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Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

Equilibrium states on right LCM semigroup C*-algebras revisited

Nadia S. Larsen

University of Oslo

4 December 2017 joint with N. Brownlowe, J. Ramagge and N. Stammeier

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

The KMS condition for finite systems

Finite quantum systems: a time evolution on Mn(C) is given by a one-parameter group of automorphisms σt(a) = eitHae−itH, where t ∈ R, a ∈ Mn(C) and H is a self-adjoint matrix.

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

The KMS condition for finite systems

Finite quantum systems: a time evolution on Mn(C) is given by a one-parameter group of automorphisms σt(a) = eitHae−itH, where t ∈ R, a ∈ Mn(C) and H is a self-adjoint matrix. The Gibbs state at β > 0 is ϕG(a) = Tr(ae−βH)

Tr(e−βH) . It satisfies

ϕG(ab) = ϕG(bσiβ(a)), (1) for a, b ∈ Mn(C) analytic, i.e. t → σt(a) extends to an entire function on C.

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

The KMS condition for finite systems

Finite quantum systems: a time evolution on Mn(C) is given by a one-parameter group of automorphisms σt(a) = eitHae−itH, where t ∈ R, a ∈ Mn(C) and H is a self-adjoint matrix. The Gibbs state at β > 0 is ϕG(a) = Tr(ae−βH)

Tr(e−βH) . It satisfies

ϕG(ab) = ϕG(bσiβ(a)), (1) for a, b ∈ Mn(C) analytic, i.e. t → σt(a) extends to an entire function on C. Partition function of (Mn(C), σ) is β → Tr(e−βH).

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

The KMS condition for finite systems

Finite quantum systems: a time evolution on Mn(C) is given by a one-parameter group of automorphisms σt(a) = eitHae−itH, where t ∈ R, a ∈ Mn(C) and H is a self-adjoint matrix. The Gibbs state at β > 0 is ϕG(a) = Tr(ae−βH)

Tr(e−βH) . It satisfies

ϕG(ab) = ϕG(bσiβ(a)), (1) for a, b ∈ Mn(C) analytic, i.e. t → σt(a) extends to an entire function on C. Partition function of (Mn(C), σ) is β → Tr(e−βH). (1) - the KMS condition, cf. Haag-Hugenholtz-Winnick (1967): equilibrium for a state on a C ∗-algebra with time evolution.

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

KMS states

By analogy with finite systems and the Gibbs state, extend the notions of KMSβ state, partition function, inverse temperature.

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

KMS states

By analogy with finite systems and the Gibbs state, extend the notions of KMSβ state, partition function, inverse temperature. A C ∗-algebra, σ : R → Aut(A) time evolution, ϕ a state on A.

1 ϕ is KMSβ (at inverse temperature β ∈ [0, ∞)) if

ϕ(ab) = ϕ(bσiβ(a)) for all a, b ∈ Aa, the dense ∗-subalgebra of analytic elements.

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

KMS states

By analogy with finite systems and the Gibbs state, extend the notions of KMSβ state, partition function, inverse temperature. A C ∗-algebra, σ : R → Aut(A) time evolution, ϕ a state on A.

1 ϕ is KMSβ (at inverse temperature β ∈ [0, ∞)) if

ϕ(ab) = ϕ(bσiβ(a)) for all a, b ∈ Aa, the dense ∗-subalgebra of analytic elements.

2 A state ϕ is a ground state if for all a, b with b analytic,

the function z → ϕ(aσz(b)) is bounded in the upper-half plane.

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

KMS states

By analogy with finite systems and the Gibbs state, extend the notions of KMSβ state, partition function, inverse temperature. A C ∗-algebra, σ : R → Aut(A) time evolution, ϕ a state on A.

1 ϕ is KMSβ (at inverse temperature β ∈ [0, ∞)) if

ϕ(ab) = ϕ(bσiβ(a)) for all a, b ∈ Aa, the dense ∗-subalgebra of analytic elements.

2 A state ϕ is a ground state if for all a, b with b analytic,

the function z → ϕ(aσz(b)) is bounded in the upper-half plane.

3 KMS∞ if ϕ = w∗ lim ϕn as βn → ∞ and ϕn is KMSβn.

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

KMS states

By analogy with finite systems and the Gibbs state, extend the notions of KMSβ state, partition function, inverse temperature. A C ∗-algebra, σ : R → Aut(A) time evolution, ϕ a state on A.

1 ϕ is KMSβ (at inverse temperature β ∈ [0, ∞)) if

ϕ(ab) = ϕ(bσiβ(a)) for all a, b ∈ Aa, the dense ∗-subalgebra of analytic elements.

2 A state ϕ is a ground state if for all a, b with b analytic,

the function z → ϕ(aσz(b)) is bounded in the upper-half plane.

3 KMS∞ if ϕ = w∗ lim ϕn as βn → ∞ and ϕn is KMSβn.

References: Bratteli-Robinson, Pedersen, Connes-Marcolli.

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

KMS∞ strictly subset of ground states

Theorem (Laca-Raeburn (2010))

C ∗(N ⋊ N×) is the universal C ∗-algebra generated by isometries s and {vp | p prime}, subject to the relations

1 vps = spvp; 2 vpvq = vqvp, 3 v∗ pvq = vqv∗ p when p = q, 4 s∗vp = sp−1vps∗, and 5 v∗ pskvp = 0 for 1 ≤ k < p.

Dynamics: σt(s) = s and σt(vp) = pitvp.

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

KMS∞ strictly subset of ground states

Theorem (Laca-Raeburn (2010))

C ∗(N ⋊ N×) is the universal C ∗-algebra generated by isometries s and {vp | p prime}, subject to the relations

1 vps = spvp; 2 vpvq = vqvp, 3 v∗ pvq = vqv∗ p when p = q, 4 s∗vp = sp−1vps∗, and 5 v∗ pskvp = 0 for 1 ≤ k < p.

Dynamics: σt(s) = s and σt(vp) = pitvp.Then, for β < 1, there are no KMS states, if β ∈ [1, 2], there is a unique KMSβ state;

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

KMS∞ strictly subset of ground states

Theorem (Laca-Raeburn (2010))

C ∗(N ⋊ N×) is the universal C ∗-algebra generated by isometries s and {vp | p prime}, subject to the relations

1 vps = spvp; 2 vpvq = vqvp, 3 v∗ pvq = vqv∗ p when p = q, 4 s∗vp = sp−1vps∗, and 5 v∗ pskvp = 0 for 1 ≤ k < p.

Dynamics: σt(s) = s and σt(vp) = pitvp.Then, for β < 1, there are no KMS states, if β ∈ [1, 2], there is a unique KMSβ state; if β ∈ (2, ∞], the KMSβ states are parametrised by probability measures on T while the ground states are parametrised by states on the Toeplitz C ∗-algebra generated by a single isometry.

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

KMS∞ strictly subset of ground states

Theorem (Afsar-Brownlowe-L-Stammeier (2016))

Let S be a right LCM monoid and N : S → N× homomorphism such that S is admissible. Consider the time evolution σt(vs) = Nit

s vs. If βc ∈ R is such that the function

ζN(β) :=

• n∈Irr(N(S))

n−(β−1), converges for β ≥ βc, then for (C ∗(S), R, σ) we have

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

KMS∞ strictly subset of ground states

Theorem (Afsar-Brownlowe-L-Stammeier (2016))

Let S be a right LCM monoid and N : S → N× homomorphism such that S is admissible. Consider the time evolution σt(vs) = Nit

s vs. If βc ∈ R is such that the function

ζN(β) :=

• n∈Irr(N(S))

n−(β−1), converges for β ≥ βc, then for (C ∗(S), R, σ) we have

1 β ∈ [0, 1): no KMSβ state;

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

KMS∞ strictly subset of ground states

Theorem (Afsar-Brownlowe-L-Stammeier (2016))

Let S be a right LCM monoid and N : S → N× homomorphism such that S is admissible. Consider the time evolution σt(vs) = Nit

s vs. If βc ∈ R is such that the function

ζN(β) :=

• n∈Irr(N(S))

n−(β−1), converges for β ≥ βc, then for (C ∗(S), R, σ) we have

1 β ∈ [0, 1): no KMSβ state; 2 β ∈ [1, βc]: unique KMSβ if action Sc S/Sc essentially

free, where Sc ⊂ S subsemigroup of elements having LCM with any t in S;

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

KMS∞ strictly subset of ground states

Theorem (Afsar-Brownlowe-L-Stammeier (2016))

Let S be a right LCM monoid and N : S → N× homomorphism such that S is admissible. Consider the time evolution σt(vs) = Nit

s vs. If βc ∈ R is such that the function

ζN(β) :=

• n∈Irr(N(S))

n−(β−1), converges for β ≥ βc, then for (C ∗(S), R, σ) we have

1 β ∈ [0, 1): no KMSβ state; 2 β ∈ [1, βc]: unique KMSβ if action Sc S/Sc essentially

free, where Sc ⊂ S subsemigroup of elements having LCM with any t in S;

3 β ∈ (βc, ∞]: KMSβ states parametrised by normalised

traces on C ∗(Sc) ;

Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen

KMS∞ strictly subset of ground states

Theorem (Afsar-Brownlowe-L-Stammeier (2016))

Let S be a right LCM monoid and N : S → N× homomorphism such that S is admissible. Consider the time evolution σt(vs) = Nit

s vs. If βc ∈ R is such that the function

ζN(β) :=

• n∈Irr(N(S))

n−(β−1), converges for β ≥ βc, then for (C ∗(S), R, σ) we have

1 β ∈ [0, 1): no KMSβ state; 2 β ∈ [1, βc]: unique KMSβ if action Sc S/Sc essentially

free, where Sc ⊂ S subsemigroup of elements having LCM with any t in S;

3 β ∈ (βc, ∞]: KMSβ states parametrised by normalised

traces on C ∗(Sc) ;

4 Ground states: parametrised by states on C ∗(Sc).