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Partial Crossed Product Description of the Cuntz-Li Algebras - - PowerPoint PPT Presentation

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Partial Crossed Product Description of the Cuntz-Li Algebras Giuliano Boava Groups, Dynamical Systems and C -Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description of the Cuntz-Li Algebras

Giuliano Boava

Groups, Dynamical Systems and C∗-Algebras

M¨ unster - August 2013.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Contents

1

Preliminaries Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

2

Partial Group Algebra Description

3

Partial Crossed Product Description

4

Application in Bost-Connes Algebra

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Contents

1

Preliminaries Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

2

Partial Group Algebra Description

3

Partial Crossed Product Description

4

Application in Bost-Connes Algebra

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Contents

1

Preliminaries Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

2

Partial Group Algebra Description

3

Partial Crossed Product Description

4

Application in Bost-Connes Algebra

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Cuntz-Li Algebras: Definition

R integral domain with finite quotients, i.e., R/(m) is finite, for all m = 0 in R, which is not a field.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Cuntz-Li Algebras: Definition

R integral domain with finite quotients, i.e., R/(m) is finite, for all m = 0 in R, which is not a field. Definition (Cuntz-Li, 2010) The Cuntz-Li algebra of R, denoted by A[R], is the universal C∗-algebra generated by isometries {sm | m ∈ R×} and unitaries {un | n ∈ R} subject to the relations (CL1) smsm′ = smm′; (CL2) unun′ = un+n′; (CL3) smun = umnsm; (CL4)

  • l+(m)∈R/(m)

ulsms∗

mu−l = 1.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Cuntz-Li Algebras: Properties

There is a natural projection pm,m′ : R/(m′) − → R/(m) whenever m ≤ m′.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Cuntz-Li Algebras: Properties

There is a natural projection pm,m′ : R/(m′) − → R/(m) whenever m ≤ m′. ˆ R = lim

← −{R/(m), pm,m′} is the profinite completion of R.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Cuntz-Li Algebras: Properties

There is a natural projection pm,m′ : R/(m′) − → R/(m) whenever m ≤ m′. ˆ R = lim

← −{R/(m), pm,m′} is the profinite completion of R.

Theorem (Cuntz-Li, 2010) span{unsms∗

mu−n | m ∈ R×, n ∈ R} is a commutative

C∗-algebra and its spectrum is homeomorphic to ˆ R.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Cuntz-Li Algebras: Properties

Theorem (Cuntz-Li, 2010) A[R] is simple and purely infinite.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Cuntz-Li Algebras: Properties

Theorem (Cuntz-Li, 2010) A[R] is simple and purely infinite. Theorem (Cuntz-Li, 2010) A[R] is a crossed product by a semigroup.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Contents

1

Preliminaries Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

2

Partial Group Algebra Description

3

Partial Crossed Product Description

4

Application in Bost-Connes Algebra

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Partial Action

Definition A partial action α of a (discrete) group G on a C∗-algebra A is a collection (Dt)t∈G of ideals of A and ∗-isomorphisms αt : Dt−1 − → Dt such that (PA1) De = A; (PA2) α−1

t

(Dt ∩ Ds−1) ⊆ D(st)−1; (PA3) αs ◦ αt(x) = αst(x), ∀ x ∈ α−1

t

(Dt ∩ Ds−1).

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Partial Crossed Product

α partial action of a group G on a C∗-algebra A.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Partial Crossed Product

α partial action of a group G on a C∗-algebra A. Let L = ⊕t∈GDt and denote an element (at)t∈G by

  • t∈G atδt.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Partial Crossed Product

α partial action of a group G on a C∗-algebra A. Let L = ⊕t∈GDt and denote an element (at)t∈G by

  • t∈G atδt.

L is a ∗-algebra with the operations (asδs)(atδt) = αs(αs−1(as)at)δst and (atδt)∗ = αt−1(a∗

t )δt−1.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Partial Crossed Product

α partial action of a group G on a C∗-algebra A. Let L = ⊕t∈GDt and denote an element (at)t∈G by

  • t∈G atδt.

L is a ∗-algebra with the operations (asδs)(atδt) = αs(αs−1(as)at)δst and (atδt)∗ = αt−1(a∗

t )δt−1.

Definition The full partial crossed product and the reduced partial crossed product of A by G through α, denoted by A⋊αG and A⋊α,rG, are the completion of L under certain C∗-norms.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Partial Representation

Definition A partial representation π of a (discrete) group G into a unital C∗-algebra B is a map π : G − → B such that, for all s, t ∈ G, (PR1) π(e) = 1; (PR2) π(t−1) = π(t)∗; (PR3) π(s)π(t)π(t−1) = π(st)π(t−1).

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Universal Property of A⋊αG

Definition Let π : G − → B be a partial representation of G into a unital C∗-algebra B and ϕ : A − → B be a ∗-homomorphism. We say that the pair (ϕ, π) is α-covariant if: (COV1) ϕ(αt(x)) = π(t)ϕ(x)π(t−1), for all t ∈ G e x ∈ Dt−1; (COV2) ϕ(x)π(t)π(t−1) = π(t)π(t−1)ϕ(x), for all x ∈ A e t ∈ G.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Universal Property of A⋊αG

Definition Let π : G − → B be a partial representation of G into a unital C∗-algebra B and ϕ : A − → B be a ∗-homomorphism. We say that the pair (ϕ, π) is α-covariant if: (COV1) ϕ(αt(x)) = π(t)ϕ(x)π(t−1), for all t ∈ G e x ∈ Dt−1; (COV2) ϕ(x)π(t)π(t−1) = π(t)π(t−1)ϕ(x), for all x ∈ A e t ∈ G. Proposition If (ϕ, π) is α-covariant pair, then there exists a unique ∗-homomorphism ϕ × π : A⋊αG − → B such that (ϕ × π)(atδt) = ϕ(at)π(t), ∀ t ∈ G, ∀ at ∈ Dt.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Contents

1

Preliminaries Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

2

Partial Group Algebra Description

3

Partial Crossed Product Description

4

Application in Bost-Connes Algebra

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Partial Group Algebra

Given a (discrete) group G, define G = {[t] | t ∈ G}.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Partial Group Algebra

Given a (discrete) group G, define G = {[t] | t ∈ G}. Definition (Exel-Laca-Quigg, 2002) The partial group algebra of G, denoted by C∗

p(G), is defined

to be the universal C∗-algebra generated by the set G subject to the relations Rp = {[e] = 1} ∪ {[t−1] = [t]∗}t∈G ∪ {[s][t][t−1] = [st][t−1]}s,t∈G.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Partial Group Algebra with Relations

Denote [t][t−1] by εt.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Partial Group Algebra with Relations

Denote [t][t−1] by εt. Let R be a set of relations on G such that every relation is

  • f the form
  • i

λi

  • j

εtij = 0.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Partial Group Algebra with Relations

Denote [t][t−1] by εt. Let R be a set of relations on G such that every relation is

  • f the form
  • i

λi

  • j

εtij = 0. Definition (Exel-Laca-Quigg, 2002) The partial group algebra of G with relations R, denoted by C∗

p(G, R), is defined to be the universal C∗-algebra generated

by the set G with the relations Rp ∪ R.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Theorems

Theorem (Exel-Laca-Quigg, 2002) C∗

p(G) ∼

= C(X) ⋊α G, where X = {ξ ⊆ G | e ∈ ξ}.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

Theorems

Theorem (Exel-Laca-Quigg, 2002) C∗

p(G) ∼

= C(X) ⋊α G, where X = {ξ ⊆ G | e ∈ ξ}. Theorem (Exel-Laca-Quigg, 2002) C∗

p(G, R) ∼

= C(Ω) ⋊α G, where Ω = {ξ ∈ X | f(t−1ξ) = 0, ∀ f ∈ R, ∀ t ∈ ξ}.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Contents

1

Preliminaries Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

2

Partial Group Algebra Description

3

Partial Crossed Product Description

4

Application in Bost-Connes Algebra

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Group Algebra Description

R integral domain with finite quotients which is not a field.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Group Algebra Description

R integral domain with finite quotients which is not a field. K field of fractions of R.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Group Algebra Description

R integral domain with finite quotients which is not a field. K field of fractions of R. Semidirect product K ⋊ K ×.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Group Algebra Description

R integral domain with finite quotients which is not a field. K field of fractions of R. Semidirect product K ⋊ K ×. Set of relations R = R1 ∪ R2 ∪ R3, where R1 =

  • ε(n,1) = 1 | n ∈ R
  • , R2 =
  • ε(0, 1

m) = 1

  • m ∈ R×

and R3 =   

  • l+(m)∈R/(m)

ε(l,m) = 1

  • m ∈ R×

  .

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Group Algebra Description

R integral domain with finite quotients which is not a field. K field of fractions of R. Semidirect product K ⋊ K ×. Set of relations R = R1 ∪ R2 ∪ R3, where R1 =

  • ε(n,1) = 1 | n ∈ R
  • , R2 =
  • ε(0, 1

m) = 1

  • m ∈ R×

and R3 =   

  • l+(m)∈R/(m)

ε(l,m) = 1

  • m ∈ R×

  . Partial group algebra C∗

p(K ⋊ K ×, R).

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Group Algebra Description

Proposition (B.-Exel, 2013) There exists a ∗-isomorphism A[R] − → C∗

p(K ⋊ K ×, R)

un − → [n, 1] sm − → [0, m] s∗

m′unsm

← −

  • n

m′ , m m′

  • .

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Sketch of the Proof

Let’s check (CL3) smun = umnsm:

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Sketch of the Proof

Let’s check (CL3) smun = umnsm: smun − → [0, m][n, 1] = [0, m][n, 1][n, 1]∗[n, 1] = [mn, m][n, 1]∗[n, 1] = [mn, m],

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Sketch of the Proof

Let’s check (CL3) smun = umnsm: smun − → [0, m][n, 1] = [0, m][n, 1][n, 1]∗[n, 1] = [mn, m][n, 1]∗[n, 1] = [mn, m], umnsm − → [mn, 1][0, m] = [mn, 1][mn, 1]∗[mn, 1][0, m] = [mn, 1][mn, 1]∗[mn, m] = [mn, m].

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Sketch of the Proof

Let’s check (PR3) [s][t][t]∗ = [st][t]∗:

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Sketch of the Proof

Let’s check (PR3) [s][t][t]∗ = [st][t]∗: With s =

  • q

p′ , p p′

  • and t =

n

m′ , m m′

  • , we have

st =

  • m′q+pn

p′m′ , pm p′m′

  • ;

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Sketch of the Proof

Let’s check (PR3) [s][t][t]∗ = [st][t]∗: With s =

  • q

p′ , p p′

  • and t =

n

m′ , m m′

  • , we have

st =

  • m′q+pn

p′m′ , pm p′m′

  • ;

[st][t]∗ − →

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Sketch of the Proof

Let’s check (PR3) [s][t][t]∗ = [st][t]∗: With s =

  • q

p′ , p p′

  • and t =

n

m′ , m m′

  • , we have

st =

  • m′q+pn

p′m′ , pm p′m′

  • ;

[st][t]∗ − → (s∗

p′m′um′q+pnspm)(s∗ m′unsm)∗

=

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Sketch of the Proof

Let’s check (PR3) [s][t][t]∗ = [st][t]∗: With s =

  • q

p′ , p p′

  • and t =

n

m′ , m m′

  • , we have

st =

  • m′q+pn

p′m′ , pm p′m′

  • ;

[st][t]∗ − → (s∗

p′m′um′q+pnspm)(s∗ m′unsm)∗

= s∗

p′uqs∗ m′spunsms∗ mu−nsm′

=

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Sketch of the Proof

Let’s check (PR3) [s][t][t]∗ = [st][t]∗: With s =

  • q

p′ , p p′

  • and t =

n

m′ , m m′

  • , we have

st =

  • m′q+pn

p′m′ , pm p′m′

  • ;

[st][t]∗ − → (s∗

p′m′um′q+pnspm)(s∗ m′unsm)∗

= s∗

p′uqs∗ m′spunsms∗ mu−nsm′

= s∗

p′uqs∗ m′sp unsms∗ mu−n

  • sm′s∗

m′

sm′ =

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Sketch of the Proof

Let’s check (PR3) [s][t][t]∗ = [st][t]∗: With s =

  • q

p′ , p p′

  • and t =

n

m′ , m m′

  • , we have

st =

  • m′q+pn

p′m′ , pm p′m′

  • ;

[st][t]∗ − → (s∗

p′m′um′q+pnspm)(s∗ m′unsm)∗

= s∗

p′uqs∗ m′spunsms∗ mu−nsm′

= s∗

p′uqs∗ m′sp unsms∗ mu−n

  • sm′s∗

m′

sm′ = s∗

p′uqs∗ m′spsm′s∗ m′unsms∗ mu−nsm′

=

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Sketch of the Proof

Let’s check (PR3) [s][t][t]∗ = [st][t]∗: With s =

  • q

p′ , p p′

  • and t =

n

m′ , m m′

  • , we have

st =

  • m′q+pn

p′m′ , pm p′m′

  • ;

[st][t]∗ − → (s∗

p′m′um′q+pnspm)(s∗ m′unsm)∗

= s∗

p′uqs∗ m′spunsms∗ mu−nsm′

= s∗

p′uqs∗ m′sp unsms∗ mu−n

  • sm′s∗

m′

sm′ = s∗

p′uqs∗ m′spsm′s∗ m′unsms∗ mu−nsm′

= (s∗

p′uqsp)(s∗ m′unsm)(s∗ mu−nsm′)

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Sketch of the Proof

Let’s check (PR3) [s][t][t]∗ = [st][t]∗: With s =

  • q

p′ , p p′

  • and t =

n

m′ , m m′

  • , we have

st =

  • m′q+pn

p′m′ , pm p′m′

  • ;

[st][t]∗ − → (s∗

p′m′um′q+pnspm)(s∗ m′unsm)∗

= s∗

p′uqs∗ m′spunsms∗ mu−nsm′

= s∗

p′uqs∗ m′sp unsms∗ mu−n

  • sm′s∗

m′

sm′ = s∗

p′uqs∗ m′spsm′s∗ m′unsms∗ mu−nsm′

= (s∗

p′uqsp)(s∗ m′unsm)(s∗ mu−nsm′)

← − [s][t][t]∗.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Contents

1

Preliminaries Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

2

Partial Group Algebra Description

3

Partial Crossed Product Description

4

Application in Bost-Connes Algebra

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

Corollary A[R] is ∗-isomorphic to C(Ω) ⋊α K ⋊ K ×.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

Corollary A[R] is ∗-isomorphic to C(Ω) ⋊α K ⋊ K ×. Now, we characterize Ω.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

Corollary A[R] is ∗-isomorphic to C(Ω) ⋊α K ⋊ K ×. Now, we characterize Ω. Extend the partial order from R× to K ×. For w, w′ ∈ K ×, w ≤ w′ if there exists r ∈ R such that w′ = wr.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

Corollary A[R] is ∗-isomorphic to C(Ω) ⋊α K ⋊ K ×. Now, we characterize Ω. Extend the partial order from R× to K ×. For w, w′ ∈ K ×, w ≤ w′ if there exists r ∈ R such that w′ = wr. Consider the fractional ideals (w) = wR, w ∈ K ×.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

Corollary A[R] is ∗-isomorphic to C(Ω) ⋊α K ⋊ K ×. Now, we characterize Ω. Extend the partial order from R× to K ×. For w, w′ ∈ K ×, w ≤ w′ if there exists r ∈ R such that w′ = wr. Consider the fractional ideals (w) = wR, w ∈ K ×. There is a natural projection pw,w′ : (R + (w′))/(w′) − → (R + (w))/(w) whenever w ≤ w′.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

Corollary A[R] is ∗-isomorphic to C(Ω) ⋊α K ⋊ K ×. Now, we characterize Ω. Extend the partial order from R× to K ×. For w, w′ ∈ K ×, w ≤ w′ if there exists r ∈ R such that w′ = wr. Consider the fractional ideals (w) = wR, w ∈ K ×. There is a natural projection pw,w′ : (R + (w′))/(w′) − → (R + (w))/(w) whenever w ≤ w′. ˆ RK = lim

← −{(R + (w))/(w), pw,w′}.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

Corollary A[R] is ∗-isomorphic to C(Ω) ⋊α K ⋊ K ×. Now, we characterize Ω. Extend the partial order from R× to K ×. For w, w′ ∈ K ×, w ≤ w′ if there exists r ∈ R such that w′ = wr. Consider the fractional ideals (w) = wR, w ∈ K ×. There is a natural projection pw,w′ : (R + (w′))/(w′) − → (R + (w))/(w) whenever w ≤ w′. ˆ RK = lim

← −{(R + (w))/(w), pw,w′}.

Clearly, ˆ RK ∼ = ˆ R.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

Proposition Ω is homeomorphic to ˆ RK and, hence, to ˆ R.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

Proposition Ω is homeomorphic to ˆ RK and, hence, to ˆ R. Corollary There exists a ∗-isomorphism A[R] − → C(ˆ RK) ⋊α K ⋊ K × un − → 1δ(n,1) sm − → 1(0,m)δ(0,m), where 1(u,w) is the characteristic function of {(uw′ + (w′))w′ ∈ ˆ RK | uw + (w) = u + (w)}.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

Proposition The partial action θ on ˆ RK is topologically free and minimal.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

Proposition The partial action θ on ˆ RK is topologically free and minimal. Corollary A[R] is simple.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Contents

1

Preliminaries Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras

2

Partial Group Algebra Description

3

Partial Crossed Product Description

4

Application in Bost-Connes Algebra

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Bost-Connes Algebra

Definition (Bost-Connes, 1995) The Bost-Connes algebra, denoted by CQ, is the universal C∗-algebra generated by isometries {µm | m ∈ N∗} and unitaries {eγ | γ ∈ Q/Z} subject to the relations (BC1) µmµm′ = µmm′; (BC2) µmµ∗

m′ = µ∗ m′µm, if (m, m′) = 1;

(BC3) eγeγ′ = eγ+γ′; (BC4) eγµm = µmemγ; (BC5) µmeγµ∗

m = 1 m

eδ, where the sum is taken over all δ ∈ Q/Z such that mδ = γ.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

Taking R = Z, we have A[Z] ∼ = C(ˆ ZQ) ⋊α Q ⋊ Q∗.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

Taking R = Z, we have A[Z] ∼ = C(ˆ ZQ) ⋊α Q ⋊ Q∗. There is a natural embedding Q∗

+ ֒

→ Q ⋊ Q∗ given by q → (0, q).

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

Taking R = Z, we have A[Z] ∼ = C(ˆ ZQ) ⋊α Q ⋊ Q∗. There is a natural embedding Q∗

+ ֒

→ Q ⋊ Q∗ given by q → (0, q). Restricting α to Q∗

+, we obtain the partial crossed product

C(ˆ ZQ) ⋊ Q∗

+.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

Taking R = Z, we have A[Z] ∼ = C(ˆ ZQ) ⋊α Q ⋊ Q∗. There is a natural embedding Q∗

+ ֒

→ Q ⋊ Q∗ given by q → (0, q). Restricting α to Q∗

+, we obtain the partial crossed product

C(ˆ ZQ) ⋊ Q∗

+.

Theorem The Bost-Connes algebra CQ is ∗-isomorphic to C(ˆ ZQ) ⋊ Q∗

+.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

One side of the isomorphism is given by CQ − → C(ˆ ZQ) ⋊ Q∗

+

µm − → 1(0,m)δm e(n/m) − →

  • l+(m)∈Z/(m)

exp

  • −ln

m · 2πi

  • 1(l,m)δ1.

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

Partial Crossed Product Description

The other side is given by C(ˆ ZQ) ⋊ Q∗

+

− → CQ δm/m′ − → µ∗

m′µm

1(n/m′,m/m′) − → 1 m

  • l+(m)∈Z/(m)

exp nl m · 2πi

  • e

lm′ m

  • .

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra

THE END!

Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras