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 12 th 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 → N k such that composition gives bijections d − 1 ( m ) ∗ d − 1 ( n ) → d − 1 ( m + n ). Λ n := d − 1 ( n ); Λ 0 = { identity morphisms } . Today, every Λ n is finite and nonempty, and v Λ w � = ∅ for all v , w ∈ Λ 0 .

Higher-rank graphs and Perron–Frobenius Definition (Kumjian–Pask, 2000) A k-graph is a countable category Λ with functor d : Λ → N k such that composition gives bijections d − 1 ( m ) ∗ d − 1 ( n ) → d − 1 ( m + n ). Λ n := d − 1 ( n ); Λ 0 = { identity morphisms } . Today, every Λ n is finite and nonempty, and v Λ w � = ∅ for all v , w ∈ Λ 0 . Matrices A n ∈ M Λ 0 ( Z ), A n ( v , w ) = v Λ n w form a multiplicative semigroup, and ∀ v , w ∃ n such that A n ( v , w ) > 0. Put A j := A e j .

Higher-rank graphs and Perron–Frobenius Definition (Kumjian–Pask, 2000) A k-graph is a countable category Λ with functor d : Λ → N k such that composition gives bijections d − 1 ( m ) ∗ d − 1 ( n ) → d − 1 ( m + n ). Λ n := d − 1 ( n ); Λ 0 = { identity morphisms } . Today, every Λ n is finite and nonempty, and v Λ w � = ∅ for all v , w ∈ Λ 0 . Matrices A n ∈ M Λ 0 ( Z ), A n ( v , w ) = v Λ n w form a multiplicative semigroup, and ∀ v , w ∃ n such that A n ( v , w ) > 0. Put A j := A e j . Proposition (Kumjian–Pask, aHLRS) • ∃ ! common positive eigenvector of the A j with unit 1-norm. • Corresponding eigenvalues are the spectral radii ρ ( A n ). • n �→ ρ ( A n ) is a homomorphism ( N k , +) → � � (0 , ∞ ) , × .

Higher-rank graphs and Perron–Frobenius Definition (Kumjian–Pask, 2000) A k-graph is a countable category Λ with functor d : Λ → N k such that composition gives bijections d − 1 ( m ) ∗ d − 1 ( n ) → d − 1 ( m + n ). Λ n := d − 1 ( n ); Λ 0 = { identity morphisms } . Today, every Λ n is finite and nonempty, and v Λ w � = ∅ for all v , w ∈ Λ 0 . Matrices A n ∈ M Λ 0 ( Z ), A n ( v , w ) = v Λ n w form a multiplicative semigroup, and ∀ v , w ∃ n such that A n ( v , w ) > 0. Put A j := A e j . Proposition (Kumjian–Pask, aHLRS) • ∃ ! common positive eigenvector of the A j with unit 1-norm. • Corresponding eigenvalues are the spectral radii ρ ( A n ). • n �→ ρ ( A n ) is a homomorphism ( N k , +) → � � (0 , ∞ ) , × . � � Define ρ (Λ) = ρ ( A 1 ) , . . . , ρ ( A k )

Higher-rank Cuntz–Krieger algebras Definition (Kumjian–Pask, 2000) The k-graph algebra C ∗ (Λ) is universal for projections { p v : v ∈ Λ 0 } and partial isometries { s f : f ∈ � j Λ e j } such that • each s ∗ f s f = p s ( f ) f ∈ v Λ ej s f s ∗ • each p v = � f • s e s f = s g s h whenever e , h ∈ Λ e j , f , g ∈ Λ e l and ef = gh .

Higher-rank Cuntz–Krieger algebras Definition (Kumjian–Pask, 2000) The k-graph algebra C ∗ (Λ) is universal for projections { p v : v ∈ Λ 0 } and partial isometries { s f : f ∈ � j Λ e j } such that • each s ∗ f s f = p s ( f ) f ∈ v Λ ej s f s ∗ • each p v = � f • s e s f = s g s h whenever e , h ∈ Λ e j , f , g ∈ Λ e l and ef = gh . First two relations say { p v : v ∈ E 0 } ∪ { s f : f ∈ Λ e j } is a Cuntz–Krieger family for the subgraph with edges Λ e j . So C ∗ (Λ) is generated by O A 1 , . . . , O A k .

Higher-rank Cuntz–Krieger algebras Definition (Kumjian–Pask, 2000) The k-graph algebra C ∗ (Λ) is universal for projections { p v : v ∈ Λ 0 } and partial isometries { s f : f ∈ � j Λ e j } such that • each s ∗ f s f = p s ( f ) f ∈ v Λ ej s f s ∗ • each p v = � f • s e s f = s g s h whenever e , h ∈ Λ e j , f , g ∈ Λ e l and ef = gh . First two relations say { p v : v ∈ E 0 } ∪ { s f : f ∈ Λ e j } is a Cuntz–Krieger family for the subgraph with edges Λ e j . So C ∗ (Λ) is generated by O A 1 , . . . , O A k . Eg: If | Λ 0 | = 1, then • Λ is a semigroup with generators � Λ e j . • C ∗ (Λ) is generated by O | Λ e 1 | , . . . , O | Λ ek | . • If, for example, ef = fe for all ef , then C ∗ (Λ) = � j O | Λ ej | .

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 �→ α t ( a ) extends to an entire function z �→ α z ( 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 �→ α t ( a ) extends to an entire function z �→ α z ( a ). Analytic elements are dense in A . Definition (Haag–Hugenholtz–Winnink, 1967) For β > 0, a state φ of A is KMS β for α if φ ( ab ) = φ ( b α i β ( a )) for all analytic a , b . A KMS 0 -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 �→ α t ( a ) extends to an entire function z �→ α z ( a ). Analytic elements are dense in A . Definition (Haag–Hugenholtz–Winnink, 1967) For β > 0, a state φ of A is KMS β for α if φ ( ab ) = φ ( b α i β ( a )) for all analytic a , b . A KMS 0 -state is an α -invariant trace. Suffices to verify KMS condition for a , b in any α -invariant set A of analytic elements such that span A = A .

The preferred dynamics on C ∗ (Λ) Take r ∈ (0 , ∞ ) k . Universal property gives α r : R � C ∗ (Λ) s.t. α r t ( s f ) = e ir j t s f for f ∈ Λ e j .

The preferred dynamics on C ∗ (Λ) Take r ∈ (0 , ∞ ) k . Universal property gives α r : R � C ∗ (Λ) s.t. α r t ( s f ) = e ir j t s f for f ∈ Λ e j . In C ∗ (Λ), s µ := s µ 1 · · · s µ l for any factorisation µ = µ 1 · · · µ l into edges is well-defined. Relations give C ∗ (Λ) = span { s µ s ∗ ν : µ, ν ∈ Λ , s ( µ ) = s ( ν ) }

The preferred dynamics on C ∗ (Λ) Take r ∈ (0 , ∞ ) k . Universal property gives α r : R � C ∗ (Λ) s.t. α r t ( s f ) = e ir j t s f for f ∈ Λ e j . In C ∗ (Λ), s µ := s µ 1 · · · s µ l for any factorisation µ = µ 1 · · · µ l into edges is well-defined. Relations give C ∗ (Λ) = span { s µ s ∗ ν : µ, ν ∈ Λ , s ( µ ) = s ( ν ) } t ( s µ s ∗ ν ) = e ir · ( d ( µ ) − d ( ν )) t s µ s ∗ ν , the s µ s ∗ Since α r ν are analytic.

The preferred dynamics on C ∗ (Λ) Take r ∈ (0 , ∞ ) k . Universal property gives α r : R � C ∗ (Λ) s.t. α r t ( s f ) = e ir j t s f for f ∈ Λ e j . In C ∗ (Λ), s µ := s µ 1 · · · s µ l for any factorisation µ = µ 1 · · · µ l into edges is well-defined. Relations give C ∗ (Λ) = span { s µ s ∗ ν : µ, ν ∈ Λ , s ( µ ) = s ( ν ) } t ( s µ s ∗ ν ) = e ir · ( d ( µ ) − d ( ν )) t s µ s ∗ ν , the s µ s ∗ Since α r ν are analytic. If φ is KMS β for α r , then f ∈ v Λ ej φ ( s f s ∗ f ) = e − β r j � � φ ( p v ) = � � w ∈ Λ 0 A j ( v , w ) φ ( p w ) , � � so φ ( p v ) v is a common positive eigenvector of the A j .

The preferred dynamics on C ∗ (Λ) Take r ∈ (0 , ∞ ) k . Universal property gives α r : R � C ∗ (Λ) s.t. α r t ( s f ) = e ir j t s f for f ∈ Λ e j . In C ∗ (Λ), s µ := s µ 1 · · · s µ l for any factorisation µ = µ 1 · · · µ l into edges is well-defined. Relations give C ∗ (Λ) = span { s µ s ∗ ν : µ, ν ∈ Λ , s ( µ ) = s ( ν ) } t ( s µ s ∗ ν ) = e ir · ( d ( µ ) − d ( ν )) t s µ s ∗ ν , the s µ s ∗ Since α r ν are analytic. If φ is KMS β for α r , then f ∈ v Λ ej φ ( s f s ∗ f ) = e − β r j � � φ ( p v ) = � � w ∈ Λ 0 A j ( v , w ) φ ( p w ) , � � so φ ( p v ) v is a common positive eigenvector of the A j . Proposition (aHLRS) If φ is KMS β for α r , then β r = ln ρ (Λ), and φ is KMS 1 for the preferred dynamics α t ( s f ) = ρ ( A j ) it s f for f ∈ Λ e j .

Groupoids from k -graphs, and periodicity Infinite paths in Λ are maps x : { ( m , n ) ∈ N k × N k | 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 ) ∈ N k × N k | 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 ∈ N k Λ n − m . Have σ : N k � Λ ∞ where σ p ( x )( m , n ) = x ( m + p , n + p ).

Groupoids from k -graphs, and periodicity Infinite paths in Λ are maps x : { ( m , n ) ∈ N k × N k | 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 ∈ N k Λ n − m . Have σ : N k � Λ ∞ where σ p ( x )( m , n ) = x ( m + p , n + p ). The periodicity group Per(Λ) := { ( m − n ) : m , n ∈ N k , σ m = σ n } . is a subgroup of Z k .