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Classification of C*-envelopes of tensor algebras arising from stochastic matrices Daniel Markiewicz (Ben-Gurion Univ. of the Negev) Joint Work with Adam Dor-On (Univ. of Waterloo) Recent Advances in Operator Theory and Operator Algebras 2016

Classification of C*-envelopes of tensor algebras arising from stochastic matrices Daniel Markiewicz (Ben-Gurion Univ. of the Negev) Joint Work with Adam Dor-On (Univ. of Waterloo) Recent Advances in Operator Theory and Operator Algebras 2016 Indian Statistical Institute, Bangalore Daniel Markiewicz Classification of C*-env. of T + ( P ) Rec. Adv. in OT & OA 2016 1 / 23
Main Goal Paper details Dor-On-M.’16 Adam Dor-On and Daniel Markiewicz, “C*-envelopes of tensor algebras arising from stochastic matrices”, arXiv:1605.03543 [math.OA]. General Problem What is the C*-envelope of the Tensor Algebra of the subproduct system over N arising from a stochastic matrix? There are some surprises when compared to the situation of product systems over N . Daniel Markiewicz Classification of C*-env. of T + ( P ) Rec. Adv. in OT & OA 2016 2 / 23
Main Goal Paper details Dor-On-M.’16 Adam Dor-On and Daniel Markiewicz, “C*-envelopes of tensor algebras arising from stochastic matrices”, arXiv:1605.03543 [math.OA]. General Problem What is the C*-envelope of the Tensor Algebra of the subproduct system over N arising from a stochastic matrix? There are some surprises when compared to the situation of product systems over N . Daniel Markiewicz Classification of C*-env. of T + ( P ) Rec. Adv. in OT & OA 2016 2 / 23
Basic framework Subproduct systems Definition (Shalit-Solel ’09, Bhat-Mukherjee ’10) Let M be a vN algebra, let X = ( X n ) n ∈ N be a family of W*-correspondences over M , and let U = ( U m,n : X m ⊗ X n → X m + n ) be a family of bounded M -linear maps. We say that X is a subproduct system over M if for all m, n, p ∈ N , 1 X 0 = M 2 U m,n is co-isometric 3 The family U “behaves like multiplication”: U m, 0 and U 0 ,n are the right/left multiplications and U m + n,p ( U m,n ⊗ I p ) = U m,n + p ( I m ⊗ U n,p ) When U m,n is unitary for all m, n we say that X is a product system. Daniel Markiewicz Classification of C*-env. of T + ( P ) Rec. Adv. in OT & OA 2016 3 / 23
Basic framework Subproduct systems Theorem (Muhly-Solel ’02, Solel-Shalit ’09) Let M be a vN algebra. Suppose that θ : M → M is a unital normal CP map. Then there exits a canonical subproduct system structure on the family of Arveson-Stinespring correspondences associated to ( θ n ) n ∈ N . Definition Given a countable (possibly infinite) set Ω , a stochastic matrix over Ω is a function P : Ω × Ω → R such that P ij ≥ 0 for all i, j and � j ∈ Ω P ij = 1 for all i . Subproduct system of a stochastic matrix There is a 1 - 1 correspondence between ucp maps of ℓ ∞ (Ω) into itself and stochastic matrices over Ω given by � θ P ( f )( i ) = P ij f ( j ) j ∈ Ω Hence, a stochastic P gives rise to a canonical subproduct system Arv ( P ) . Daniel Markiewicz Classification of C*-env. of T + ( P ) Rec. Adv. in OT & OA 2016 4 / 23
Basic framework Subproduct systems Theorem (Muhly-Solel ’02, Solel-Shalit ’09) Let M be a vN algebra. Suppose that θ : M → M is a unital normal CP map. Then there exits a canonical subproduct system structure on the family of Arveson-Stinespring correspondences associated to ( θ n ) n ∈ N . Definition Given a countable (possibly infinite) set Ω , a stochastic matrix over Ω is a function P : Ω × Ω → R such that P ij ≥ 0 for all i, j and � j ∈ Ω P ij = 1 for all i . Subproduct system of a stochastic matrix There is a 1 - 1 correspondence between ucp maps of ℓ ∞ (Ω) into itself and stochastic matrices over Ω given by � θ P ( f )( i ) = P ij f ( j ) j ∈ Ω Hence, a stochastic P gives rise to a canonical subproduct system Arv ( P ) . Daniel Markiewicz Classification of C*-env. of T + ( P ) Rec. Adv. in OT & OA 2016 4 / 23
Basic framework Subproduct systems Theorem (Muhly-Solel ’02, Solel-Shalit ’09) Let M be a vN algebra. Suppose that θ : M → M is a unital normal CP map. Then there exits a canonical subproduct system structure on the family of Arveson-Stinespring correspondences associated to ( θ n ) n ∈ N . Definition Given a countable (possibly infinite) set Ω , a stochastic matrix over Ω is a function P : Ω × Ω → R such that P ij ≥ 0 for all i, j and � j ∈ Ω P ij = 1 for all i . Subproduct system of a stochastic matrix There is a 1 - 1 correspondence between ucp maps of ℓ ∞ (Ω) into itself and stochastic matrices over Ω given by � θ P ( f )( i ) = P ij f ( j ) j ∈ Ω Hence, a stochastic P gives rise to a canonical subproduct system Arv ( P ) . Daniel Markiewicz Classification of C*-env. of T + ( P ) Rec. Adv. in OT & OA 2016 4 / 23
Basic framework Tensor, Toeplitz and Cuntz-Pimsner algebras Given a subproduct system ( X, U ) , we define the Fock W*-correspondence ∞ � F X = X n n =0 Define for every ξ ∈ X m the shift operator S ( m ) ψ = U m,n ( ξ ⊗ ψ ) , ψ ∈ X n ξ Tensor algebra (not self-adjoint): �·� M ∪ { S ( m ) T + ( X ) = Alg | ∀ ξ ∈ X m , ∀ m } ξ Toeplitz algebra: T ( X ) = C ∗ ( T + ( X )) Cuntz-Pimsner algebra: O ( X ) = T ( X ) / J ( X ) for appropriate J ( X ) For the case of subproduct systems, Viselter ’12 defined the ideal J ( X ) as follows: let Q n denote the orthogonal projection onto the n th summand of Fock module: J ( X ) = { T ∈ T ( X ) : lim n →∞ � TQ n � = 0 } . Daniel Markiewicz Classification of C*-env. of T + ( P ) Rec. Adv. in OT & OA 2016 5 / 23
Basic framework Tensor, Toeplitz and Cuntz-Pimsner algebras Given a subproduct system ( X, U ) , we define the Fock W*-correspondence ∞ � F X = X n n =0 Define for every ξ ∈ X m the shift operator S ( m ) ψ = U m,n ( ξ ⊗ ψ ) , ψ ∈ X n ξ Tensor algebra (not self-adjoint): �·� M ∪ { S ( m ) T + ( X ) = Alg | ∀ ξ ∈ X m , ∀ m } ξ Toeplitz algebra: T ( X ) = C ∗ ( T + ( X )) Cuntz-Pimsner algebra: O ( X ) = T ( X ) / J ( X ) for appropriate J ( X ) For the case of subproduct systems, Viselter ’12 defined the ideal J ( X ) as follows: let Q n denote the orthogonal projection onto the n th summand of Fock module: J ( X ) = { T ∈ T ( X ) : lim n →∞ � TQ n � = 0 } . Daniel Markiewicz Classification of C*-env. of T + ( P ) Rec. Adv. in OT & OA 2016 5 / 23
Basic framework Tensor, Toeplitz and Cuntz-Pimsner algebras Given a subproduct system ( X, U ) , we define the Fock W*-correspondence ∞ � F X = X n n =0 Define for every ξ ∈ X m the shift operator S ( m ) ψ = U m,n ( ξ ⊗ ψ ) , ψ ∈ X n ξ Tensor algebra (not self-adjoint): �·� M ∪ { S ( m ) T + ( X ) = Alg | ∀ ξ ∈ X m , ∀ m } ξ Toeplitz algebra: T ( X ) = C ∗ ( T + ( X )) Cuntz-Pimsner algebra: O ( X ) = T ( X ) / J ( X ) for appropriate J ( X ) For the case of subproduct systems, Viselter ’12 defined the ideal J ( X ) as follows: let Q n denote the orthogonal projection onto the n th summand of Fock module: J ( X ) = { T ∈ T ( X ) : lim n →∞ � TQ n � = 0 } . Daniel Markiewicz Classification of C*-env. of T + ( P ) Rec. Adv. in OT & OA 2016 5 / 23
Basic framework Tensor, Toeplitz and Cuntz-Pimsner algebras Given a subproduct system ( X, U ) , we define the Fock W*-correspondence ∞ � F X = X n n =0 Define for every ξ ∈ X m the shift operator S ( m ) ψ = U m,n ( ξ ⊗ ψ ) , ψ ∈ X n ξ Tensor algebra (not self-adjoint): �·� M ∪ { S ( m ) T + ( X ) = Alg | ∀ ξ ∈ X m , ∀ m } ξ Toeplitz algebra: T ( X ) = C ∗ ( T + ( X )) Cuntz-Pimsner algebra: O ( X ) = T ( X ) / J ( X ) for appropriate J ( X ) For the case of subproduct systems, Viselter ’12 defined the ideal J ( X ) as follows: let Q n denote the orthogonal projection onto the n th summand of Fock module: J ( X ) = { T ∈ T ( X ) : lim n →∞ � TQ n � = 0 } . Daniel Markiewicz Classification of C*-env. of T + ( P ) Rec. Adv. in OT & OA 2016 5 / 23
Basic framework Tensor, Toeplitz and Cuntz-Pimsner algebras Given a subproduct system ( X, U ) , we define the Fock W*-correspondence ∞ � F X = X n n =0 Define for every ξ ∈ X m the shift operator S ( m ) ψ = U m,n ( ξ ⊗ ψ ) , ψ ∈ X n ξ Tensor algebra (not self-adjoint): �·� M ∪ { S ( m ) T + ( X ) = Alg | ∀ ξ ∈ X m , ∀ m } ξ Toeplitz algebra: T ( X ) = C ∗ ( T + ( X )) Cuntz-Pimsner algebra: O ( X ) = T ( X ) / J ( X ) for appropriate J ( X ) For the case of subproduct systems, Viselter ’12 defined the ideal J ( X ) as follows: let Q n denote the orthogonal projection onto the n th summand of Fock module: J ( X ) = { T ∈ T ( X ) : lim n →∞ � TQ n � = 0 } . Daniel Markiewicz Classification of C*-env. of T + ( P ) Rec. Adv. in OT & OA 2016 5 / 23
Basic framework Tensor, Toeplitz and Cuntz-Pimsner algebras Given a subproduct system ( X, U ) , we define the Fock W*-correspondence ∞ � F X = X n n =0 Define for every ξ ∈ X m the shift operator S ( m ) ψ = U m,n ( ξ ⊗ ψ ) , ψ ∈ X n ξ Tensor algebra (not self-adjoint): �·� M ∪ { S ( m ) T + ( X ) = Alg | ∀ ξ ∈ X m , ∀ m } ξ Toeplitz algebra: T ( X ) = C ∗ ( T + ( X )) Cuntz-Pimsner algebra: O ( X ) = T ( X ) / J ( X ) for appropriate J ( X ) For the case of subproduct systems, Viselter ’12 defined the ideal J ( X ) as follows: let Q n denote the orthogonal projection onto the n th summand of Fock module: J ( X ) = { T ∈ T ( X ) : lim n →∞ � TQ n � = 0 } . Daniel Markiewicz Classification of C*-env. of T + ( P ) Rec. Adv. in OT & OA 2016 5 / 23

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