordered abelian group (Douglas 1972, Murphy 1987).

32 Any cancellative semigroup has a Toeplitz representation T on l 2 (P) such that T x e y = e xy. The Toeplitz algebra of P is T (P) := C (T x : x P) B(l 2 (P)). Examples. T (N) is generated by the unilateral shift, and is the universal C -algebra generated by an isometry (Coburn 1967). Roughly, the same is true when P is the positive cone in a totally ordered abelian group (Douglas 1972, Murphy 1987). T (N 2 ) is generated by the isometries T e1, T e2, but is not universal for such pairs: T satisfies the extra relation T e1 T e 2 = T e 2 T e1.

33 Consider a subsemigroup P of a group G which satisfies P P 1 = {e} and which generates G. Examples. (Z, N), (Z 2, N 2 ), (Q +, N ).

34 Consider a subsemigroup P of a group G which satisfies P P 1 = {e} and which generates G. Examples. (Z, N), (Z 2, N 2 ), (Q +, N ). For x, y G we define x y x 1 y P y xp, and then is a partial order on G. Def. [Nica, 92]. (G, P) is quasi-lattice ordered if every pair x, y G with a common upper bound in P has a least upper bound x y in P.

35 Consider a subsemigroup P of a group G which satisfies P P 1 = {e} and which generates G. Examples. (Z, N), (Z 2, N 2 ), (Q +, N ). For x, y G we define x y x 1 y P y xp, and then is a partial order on G. Def. [Nica, 92]. (G, P) is quasi-lattice ordered if every pair x, y G with a common upper bound in P has a least upper bound x y in P. Examples. (1) In (Z 2, N 2 ) we have (m 1, m 2 ) (n 1, n 2 ) = (max(m 1, n 1 ), max(m 2, n 2 )).

36 Consider a subsemigroup P of a group G which satisfies P P 1 = {e} and which generates G. Examples. (Z, N), (Z 2, N 2 ), (Q +, N ). For x, y G we define x y x 1 y P y xp, and then is a partial order on G. Def. [Nica, 92]. (G, P) is quasi-lattice ordered if every pair x, y G with a common upper bound in P has a least upper bound x y in P. Examples. (1) In (Z 2, N 2 ) we have (m 1, m 2 ) (n 1, n 2 ) = (max(m 1, n 1 ), max(m 2, n 2 )). (2) In (Q +, N ), m n m n, and m n = [m, n].

37 Consider a subsemigroup P of a group G which satisfies P P 1 = {e} and which generates G. Examples. (Z, N), (Z 2, N 2 ), (Q +, N ). For x, y G we define x y x 1 y P y xp, and then is a partial order on G. Def. [Nica, 92]. (G, P) is quasi-lattice ordered if every pair x, y G with a common upper bound in P has a least upper bound x y in P. Examples. (1) In (Z 2, N 2 ) we have (m 1, m 2 ) (n 1, n 2 ) = (max(m 1, n 1 ), max(m 2, n 2 )). (2) In (Q +, N ), m n m n, and m n = [m, n]. (3) F 2 is the free group with generators a, b, and P = a, b. Then x y means that x is an initial segment of y, and the rest of y has no factors of a 1 or b 1. Here x y = often.

38 An isometric representation V : P Isom(H) is Nica covariant if { (V x Vx )(V y Vy V x y Vx y if x y < ) = 0 if x y =. For example, T : P Isom(l 2 (P)).

39 An isometric representation V : P Isom(H) is Nica covariant if { (V x Vx )(V y Vy V x y Vx y if x y < ) = 0 if x y =. For example, T : P Isom(l 2 (P)). Nica covariance implies V x V y = (V x V x )V x V y (V y V y ) = V x (V x V x )(V y V y )V y = V x (V x y V x y)v y = V x V x V x 1 (x y) V y 1 (x y) V y V y = V x 1 (x y) V y 1 (x y), so C (V ) = span{v x V y : x, y P}.

40 An isometric representation V : P Isom(H) is Nica covariant if { (V x Vx )(V y Vy V x y Vx y if x y < ) = 0 if x y =. For example, T : P Isom(l 2 (P)). Nica covariance implies V x V y = (V x V x )V x V y (V y V y ) = V x (V x V x )(V y V y )V y = V x (V x y V x y)v y = V x V x V x 1 (x y) V y 1 (x y) V y V y = V x 1 (x y) V y 1 (x y), so C (V ) = span{v x V y : x, y P}. Example. V : N 2 Isom(H) is Nica covariant if and only if V e1 V e 2 = V e 2 V e1.

41 Example. Nica-covariant representations of (F 2, a, b ) are given by pairs of isometries such that (S a Sa)(S b Sb ) = 0, or equivalently S a Sa + S b Sb.

42 Example. Nica-covariant representations of (F 2, a, b ) are given by pairs of isometries such that (S a Sa)(S b Sb ) = 0, or equivalently S a Sa + S b Sb. Theorem [Nica 1992, Laca-R 1996]. If (G, P) is suitably amenable, then (T (P), T ) = span{t x T y } is universal for Nica-covariant isometric representations of P.

43 Example. Nica-covariant representations of (F 2, a, b ) are given by pairs of isometries such that (S a Sa)(S b Sb ) = 0, or equivalently S a Sa + S b Sb. Theorem [Nica 1992, Laca-R 1996]. If (G, P) is suitably amenable, then (T (P), T ) = span{t x T y } is universal for Nica-covariant isometric representations of P. For (G, P) = (F 2, P) we recover the uniqueness of the Toeplitz-Cuntz algebra T O 2 (Cuntz 1977).

44 Example. Nica-covariant representations of (F 2, a, b ) are given by pairs of isometries such that (S a Sa)(S b Sb ) = 0, or equivalently S a Sa + S b Sb. Theorem [Nica 1992, Laca-R 1996]. If (G, P) is suitably amenable, then (T (P), T ) = span{t x T y } is universal for Nica-covariant isometric representations of P. For (G, P) = (F 2, P) we recover the uniqueness of the Toeplitz-Cuntz algebra T O 2 (Cuntz 1977). The new work with Marcelo concerns the following semigroup recently studied by Cuntz: Example. N N with (m, a)(n, b) = (m + an, ab).

45 Example. Nica-covariant representations of (F 2, a, b ) are given by pairs of isometries such that (S a Sa)(S b Sb ) = 0, or equivalently S a Sa + S b Sb. Theorem [Nica 1992, Laca-R 1996]. If (G, P) is suitably amenable, then (T (P), T ) = span{t x T y } is universal for Nica-covariant isometric representations of P. For (G, P) = (F 2, P) we recover the uniqueness of the Toeplitz-Cuntz algebra T O 2 (Cuntz 1977). The new work with Marcelo concerns the following semigroup recently studied by Cuntz: Example. N N with (m, a)(n, b) = (m + an, ab). Question 1. Is (Q Q +, N N ) quasi-lattice ordered?

46 We need to understand the partial order on N N. We have (m, a) (k, c) (l, d) P with (k, c) = (m + al, ad) a c and k m + an.

47 We need to understand the partial order on N N. We have (m, a) (k, c) (l, d) P with (k, c) = (m + al, ad) a c and k m + an. Proposition. (Q Q +, N N ) is quasi-lattice ordered, and for (m, a), (n, b) P = N N we have { unless (a, b) divides m n, (m, a) (n, b) = (l, [a, b]) if it does, where l = min((m + an) (n + bn)) (which is non-empty if and only if (a, b) divides m n).

48 We need to understand the partial order on N N. We have (m, a) (k, c) (l, d) P with (k, c) = (m + al, ad) a c and k m + an. Proposition. (Q Q +, N N ) is quasi-lattice ordered, and for (m, a), (n, b) P = N N we have { unless (a, b) divides m n, (m, a) (n, b) = (l, [a, b]) if it does, where l = min((m + an) (n + bn)) (which is non-empty if and only if (a, b) divides m n). In particular, T (N N ) = span{t (m,a) T (n,b) }.

49 We set S = T (1,1) and V a = T (0,a). Then (a) V a V b = V b V a for all a, b, and Va V b = V b Va for (a, b) = 1, (b) V a S = S a V a, (c) S V a = S p 1 V a S, and (d) Va S l V a = 0 for 0 < l < a.

50 We set S = T (1,1) and V a = T (0,a). Then (a) V a V b = V b V a for all a, b, and V a V b = V b V a for (a, b) = 1, (b) V a S = S a V a, (c) S V a = S p 1 V a S, and (d) V a S l V a = 0 for 0 < l < a. (a) says that V is a Nica-covariant representation of N ; (b) says T (0,p) T (1,1) = T (p,p) ; (c) and (d) are Nica covariance for the pairs x = (1, 1), y = (0, p) and x = (0, p), y = (1, p).

51 We set S = T (1,1) and V a = T (0,a). Then (a) V a V b = V b V a for all a, b, and V a V b = V b V a for (a, b) = 1, (b) V a S = S a V a, (c) S V a = S p 1 V a S, and (d) V a S l V a = 0 for 0 < l < a. (a) says that V is a Nica-covariant representation of N ; (b) says T (0,p) T (1,1) = T (p,p) ; (c) and (d) are Nica covariance for the pairs x = (1, 1), y = (0, p) and x = (0, p), y = (1, p). (d) implies that {S k V a : 0 k < a} is a Toeplitz-Cuntz family; in Cuntz s algebra, SS = 1 and {S k V a } is a Cuntz family.

52 We set S = T (1,1) and V a = T (0,a). Then (a) V a V b = V b V a for all a, b, and V a V b = V b V a for (a, b) = 1, (b) V a S = S a V a, (c) S V a = S p 1 V a S, and (d) V a S l V a = 0 for 0 < l < a. (a) says that V is a Nica-covariant representation of N ; (b) says T (0,p) T (1,1) = T (p,p) ; (c) and (d) are Nica covariance for the pairs x = (1, 1), y = (0, p) and x = (0, p), y = (1, p). (d) implies that {S k V a : 0 k < a} is a Toeplitz-Cuntz family; in Cuntz s algebra, SS = 1 and {S k V a } is a Cuntz family. Theorem. (T (N N ), S, V a ) = span{s m V a V b S n } is universal for families satisfying (a) (d).

53 We set S = T (1,1) and V a = T (0,a). Then (a) V a V b = V b V a for all a, b, and V a V b = V b V a for (a, b) = 1, (b) V a S = S a V a, (c) S V a = S p 1 V a S, and (d) V a S l V a = 0 for 0 < l < a. (a) says that V is a Nica-covariant representation of N ; (b) says T (0,p) T (1,1) = T (p,p) ; (c) and (d) are Nica covariance for the pairs x = (1, 1), y = (0, p) and x = (0, p), y = (1, p). (d) implies that {S k V a : 0 k < a} is a Toeplitz-Cuntz family; in Cuntz s algebra, SS = 1 and {S k V a } is a Cuntz family. Theorem. (T (N N ), S, V a ) = span{s m V a V b S n } is universal for families satisfying (a) (d). Corollary. There is a continuous action σ : R Aut T (N N ) such that σ t (S) = S and σ t (V a ) = a it V a.

54 We have σ t (S m V a V b S n ) = a it b it S m V a V b S n = e (log a log b)it S m V a V b S n, so all the spanning elements are analytic.

55 We have σ t (S m V a V b S n ) = a it b it S m V a V b S n = e (log a log b)it S m V a V b S n, so all the spanning elements are analytic.the following lemma looks disarmingly easy: Lemma. A state φ of T (N N ) is a KMS β state if and only if 0 if a b or m n (mod a) φ(s m V a Vb S n ) = a β φ(s a 1 (m n) ) if a = b, m n an a β φ(s a 1 (n m) ) if a = b, n m an.

56 We have σ t (S m V a V b S n ) = a it b it S m V a V b S n = e (log a log b)it S m V a V b S n, so all the spanning elements are analytic.the following lemma looks disarmingly easy: Lemma. A state φ of T (N N ) is a KMS β state if and only if 0 if a b or m n (mod a) φ(s m V a Vb S n ) = a β φ(s a 1 (m n) ) if a = b, m n an a β φ(s a 1 (n m) ) if a = b, n m an. To prove it, though, we have to show that the condition implies φ ( (S m V a V b S n )(S k V c V d S l ) ) = a it b it φ ( (S k V c V d S l )(S m V a V b S n ) ) and this involves being able to compute least upper bounds in N N.

57 Theorem (Laca R) Consider the system (C (N N ), σ) described above. Then: For β < 1, there are no KMS β states. For 1 β 2, there is a unique KMS β state. For β > 2, the simplex of KMS β states is isomorphic to the simplex P(T) of probability measures on the unit circle T.

58 Theorem (Laca R) Consider the system (C (N N ), σ) described above. Then: For β < 1, there are no KMS β states. For 1 β 2, there is a unique KMS β state. For β > 2, the simplex of KMS β states is isomorphic to the simplex P(T) of probability measures on the unit circle T. An unusual feature is that the KMS β states for β > 2 do not factor through an expectation onto a commutative subalgebra. They do factor through an expectation onto C (V a V a, S): the elements V a V a span a commutative algebra, but the KMS β states for β > 2 need not vanish on powers of the generator S.

59 Theorem (Laca R) Consider the system (C (N N ), σ) described above. Then: For β < 1, there are no KMS β states. For 1 β 2, there is a unique KMS β state. For β > 2, the simplex of KMS β states is isomorphic to the simplex P(T) of probability measures on the unit circle T. An unusual feature is that the KMS β states for β > 2 do not factor through an expectation onto a commutative subalgebra. They do factor through an expectation onto C (V a V a, S): the elements V a V a span a commutative algebra, but the KMS β states for β > 2 need not vanish on powers of the generator S. We have spontaneous symmetry breaking as β increases through 2, but the circular symmetry which is being broken does not come from an action of T on T (N N ).