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

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

Preliminaries Partial Group Algebra Description Partial Crossed Product Description Application in Bost-Connes Algebra Contents Preliminaries 1 Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras Partial Group Algebra Description 2 Partial Crossed Product Description 3 Application in Bost-Connes Algebra 4 Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

Preliminaries Cuntz-Li Algebras Partial Group Algebra Description Partial Crossed Products Partial Crossed Product Description Partial Group Algebras Application in Bost-Connes Algebra Contents Preliminaries 1 Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras Partial Group Algebra Description 2 Partial Crossed Product Description 3 Application in Bost-Connes Algebra 4 Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

Preliminaries Cuntz-Li Algebras Partial Group Algebra Description Partial Crossed Products Partial Crossed Product Description Partial Group Algebras Application in Bost-Connes Algebra Contents Preliminaries 1 Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras Partial Group Algebra Description 2 Partial Crossed Product Description 3 Application in Bost-Connes Algebra 4 Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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

Preliminaries Cuntz-Li Algebras Partial Group Algebra Description Partial Crossed Products Partial Crossed Product Description Partial Group Algebras Application in Bost-Connes Algebra 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 { s m | m ∈ R × } and unitaries { u n | n ∈ R } subject to the relations (CL1) s m s m ′ = s mm ′ ; (CL2) u n u n ′ = u n + n ′ ; (CL3) s m u n = u mn s m ; � m u − l = 1. u l s m s ∗ (CL4) l +( m ) ∈ R / ( m ) Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

Preliminaries Cuntz-Li Algebras Partial Group Algebra Description Partial Crossed Products Partial Crossed Product Description Partial Group Algebras Application in Bost-Connes Algebra Cuntz-Li Algebras: Properties There is a natural projection p m , m ′ : R / ( m ′ ) − → R / ( m ) whenever m ≤ m ′ . Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

Preliminaries Cuntz-Li Algebras Partial Group Algebra Description Partial Crossed Products Partial Crossed Product Description Partial Group Algebras Application in Bost-Connes Algebra Cuntz-Li Algebras: Properties There is a natural projection p m , m ′ : R / ( m ′ ) − → R / ( m ) whenever m ≤ m ′ . ˆ R = lim − { R / ( m ) , p m , m ′ } is the profinite completion of R . ← Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

Preliminaries Cuntz-Li Algebras Partial Group Algebra Description Partial Crossed Products Partial Crossed Product Description Partial Group Algebras Application in Bost-Connes Algebra Cuntz-Li Algebras: Properties There is a natural projection p m , m ′ : R / ( m ′ ) − → R / ( m ) whenever m ≤ m ′ . ˆ R = lim − { R / ( m ) , p m , m ′ } is the profinite completion of R . ← Theorem (Cuntz-Li, 2010) m u − n | m ∈ R × , n ∈ R } is a commutative span { u n s m s ∗ C ∗ -algebra and its spectrum is homeomorphic to ˆ R. Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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

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

Preliminaries Cuntz-Li Algebras Partial Group Algebra Description Partial Crossed Products Partial Crossed Product Description Partial Group Algebras Application in Bost-Connes Algebra Contents Preliminaries 1 Cuntz-Li Algebras Partial Crossed Products Partial Group Algebras Partial Group Algebra Description 2 Partial Crossed Product Description 3 Application in Bost-Connes Algebra 4 Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

Preliminaries Cuntz-Li Algebras Partial Group Algebra Description Partial Crossed Products Partial Crossed Product Description Partial Group Algebras Application in Bost-Connes Algebra Partial Action Definition A partial action α of a (discrete) group G on a C ∗ -algebra A is a collection ( D t ) t ∈ G of ideals of A and ∗ -isomorphisms α t : D t − 1 − → D t such that (PA1) D e = A ; (PA2) α − 1 ( D t ∩ D s − 1 ) ⊆ D ( st ) − 1 ; t (PA3) α s ◦ α t ( x ) = α st ( x ) , ∀ x ∈ α − 1 ( D t ∩ D s − 1 ) . t Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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

Preliminaries Cuntz-Li Algebras Partial Group Algebra Description Partial Crossed Products Partial Crossed Product Description Partial Group Algebras Application in Bost-Connes Algebra Partial Crossed Product α partial action of a group G on a C ∗ -algebra A . Let L = ⊕ t ∈ G D t and denote an element ( a t ) t ∈ G by � t ∈ G a t δ t . Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

Preliminaries Cuntz-Li Algebras Partial Group Algebra Description Partial Crossed Products Partial Crossed Product Description Partial Group Algebras Application in Bost-Connes Algebra Partial Crossed Product α partial action of a group G on a C ∗ -algebra A . Let L = ⊕ t ∈ G D t and denote an element ( a t ) t ∈ G by � t ∈ G a t δ t . L is a ∗ -algebra with the operations ( a s δ s )( a t δ t ) = α s ( α s − 1 ( a s ) a t ) δ st and ( a t δ t ) ∗ = α t − 1 ( a ∗ t ) δ t − 1 . Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

Preliminaries Cuntz-Li Algebras Partial Group Algebra Description Partial Crossed Products Partial Crossed Product Description Partial Group Algebras Application in Bost-Connes Algebra Partial Crossed Product α partial action of a group G on a C ∗ -algebra A . Let L = ⊕ t ∈ G D t and denote an element ( a t ) t ∈ G by � t ∈ G a t δ t . L is a ∗ -algebra with the operations ( a s δ s )( a t δ t ) = α s ( α s − 1 ( a s ) a t ) δ st and ( a t δ t ) ∗ = α t − 1 ( a ∗ t ) δ t − 1 . Definition The full partial crossed product and the reduced partial crossed product of A by G through α , denoted by A ⋊ α G and A ⋊ α, r G , are the completion of L under certain C ∗ -norms. Giuliano Boava Partial Crossed Product Description of the Cuntz-Li Algebras

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