KMS states and von Neumman factors from higher-rank graphs The 12 th - - PowerPoint PPT Presentation

KMS states and von Neumman factors from higher-rank graphs

The 12th Abel Symposium Aidan Sims University of Wollongong (with an Huef–Laca–Raeburn, and with Laca–Larsen–Neshveyev–Webster) August 7–11, 2015

Higher-rank graphs and Perron–Frobenius

Definition (Kumjian–Pask, 2000) A k-graph is a countable category Λ with functor d : Λ → Nk such that composition gives bijections d−1(m) ∗ d−1(n) → d−1(m + n). Λn := d−1(n); Λ0 = {identity morphisms}. Today, every Λn is finite and nonempty, and vΛw = ∅ for all v, w ∈ Λ0.

Higher-rank graphs and Perron–Frobenius

Definition (Kumjian–Pask, 2000) A k-graph is a countable category Λ with functor d : Λ → Nk such that composition gives bijections d−1(m) ∗ d−1(n) → d−1(m + n). Λn := d−1(n); Λ0 = {identity morphisms}. Today, every Λn is finite and nonempty, and vΛw = ∅ for all v, w ∈ Λ0. Matrices An ∈ MΛ0(Z), An(v, w) = vΛnw form a multiplicative semigroup, and ∀v, w ∃n such that An(v, w) > 0. Put Aj := Aej.

Higher-rank graphs and Perron–Frobenius

Definition (Kumjian–Pask, 2000) A k-graph is a countable category Λ with functor d : Λ → Nk such that composition gives bijections d−1(m) ∗ d−1(n) → d−1(m + n). Λn := d−1(n); Λ0 = {identity morphisms}. Today, every Λn is finite and nonempty, and vΛw = ∅ for all v, w ∈ Λ0. Matrices An ∈ MΛ0(Z), An(v, w) = vΛnw form a multiplicative semigroup, and ∀v, w ∃n such that An(v, w) > 0. Put Aj := Aej. Proposition (Kumjian–Pask, aHLRS)

• ∃! common positive eigenvector of the Aj with unit 1-norm.
• Corresponding eigenvalues are the spectral radii ρ(An).
• n → ρ(An) is a homomorphism (Nk, +) →
• (0, ∞), ×
• .

Higher-rank graphs and Perron–Frobenius

Definition (Kumjian–Pask, 2000) A k-graph is a countable category Λ with functor d : Λ → Nk such that composition gives bijections d−1(m) ∗ d−1(n) → d−1(m + n). Λn := d−1(n); Λ0 = {identity morphisms}. Today, every Λn is finite and nonempty, and vΛw = ∅ for all v, w ∈ Λ0. Matrices An ∈ MΛ0(Z), An(v, w) = vΛnw form a multiplicative semigroup, and ∀v, w ∃n such that An(v, w) > 0. Put Aj := Aej. Proposition (Kumjian–Pask, aHLRS)

• ∃! common positive eigenvector of the Aj with unit 1-norm.
• Corresponding eigenvalues are the spectral radii ρ(An).
• n → ρ(An) is a homomorphism (Nk, +) →
• (0, ∞), ×
• .

Define ρ(Λ) =

• ρ(A1), . . . , ρ(Ak)

Higher-rank Cuntz–Krieger algebras

Definition (Kumjian–Pask, 2000) The k-graph algebra C ∗(Λ) is universal for projections {pv : v ∈ Λ0} and partial isometries {sf : f ∈

j Λej} such that

• each s∗

f sf = ps(f )

• each pv =

f ∈vΛej sf s∗ f

• sesf = sgsh whenever e, h ∈ Λej, f , g ∈ Λel and ef = gh.

Higher-rank Cuntz–Krieger algebras

Definition (Kumjian–Pask, 2000) The k-graph algebra C ∗(Λ) is universal for projections {pv : v ∈ Λ0} and partial isometries {sf : f ∈

j Λej} such that

• each s∗

f sf = ps(f )

• each pv =

f ∈vΛej sf s∗ f

• sesf = sgsh whenever e, h ∈ Λej, f , g ∈ Λel and ef = gh.

First two relations say {pv : v ∈ E 0} ∪ {sf : f ∈ Λej} is a Cuntz–Krieger family for the subgraph with edges Λej. So C ∗(Λ) is generated by OA1, . . . , OAk.

Higher-rank Cuntz–Krieger algebras

Definition (Kumjian–Pask, 2000) The k-graph algebra C ∗(Λ) is universal for projections {pv : v ∈ Λ0} and partial isometries {sf : f ∈

j Λej} such that

• each s∗

f sf = ps(f )

• each pv =

f ∈vΛej sf s∗ f

• sesf = sgsh whenever e, h ∈ Λej, f , g ∈ Λel and ef = gh.

First two relations say {pv : v ∈ E 0} ∪ {sf : f ∈ Λej} is a Cuntz–Krieger family for the subgraph with edges Λej. So C ∗(Λ) is generated by OA1, . . . , OAk. Eg: If |Λ0| = 1, then

• Λ is a semigroup with generators Λej.
• C ∗(Λ) is generated by O|Λe1|, . . . , O|Λek |.
• If, for example, ef = fe for all ef , then C ∗(Λ) =

j O|Λej |.

KMS states

Consider an action α of R on a C ∗-algebra A.

KMS states

Consider an action α of R on a C ∗-algebra A. a ∈ A is analytic (for α) if t → αt(a) extends to an entire function z → αz(a). Analytic elements are dense in A.

KMS states

Consider an action α of R on a C ∗-algebra A. a ∈ A is analytic (for α) if t → αt(a) extends to an entire function z → αz(a). Analytic elements are dense in A. Definition (Haag–Hugenholtz–Winnink, 1967) For β > 0, a state φ of A is KMSβ for α if φ(ab) = φ(bαiβ(a)) for all analytic a, b. A KMS0-state is an α-invariant trace.

KMS states

Consider an action α of R on a C ∗-algebra A. a ∈ A is analytic (for α) if t → αt(a) extends to an entire function z → αz(a). Analytic elements are dense in A. Definition (Haag–Hugenholtz–Winnink, 1967) For β > 0, a state φ of A is KMSβ for α if φ(ab) = φ(bαiβ(a)) for all analytic a, b. A KMS0-state is an α-invariant trace. Suffices to verify KMS condition for a, b in any α-invariant set A

• f analytic elements such that spanA = A.

The preferred dynamics on C ∗(Λ)

Take r ∈ (0, ∞)k. Universal property gives αr : R C ∗(Λ) s.t. αr

t(sf ) = eirjtsf for f ∈ Λej.

The preferred dynamics on C ∗(Λ)

Take r ∈ (0, ∞)k. Universal property gives αr : R C ∗(Λ) s.t. αr

t(sf ) = eirjtsf for f ∈ Λej.

In C ∗(Λ), sµ := sµ1 · · · sµl for any factorisation µ = µ1 · · · µl into edges is well-defined. Relations give C ∗(Λ) = span{sµs∗

ν : µ, ν ∈ Λ, s(µ) = s(ν)}

The preferred dynamics on C ∗(Λ)

Take r ∈ (0, ∞)k. Universal property gives αr : R C ∗(Λ) s.t. αr

t(sf ) = eirjtsf for f ∈ Λej.

In C ∗(Λ), sµ := sµ1 · · · sµl for any factorisation µ = µ1 · · · µl into edges is well-defined. Relations give C ∗(Λ) = span{sµs∗

ν : µ, ν ∈ Λ, s(µ) = s(ν)}

Since αr

t(sµs∗ ν) = eir·(d(µ)−d(ν))tsµs∗ ν, the sµs∗ ν are analytic.

The preferred dynamics on C ∗(Λ)

Take r ∈ (0, ∞)k. Universal property gives αr : R C ∗(Λ) s.t. αr

t(sf ) = eirjtsf for f ∈ Λej.

In C ∗(Λ), sµ := sµ1 · · · sµl for any factorisation µ = µ1 · · · µl into edges is well-defined. Relations give C ∗(Λ) = span{sµs∗

ν : µ, ν ∈ Λ, s(µ) = s(ν)}

Since αr

t(sµs∗ ν) = eir·(d(µ)−d(ν))tsµs∗ ν, the sµs∗ ν are analytic.

If φ is KMSβ for αr, then φ(pv) =

f ∈vΛej φ(sf s∗ f ) = e−βrj w∈Λ0 Aj(v, w)φ(pw)

• ,

so

• φ(pv)
• v is a common positive eigenvector of the Aj.

The preferred dynamics on C ∗(Λ)

Take r ∈ (0, ∞)k. Universal property gives αr : R C ∗(Λ) s.t. αr

t(sf ) = eirjtsf for f ∈ Λej.

In C ∗(Λ), sµ := sµ1 · · · sµl for any factorisation µ = µ1 · · · µl into edges is well-defined. Relations give C ∗(Λ) = span{sµs∗

ν : µ, ν ∈ Λ, s(µ) = s(ν)}

Since αr

t(sµs∗ ν) = eir·(d(µ)−d(ν))tsµs∗ ν, the sµs∗ ν are analytic.

If φ is KMSβ for αr, then φ(pv) =

f ∈vΛej φ(sf s∗ f ) = e−βrj w∈Λ0 Aj(v, w)φ(pw)

• ,

so

• φ(pv)
• v is a common positive eigenvector of the Aj.

Proposition (aHLRS) If φ is KMSβ for αr, then βr = ln ρ(Λ), and φ is KMS1 for the preferred dynamics αt(sf ) = ρ(Aj)itsf for f ∈ Λej.

Groupoids from k-graphs, and periodicity

Infinite paths in Λ are maps x : {(m, n) ∈ Nk × Nk | m ≤ n} → Λ with d(x(m, n)) = n − m and x(m, n)x(n, p) = x(m, p). Λ∞ = {infinite paths}.

Groupoids from k-graphs, and periodicity

Infinite paths in Λ are maps x : {(m, n) ∈ Nk × Nk | m ≤ n} → Λ with d(x(m, n)) = n − m and x(m, n)x(n, p) = x(m, p). Λ∞ = {infinite paths}. Λ∞ is a locally compact Hausdorff subspace of

m≤n∈Nk Λn−m.

Have σ : Nk Λ∞ where σp(x)(m, n) = x(m + p, n + p).

Groupoids from k-graphs, and periodicity

Infinite paths in Λ are maps x : {(m, n) ∈ Nk × Nk | m ≤ n} → Λ with d(x(m, n)) = n − m and x(m, n)x(n, p) = x(m, p). Λ∞ = {infinite paths}. Λ∞ is a locally compact Hausdorff subspace of

m≤n∈Nk Λn−m.

Have σ : Nk Λ∞ where σp(x)(m, n) = x(m + p, n + p). The periodicity group Per(Λ) := {(m − n) : m, n ∈ Nk, σm = σn}. is a subgroup of Zk.

Groupoids from k-graphs, and periodicity

Infinite paths in Λ are maps x : {(m, n) ∈ Nk × Nk | m ≤ n} → Λ with d(x(m, n)) = n − m and x(m, n)x(n, p) = x(m, p). Λ∞ = {infinite paths}. Λ∞ is a locally compact Hausdorff subspace of

m≤n∈Nk Λn−m.

Have σ : Nk Λ∞ where σp(x)(m, n) = x(m + p, n + p). The periodicity group Per(Λ) := {(m − n) : m, n ∈ Nk, σm = σn}. is a subgroup of Zk. The k-graph groupoid is the Deaconu–Renault groupoid GΛ = {(x, p − q, y) ∈ Λ∞ × Zk × Λ∞ | σp(x) = σq(y)}

Groupoids from k-graphs, and periodicity

Infinite paths in Λ are maps x : {(m, n) ∈ Nk × Nk | m ≤ n} → Λ with d(x(m, n)) = n − m and x(m, n)x(n, p) = x(m, p). Λ∞ = {infinite paths}. Λ∞ is a locally compact Hausdorff subspace of

m≤n∈Nk Λn−m.

Have σ : Nk Λ∞ where σp(x)(m, n) = x(m + p, n + p). The periodicity group Per(Λ) := {(m − n) : m, n ∈ Nk, σm = σn}. is a subgroup of Zk. The k-graph groupoid is the Deaconu–Renault groupoid GΛ = {(x, p − q, y) ∈ Λ∞ × Zk × Λ∞ | σp(x) = σq(y)} Int(Iso(GΛ)) = {(x, m, x) : m ∈ Per(Λ)}, and C ∗(Λ) ∼ = C ∗(GΛ).

Periodicity, and KMS states of C ∗(Λ)

We’re interested in the KMS states for the preferred dynamics.

Periodicity, and KMS states of C ∗(Λ)

We’re interested in the KMS states for the preferred dynamics. Theorem (aHLRS)

• The map g → 1{(x,g,x):x∈Λ∞} ∈ Cc(GΛ) determines a unital

injection ι : C ∗(Per(Λ)) → Z(C ∗(Λ))

• The map ι∗ : φ → φ ◦ ι is an affine isomorphism from the KMS1

simplex of C ∗(Λ) to the state-space of C ∗(Per(Λ)).

• The following are equivalent:

– Per(Λ) = {0}; – C ∗(Λ) is simple; – (ι∗)−1(Tr) is a factor state; – (ι∗)−1(Tr) is the only KMS1 state.

More Perron–Frobenius theory

General theory says that for extremal KMS states φ (and only extremal ones), πφ(C ∗(Λ))′′ is a factor. To analyse these factors, we need a bit more Perron–Frobenius theory.

More Perron–Frobenius theory

General theory says that for extremal KMS states φ (and only extremal ones), πφ(C ∗(Λ))′′ is a factor. To analyse these factors, we need a bit more Perron–Frobenius theory. A cycle in a k-graph is λ ∈ Λ with r(λ) = s(λ). Proposition (LLNSW)

• PΛ := {d(µ) − d(ν) : µ, ν are cycles} is a subgroup of Zk.
• There is a function C : Λ0 × Λ0 → Zk/PΛ such that

C(r(λ), s(λ)) = d(λ) + PΛ for all λ.

More Perron–Frobenius theory

General theory says that for extremal KMS states φ (and only extremal ones), πφ(C ∗(Λ))′′ is a factor. To analyse these factors, we need a bit more Perron–Frobenius theory. A cycle in a k-graph is λ ∈ Λ with r(λ) = s(λ). Proposition (LLNSW)

• PΛ := {d(µ) − d(ν) : µ, ν are cycles} is a subgroup of Zk.
• There is a function C : Λ0 × Λ0 → Zk/PΛ such that

C(r(λ), s(λ)) = d(λ) + PΛ for all λ. Obtain equivalence relation on Λ0: v ∼ w iff C(v, w) = 0

A Borel equivalence relation

Lemma (aHLRS) The KMS states of C ∗(Λ) all induce the same measure µ on Λ∞ = G(0)

Λ . We have Gx x = {x} × Per(Λ) × {x} for µ-a.e. x.

A Borel equivalence relation

Lemma (aHLRS) The KMS states of C ∗(Λ) all induce the same measure µ on Λ∞ = G(0)

Λ . We have Gx x = {x} × Per(Λ) × {x} for µ-a.e. x.

The relation R := {(x, y) : (x, m, y) ∈ GΛ} is a Borel equivalence relation on Λ∞

A Borel equivalence relation

Lemma (aHLRS) The KMS states of C ∗(Λ) all induce the same measure µ on Λ∞ = G(0)

Λ . We have Gx x = {x} × Per(Λ) × {x} for µ-a.e. x.

The relation R := {(x, y) : (x, m, y) ∈ GΛ} is a Borel equivalence relation on Λ∞ Each πφ(C ∗(Λ))′′ ∼ = W ∗(R).

A Borel equivalence relation

Lemma (aHLRS) The KMS states of C ∗(Λ) all induce the same measure µ on Λ∞ = G(0)

Λ . We have Gx x = {x} × Per(Λ) × {x} for µ-a.e. x.

The relation R := {(x, y) : (x, m, y) ∈ GΛ} is a Borel equivalence relation on Λ∞ Each πφ(C ∗(Λ))′′ ∼ = W ∗(R). There is an R-invariant µ-conull X ⊆ Λ∞ on which the Radon-Nikodym cocycle D for R satisfies D(x, y) = ρ(Λ)n−m if σn(x) = σm(y).

Analysis of R

We define RD := {(x, y) ∈ R : D(x, y) = 1}, and

Analysis of R

We define RD := {(x, y) ∈ R : D(x, y) = 1}, and Rγ := {(x, y) ∈ R : σn(x) = σn(y) for some n}.

Analysis of R

We define RD := {(x, y) ∈ R : D(x, y) = 1}, and Rγ := {(x, y) ∈ R : σn(x) = σn(y) for some n}. Then Rγ ∼ = {(x, m, y) ∈ GΛ : m = 0} is a topological groupoid and C ∗(Rγ) ∼ = C ∗(Λ)γ.

Analysis of R

We define RD := {(x, y) ∈ R : D(x, y) = 1}, and Rγ := {(x, y) ∈ R : σn(x) = σn(y) for some n}. Then Rγ ∼ = {(x, m, y) ∈ GΛ : m = 0} is a topological groupoid and C ∗(Rγ) ∼ = C ∗(Λ)γ. We have Rγ ⊆ RD.

Analysis of R

We define RD := {(x, y) ∈ R : D(x, y) = 1}, and Rγ := {(x, y) ∈ R : σn(x) = σn(y) for some n}. Then Rγ ∼ = {(x, m, y) ∈ GΛ : m = 0} is a topological groupoid and C ∗(Rγ) ∼ = C ∗(Λ)γ. We have Rγ ⊆ RD. Lemma (LLNSW) The ergodic components of Rγ are {x ∈ Λ∞ : r(x) ∈ ω} for ω ∈ Λ0/∼. In particular, the ergodic components of RD are unions

• f these sets.

The von Neumann factors

We can apply Connes’ theorem (1973) about the Connes invariant

• f a factor with a faithful normal state to see that

RD ergodic = ⇒ S(W ∗(R)) = essential range of D.

The von Neumann factors

We can apply Connes’ theorem (1973) about the Connes invariant

• f a factor with a faithful normal state to see that

RD ergodic = ⇒ S(W ∗(R)) = essential range of D. Theorem (LLNSW) Let M := πφ(C ∗(Λ))′′. If ρ(Λ) = (1, . . . , 1), then M = MΛ0(C). Otherwise, M is the injective type IIIλ factor, where λ = sup

• {ρ(Λ)g : g ∈ PΛ} ∩ (0, 1)
• .

The von Neumann factors

We can apply Connes’ theorem (1973) about the Connes invariant

• f a factor with a faithful normal state to see that

RD ergodic = ⇒ S(W ∗(R)) = essential range of D. Theorem (LLNSW) Let M := πφ(C ∗(Λ))′′. If ρ(Λ) = (1, . . . , 1), then M = MΛ0(C). Otherwise, M is the injective type IIIλ factor, where λ = sup

• {ρ(Λ)g : g ∈ PΛ} ∩ (0, 1)
• .

Corollary If any two ln ρ(Λ) are rationally independent, then M is type III1. It is never type III0.

The von Neumann factors

We can apply Connes’ theorem (1973) about the Connes invariant

• f a factor with a faithful normal state to see that

RD ergodic = ⇒ S(W ∗(R)) = essential range of D. Theorem (LLNSW) Let M := πφ(C ∗(Λ))′′. If ρ(Λ) = (1, . . . , 1), then M = MΛ0(C). Otherwise, M is the injective type IIIλ factor, where λ = sup

• {ρ(Λ)g : g ∈ PΛ} ∩ (0, 1)
• .

Corollary If any two ln ρ(Λ) are rationally independent, then M is type III1. It is never type III0.

• Eg. (Yang 2012,2013) suppose |Λ0| = 1 and Per(Λ) = {0}. Each

ρ(Aj) = ρ

• (|Λej|)
• = |Λej|, and PΛ = Zk because there is a loop of

each degree. So λ = sup

• {

j |Λej|gj : g ∈ Zk} ∩ (0, 1)

• .