Subproduct systems and superproduct systems (or: behind the scenes - PowerPoint PPT Presentation

Subproduct systems and dilations The GNS representation ( E , ξ ) of a CP map Let T : B → B be a CP map. Then there exists a unique W*-correspondence 2 E over B , and a vector ξ ∈ E , such that s = E span B ξ B and � ξ, b ξ � = T ( b ) for all b ∈ B . Construction : on E 0 = B ⊗ B put inner product � a ⊗ b , c ⊗ d � = b ∗ T ( a ∗ c ) d and bimodule operation a ( x ⊗ y ) d = ax ⊗ yd . Complete the quotient, and put ξ = 1 ⊗ 1. This works: � ξ, b ξ � 2 A bimodule over B , that has a B -valued inner product. Equivalently, one may use Skeide’s von Neumann modules (and we do). 10 / 27

Subproduct systems and dilations The GNS representation ( E , ξ ) of a CP map Let T : B → B be a CP map. Then there exists a unique W*-correspondence 2 E over B , and a vector ξ ∈ E , such that s = E span B ξ B and � ξ, b ξ � = T ( b ) for all b ∈ B . Construction : on E 0 = B ⊗ B put inner product � a ⊗ b , c ⊗ d � = b ∗ T ( a ∗ c ) d and bimodule operation a ( x ⊗ y ) d = ax ⊗ yd . Complete the quotient, and put ξ = 1 ⊗ 1. This works: � ξ, b ξ � = � 1 ⊗ 1 , b ⊗ 1 � 2 A bimodule over B , that has a B -valued inner product. Equivalently, one may use Skeide’s von Neumann modules (and we do). 10 / 27

Subproduct systems and dilations The GNS representation ( E , ξ ) of a CP map Let T : B → B be a CP map. Then there exists a unique W*-correspondence 2 E over B , and a vector ξ ∈ E , such that s = E span B ξ B and � ξ, b ξ � = T ( b ) for all b ∈ B . Construction : on E 0 = B ⊗ B put inner product � a ⊗ b , c ⊗ d � = b ∗ T ( a ∗ c ) d and bimodule operation a ( x ⊗ y ) d = ax ⊗ yd . Complete the quotient, and put ξ = 1 ⊗ 1. This works: � ξ, b ξ � = � 1 ⊗ 1 , b ⊗ 1 � = 1 ∗ T ( 1 ∗ b ) 1 = T ( b ) . 2 A bimodule over B , that has a B -valued inner product. Equivalently, one may use Skeide’s von Neumann modules (and we do). 10 / 27

Subproduct systems and dilations The GNS representation ( E , ξ ) of a CP map Let T : B → B be a CP map. Then there exists a unique W*-correspondence 3 E over B , and a vector ξ ∈ E , such that s = E span B ξ B and � ξ, b ξ � = T ( b ) for all b ∈ B . Construction : on E 0 = B ⊗ B put inner product � a ⊗ b , c ⊗ d � = b ∗ T ( a ∗ c ) d and bimodule operation a ( x ⊗ y ) d = ax ⊗ yd . Complete the quotient, and put ξ = 1 ⊗ 1. This works: � ξ, b ξ � = � 1 ⊗ 1 , b ⊗ 1 � = 1 ∗ T ( 1 ∗ b ) 1 = T ( b ) . 3 A bimodule over B , that has a B -valued inner product. Equivalently, one may use Skeide’s von Neumann modules (and we do). 11 / 27

Subproduct systems and dilations The GNS representation of a CP-semigroup Let T = ( T s ) s ∈ S be a CP-semigroup on B . For every s , let ( E s , ξ s ) be the GNS representation of T s . 12 / 27

Subproduct systems and dilations The GNS representation of a CP-semigroup Let T = ( T s ) s ∈ S be a CP-semigroup on B . For every s , let ( E s , ξ s ) be the GNS representation of T s . For s , t ∈ S , define w s , t : E s + t → E s ⊙ E t ( really E s ⊙ s E t ) by w s , t : a ξ s + t b �→ a ξ s ⊙ ξ t b , and then extend linearly. 12 / 27

Subproduct systems and dilations The GNS representation of a CP-semigroup Let T = ( T s ) s ∈ S be a CP-semigroup on B . For every s , let ( E s , ξ s ) be the GNS representation of T s . For s , t ∈ S , define w s , t : E s + t → E s ⊙ E t ( really E s ⊙ s E t ) by w s , t : a ξ s + t b �→ a ξ s ⊙ ξ t b , and then extend linearly. We check: � a ξ s ⊙ ξ t b , a ξ s ⊙ ξ t b � 12 / 27

Subproduct systems and dilations The GNS representation of a CP-semigroup Let T = ( T s ) s ∈ S be a CP-semigroup on B . For every s , let ( E s , ξ s ) be the GNS representation of T s . For s , t ∈ S , define w s , t : E s + t → E s ⊙ E t ( really E s ⊙ s E t ) by w s , t : a ξ s + t b �→ a ξ s ⊙ ξ t b , and then extend linearly. We check: � a ξ s ⊙ ξ t b , a ξ s ⊙ ξ t b � = � ξ t b , � a ξ s , a ξ s � ξ t b � 12 / 27

Subproduct systems and dilations The GNS representation of a CP-semigroup Let T = ( T s ) s ∈ S be a CP-semigroup on B . For every s , let ( E s , ξ s ) be the GNS representation of T s . For s , t ∈ S , define w s , t : E s + t → E s ⊙ E t ( really E s ⊙ s E t ) by w s , t : a ξ s + t b �→ a ξ s ⊙ ξ t b , and then extend linearly. We check: � a ξ s ⊙ ξ t b , a ξ s ⊙ ξ t b � = � ξ t b , � a ξ s , a ξ s � ξ t b � = b ∗ � ξ t , T s ( a ∗ a ) ξ t � b 12 / 27

Subproduct systems and dilations The GNS representation of a CP-semigroup Let T = ( T s ) s ∈ S be a CP-semigroup on B . For every s , let ( E s , ξ s ) be the GNS representation of T s . For s , t ∈ S , define w s , t : E s + t → E s ⊙ E t ( really E s ⊙ s E t ) by w s , t : a ξ s + t b �→ a ξ s ⊙ ξ t b , and then extend linearly. We check: � a ξ s ⊙ ξ t b , a ξ s ⊙ ξ t b � = � ξ t b , � a ξ s , a ξ s � ξ t b � = b ∗ � ξ t , T s ( a ∗ a ) ξ t � b = = b ∗ T t ( T s ( a ∗ a )) b 12 / 27

Subproduct systems and dilations The GNS representation of a CP-semigroup Let T = ( T s ) s ∈ S be a CP-semigroup on B . For every s , let ( E s , ξ s ) be the GNS representation of T s . For s , t ∈ S , define w s , t : E s + t → E s ⊙ E t ( really E s ⊙ s E t ) by w s , t : a ξ s + t b �→ a ξ s ⊙ ξ t b , and then extend linearly. We check: � a ξ s ⊙ ξ t b , a ξ s ⊙ ξ t b � = � ξ t b , � a ξ s , a ξ s � ξ t b � = b ∗ � ξ t , T s ( a ∗ a ) ξ t � b = = b ∗ T t ( T s ( a ∗ a )) b = b ∗ T t + s ( a ∗ a ) b 12 / 27

Subproduct systems and dilations The GNS representation of a CP-semigroup Let T = ( T s ) s ∈ S be a CP-semigroup on B . For every s , let ( E s , ξ s ) be the GNS representation of T s . For s , t ∈ S , define w s , t : E s + t → E s ⊙ E t ( really E s ⊙ s E t ) by w s , t : a ξ s + t b �→ a ξ s ⊙ ξ t b , and then extend linearly. We check: � a ξ s ⊙ ξ t b , a ξ s ⊙ ξ t b � = � ξ t b , � a ξ s , a ξ s � ξ t b � = b ∗ � ξ t , T s ( a ∗ a ) ξ t � b = = b ∗ T t ( T s ( a ∗ a )) b = b ∗ T t + s ( a ∗ a ) b = � a ξ s + t b , a ξ s + t b � . w s , t is an isometry! 12 / 27

Subproduct systems and dilations Subproduct systems 4 Definition A subproduct system is a family E � = ( E s ) s ∈ S of B -correspondences, together with a family { w s , t : E s + t → E s ⊙ E t } of isometric bimodule maps, which iterate associatively 4 Inclusion systems by Bhat-Mukherjee. 13 / 27

� � � Subproduct systems and dilations Subproduct systems 4 Definition A subproduct system is a family E � = ( E s ) s ∈ S of B -correspondences, together with a family { w s , t : E s + t → E s ⊙ E t } of isometric bimodule maps, which iterate associatively, i.e., the following diagram is commutative ( ∀ r , s , t ): E r + s + t E r ⊙ E s + t � E r ⊙ E s ⊙ E t E r + s ⊙ E t 4 Inclusion systems by Bhat-Mukherjee. 13 / 27

� � � Subproduct systems and dilations Subproduct systems 4 Definition A subproduct system is a family E � = ( E s ) s ∈ S of B -correspondences, together with a family { w s , t : E s + t → E s ⊙ E t } of isometric bimodule maps, which iterate associatively, i.e., the following diagram is commutative ( ∀ r , s , t ): E r + s + t E r ⊙ E s + t � E r ⊙ E s ⊙ E t E r + s ⊙ E t A product system is a subproduct system in which w s , t are all unitaries. 4 Inclusion systems by Bhat-Mukherjee. 13 / 27

� � � Subproduct systems and dilations Subproduct systems 4 Definition A subproduct system is a family E � = ( E s ) s ∈ S of B -correspondences, together with a family { w s , t : E s + t → E s ⊙ E t } of isometric bimodule maps, which iterate associatively, i.e., the following diagram is commutative ( ∀ r , s , t ): E r + s + t E r ⊙ E s + t � E r ⊙ E s ⊙ E t E r + s ⊙ E t A product system is a subproduct system in which w s , t are all unitaries. Definition A family { ξ s ∈ E s } s ∈ S is called a unit if w s , t ξ s + t = ξ s ⊙ ξ t for all s , t . 4 Inclusion systems by Bhat-Mukherjee. 13 / 27

Subproduct systems and dilations Recap E s ⊙ E t ⊇ E s + t (or E s ⊙ s E t , etc.) Subproduct system: 14 / 27

Subproduct systems and dilations Recap E s ⊙ E t ⊇ E s + t (or E s ⊙ s E t , etc.) Subproduct system: Product system: E s ⊙ E t = E s + t 14 / 27

Subproduct systems and dilations Recap E s ⊙ E t ⊇ E s + t (or E s ⊙ s E t , etc.) Subproduct system: Product system: E s ⊙ E t = E s + t Unit: ξ s ⊙ ξ t = ξ s + t 14 / 27

Subproduct systems and dilations Recap E s ⊙ E t ⊇ E s + t (or E s ⊙ s E t , etc.) Subproduct system: Product system: E s ⊙ E t = E s + t Unit: ξ s ⊙ ξ t = ξ s + t For every CP-semigroup on B , there exists a subproduct system E � = ( E s ) s ∈ S of B -correspondences (called the GNS subproduct system ) and a unit ( ξ s ) s ∈ S such that T s ( b ) = � ξ s , b ξ s � for all s ∈ S , b ∈ B . 14 / 27

Subproduct systems and dilations Recap E s ⊙ E t ⊇ E s + t (or E s ⊙ s E t , etc.) Subproduct system: Product system: E s ⊙ E t = E s + t Unit: ξ s ⊙ ξ t = ξ s + t For every CP-semigroup on B , there exists a subproduct system E � = ( E s ) s ∈ S of B -correspondences (called the GNS subproduct system ) and a unit ( ξ s ) s ∈ S such that T s ( b ) = � ξ s , b ξ s � for all s ∈ S , b ∈ B . Theorem (Following Bhat-Skeide, 2000) Let T be a Markov semigroup. If the GNS subproduct system of T can be embedded in a product system , then T has a unital dilation ( A , ϑ, p ) . 14 / 27

Subproduct systems and dilations Recap E s ⊙ E t ⊇ E s + t (or E s ⊙ s E t , etc.) Subproduct system: Product system: E s ⊙ E t = E s + t Unit: ξ s ⊙ ξ t = ξ s + t For every CP-semigroup on B , there exists a subproduct system E � = ( E s ) s ∈ S of B -correspondences (called the GNS subproduct system ) and a unit ( ξ s ) s ∈ S such that T s ( b ) = � ξ s , b ξ s � for all s ∈ S , b ∈ B . Theorem (Following Bhat-Skeide, 2000) Let T be a Markov semigroup. If the GNS subproduct system of T can be embedded in a product system , then T has a unital dilation ( A , ϑ, p ) . In fact, one can take A = B a ( E ) , where E is some (full) B -correspondence. Markov semigroup = unital CP-semigroup. 14 / 27

Subproduct systems and dilations An application Theorem (S.-Skeide, see also Bhat 98, Solel 2006) Every Markov semigroup over N 2 has a unital dilation: 15 / 27

Subproduct systems and dilations An application Theorem (S.-Skeide, see also Bhat 98, Solel 2006) Every Markov semigroup over N 2 has a unital dilation: If T 1 , T 2 are two commuting normal unital CP maps on a vN algebra B , then there exist two commuting normal unital *-endomorphisms ϑ 1 , ϑ 2 on a vN algebra A containing B , a projection p ∈ A such that B = p A p, 15 / 27

Subproduct systems and dilations An application Theorem (S.-Skeide, see also Bhat 98, Solel 2006) Every Markov semigroup over N 2 has a unital dilation: If T 1 , T 2 are two commuting normal unital CP maps on a vN algebra B , then there exist two commuting normal unital *-endomorphisms ϑ 1 , ϑ 2 on a vN algebra A containing B , a projection p ∈ A such that B = p A p, and T n 1 1 ◦ T n 2 2 ( b ) = p ϑ n 1 1 ◦ ϑ n 2 2 ( b ) p for all b ∈ B , n 1 , n 2 ∈ N . 15 / 27

Subproduct systems and dilations An application Theorem (S.-Skeide, see also Bhat 98, Solel 2006) Every Markov semigroup over N 2 has a unital dilation: If T 1 , T 2 are two commuting normal unital CP maps on a vN algebra B , then there exist two commuting normal unital *-endomorphisms ϑ 1 , ϑ 2 on a vN algebra A containing B , a projection p ∈ A such that B = p A p, and T n 1 1 ◦ T n 2 2 ( b ) = p ϑ n 1 1 ◦ ϑ n 2 2 ( b ) p for all b ∈ B , n 1 , n 2 ∈ N . Proof. Given a Markov semigroup over N 2 , we construct a product system that contains the GNS subproduct system of that semigroup. 15 / 27

Subproduct systems and dilations An application Theorem (S.-Skeide, see also Bhat 98, Solel 2006) Every Markov semigroup over N 2 has a unital dilation: If T 1 , T 2 are two commuting normal unital CP maps on a vN algebra B , then there exist two commuting normal unital *-endomorphisms ϑ 1 , ϑ 2 on a vN algebra A containing B , a projection p ∈ A such that B = p A p, and T n 1 1 ◦ T n 2 2 ( b ) = p ϑ n 1 1 ◦ ϑ n 2 2 ( b ) p for all b ∈ B , n 1 , n 2 ∈ N . Proof. Given a Markov semigroup over N 2 , we construct a product system that contains the GNS subproduct system of that semigroup. Then apply previous theorem. 15 / 27

Subproduct systems and dilations An application Theorem (S.-Skeide, see also Bhat 98, Solel 2006) Every Markov semigroup over N 2 has a unital dilation: If T 1 , T 2 are two commuting normal unital CP maps on a vN algebra B , then there exist two commuting normal unital *-endomorphisms ϑ 1 , ϑ 2 on a vN algebra A containing B , a projection p ∈ A such that B = p A p, and T n 1 1 ◦ T n 2 2 ( b ) = p ϑ n 1 1 ◦ ϑ n 2 2 ( b ) p for all b ∈ B , n 1 , n 2 ∈ N . Proof. Given a Markov semigroup over N 2 , we construct a product system that contains the GNS subproduct system of that semigroup. Then apply previous theorem. s = B a ( E ) , where E = A p s . In Remark: In fact we have A = A p A particular, A is Morita equivalent to B (in the sense of Rieffel). 15 / 27

Subproduct systems and dilations The converse direction A sufficient condition for the existence of a dilation for a unital CP-semigroup T is that its GNS subproduct system embeds into a product system. 16 / 27

Subproduct systems and dilations The converse direction A sufficient condition for the existence of a dilation for a unital CP-semigroup T is that its GNS subproduct system embeds into a product system. What about the converse direction? 16 / 27

Subproduct systems and dilations The converse direction A sufficient condition for the existence of a dilation for a unital CP-semigroup T is that its GNS subproduct system embeds into a product system. What about the converse direction? Theorem (S.-Skeide) • If a Markov semigroup T = ( T s ) s ∈ S has a minimal dilation then its GNS subproduct system embeds into a product system. 16 / 27

Subproduct systems and dilations The converse direction A sufficient condition for the existence of a dilation for a unital CP-semigroup T is that its GNS subproduct system embeds into a product system. What about the converse direction? Theorem (S.-Skeide) • If a Markov semigroup T = ( T s ) s ∈ S has a minimal dilation then its GNS subproduct system embeds into a product system. • A Markov semigroup T = ( T s ) s ∈ S has a dilation ( B a ( E ) , ϑ, p ) where E is a (full) B -correspondence, if and only if its GNS subproduct system embeds into a product system. 16 / 27

Subproduct systems and dilations The converse direction A sufficient condition for the existence of a dilation for a unital CP-semigroup T is that its GNS subproduct system embeds into a product system. What about the converse direction? Theorem (S.-Skeide) • If a Markov semigroup T = ( T s ) s ∈ S has a minimal dilation then its GNS subproduct system embeds into a product system. • A Markov semigroup T = ( T s ) s ∈ S has a dilation ( B a ( E ) , ϑ, p ) where E is a (full) B -correspondence, if and only if its GNS subproduct system embeds into a product system. Caveats: 1. We did not define what "minimal" means. 16 / 27

Subproduct systems and dilations The converse direction A sufficient condition for the existence of a dilation for a unital CP-semigroup T is that its GNS subproduct system embeds into a product system. What about the converse direction? Theorem (S.-Skeide) • If a Markov semigroup T = ( T s ) s ∈ S has a minimal dilation then its GNS subproduct system embeds into a product system. • A Markov semigroup T = ( T s ) s ∈ S has a dilation ( B a ( E ) , ϑ, p ) where E is a (full) B -correspondence, if and only if its GNS subproduct system embeds into a product system. Caveats: 1. We did not define what "minimal" means. 2. Over N k ( k ≥ 2), minimal dilations are not unique. 16 / 27

Subproduct systems and dilations The converse direction A sufficient condition for the existence of a dilation for a unital CP-semigroup T is that its GNS subproduct system embeds into a product system. What about the converse direction? Theorem (S.-Skeide) • If a Markov semigroup T = ( T s ) s ∈ S has a minimal dilation then its GNS subproduct system embeds into a product system. • A Markov semigroup T = ( T s ) s ∈ S has a dilation ( B a ( E ) , ϑ, p ) where E is a (full) B -correspondence, if and only if its GNS subproduct system embeds into a product system. Caveats: 1. We did not define what "minimal" means. 2. Over N k ( k ≥ 2), minimal dilations are not unique. 3. Over N k ( k ≥ 2), a given dilation might not be "minimalizable", that is, cannot be compressed or restricted to a minimal one (new and weird). 16 / 27

Subproduct systems and dilations The converse direction A sufficient condition for the existence of a dilation for a unital CP-semigroup T is that its GNS subproduct system embeds into a product system. What about the converse direction? Theorem (S.-Skeide) • If a Markov semigroup T = ( T s ) s ∈ S has a minimal dilation then its GNS subproduct system embeds into a product system. • A Markov semigroup T = ( T s ) s ∈ S has a dilation ( B a ( E ) , ϑ, p ) where E is a (full) B -correspondence, if and only if its GNS subproduct system embeds into a product system. Caveats: 1. We did not define what "minimal" means. 2. Over N k ( k ≥ 2), minimal dilations are not unique. 3. Over N k ( k ≥ 2), a given dilation might not be "minimalizable", that is, cannot be compressed or restricted to a minimal one (new and weird). 4. What about dilations ( A , ϑ, p ) , where A � = B a ( E ) ? 16 / 27

Subproduct systems and dilations The converse direction II Theorem (S.-Skeide) • If a Markov semigroup T = ( T s ) s ∈ S has a minimal dilation then its GNS subproduct system embeds into a product system. Corollary (S.-Skeide) There exist CP and Markov semigroups over N 3 for which there is no minimal dilation. 17 / 27

Subproduct systems and dilations The converse direction II Theorem (S.-Skeide) • If a Markov semigroup T = ( T s ) s ∈ S has a minimal dilation then its GNS subproduct system embeds into a product system. Corollary (S.-Skeide) There exist CP and Markov semigroups over N 3 for which there is no minimal dilation. "Proof" (not really...) [S.-Solel] construct a subproduct system over N 3 that cannot be embedded into a product system. We apply the above theorem to that subproduct system. 17 / 27

Subproduct systems and dilations The converse direction II Theorem (S.-Skeide) • If a Markov semigroup T = ( T s ) s ∈ S has a minimal dilation then its GNS subproduct system embeds into a product system. Corollary (S.-Skeide) There exist CP and Markov semigroups over N 3 for which there is no minimal dilation. "Proof" (not really...) [S.-Solel] construct a subproduct system over N 3 that cannot be embedded into a product system. We apply the above theorem to that subproduct system. Problem: this does not rule out the existence of non-minimal dilations. 17 / 27

Minimality Minimality Let T = ( T s ) s ∈ S be a CP-semigroup over S , and ( A , ϑ, p ) a dilation. Suppose that B ⊆ B ( H ) and that A ⊆ B ( K ) , so that p = P H . 18 / 27

Minimality Minimality Let T = ( T s ) s ∈ S be a CP-semigroup over S , and ( A , ϑ, p ) a dilation. Suppose that B ⊆ B ( H ) and that A ⊆ B ( K ) , so that p = P H . There are three properties that one may require for "minimality": 18 / 27

Minimality Minimality Let T = ( T s ) s ∈ S be a CP-semigroup over S , and ( A , ϑ, p ) a dilation. Suppose that B ⊆ B ( H ) and that A ⊆ B ( K ) , so that p = P H . There are three properties that one may require for "minimality": 1. "Algebraic minimality", that is A = W ∗ ( ∪ s ∈ S ϑ s ( B )) . 18 / 27

Minimality Minimality Let T = ( T s ) s ∈ S be a CP-semigroup over S , and ( A , ϑ, p ) a dilation. Suppose that B ⊆ B ( H ) and that A ⊆ B ( K ) , so that p = P H . There are three properties that one may require for "minimality": 1. "Algebraic minimality", that is A = W ∗ ( ∪ s ∈ S ϑ s ( B )) . s . Assuming 1, same as: 2. "Spatial minimality", that is, A = A p A K = span { ϑ s 1 ( b 1 ) · · · ϑ s n ( b n ) h : s i ∈ S , b i ∈ B , h ∈ H } . 18 / 27

Minimality Minimality Let T = ( T s ) s ∈ S be a CP-semigroup over S , and ( A , ϑ, p ) a dilation. Suppose that B ⊆ B ( H ) and that A ⊆ B ( K ) , so that p = P H . There are three properties that one may require for "minimality": 1. "Algebraic minimality", that is A = W ∗ ( ∪ s ∈ S ϑ s ( B )) . s . Assuming 1, same as: 2. "Spatial minimality", that is, A = A p A K = span { ϑ s 1 ( b 1 ) · · · ϑ s n ( b n ) h : s i ∈ S , b i ∈ B , h ∈ H } . 3. "Incompressibility": there is no nontrivial projection p ≤ q ∈ A s.t. q ϑ s ( · ) q : q A q → q A q , q ϑ s ( · ) q : qaq �→ q ϑ s ( qaq ) q , is an E-semigroup, and a dilation of T . 18 / 27

Minimality Minimality II 1. A = W ∗ ( ∪ s ∈ S ϑ s ( B )) . s . 2. A = A p A 3. No nontrivial projection p ≤ q � = 1 in A s.t. q ϑ s ( · ) q is a dilation. 19 / 27

Minimality Minimality II 1. A = W ∗ ( ∪ s ∈ S ϑ s ( B )) . s . 2. A = A p A 3. No nontrivial projection p ≤ q � = 1 in A s.t. q ϑ s ( · ) q is a dilation. The notion of minimality referred to in theorem and corollary above is the strongest one: 1+2. (This also implies 3). 19 / 27

Minimality Minimality II 1. A = W ∗ ( ∪ s ∈ S ϑ s ( B )) . s . 2. A = A p A 3. No nontrivial projection p ≤ q � = 1 in A s.t. q ϑ s ( · ) q is a dilation. The notion of minimality referred to in theorem and corollary above is the strongest one: 1+2. (This also implies 3). It is easy to restrict to a semigroup satisfying 1, and not hard to compress to obtain 1+3, but that is not the notion that works best (M.E.). 19 / 27

Minimality Minimality II 1. A = W ∗ ( ∪ s ∈ S ϑ s ( B )) . s . 2. A = A p A 3. No nontrivial projection p ≤ q � = 1 in A s.t. q ϑ s ( · ) q is a dilation. The notion of minimality referred to in theorem and corollary above is the strongest one: 1+2. (This also implies 3). It is easy to restrict to a semigroup satisfying 1, and not hard to compress to obtain 1+3, but that is not the notion that works best (M.E.). Over R + (and N ), 1+2 is equivalent to 1+3. (non-trivial!) 19 / 27

Minimality Minimality II 1. A = W ∗ ( ∪ s ∈ S ϑ s ( B )) . s . 2. A = A p A 3. No nontrivial projection p ≤ q � = 1 in A s.t. q ϑ s ( · ) q is a dilation. The notion of minimality referred to in theorem and corollary above is the strongest one: 1+2. (This also implies 3). It is easy to restrict to a semigroup satisfying 1, and not hard to compress to obtain 1+3, but that is not the notion that works best (M.E.). Over R + (and N ), 1+2 is equivalent to 1+3. (non-trivial!) We have an example of a dilation ( A , ϑ, p ) over N 2 , which satisfies 2, but not 1. After restricting to W ∗ ( ∪ s ∈ S ϑ s ( B )) , and then compressing to the minimal compressing q , one obtains an algebraically minimal and incompressible dilation (1+3), which does not satisfy 2. 19 / 27

Dilations and superproduct systems Dilation ⇒ what? Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. Following a construction from [Skeide02], we see what structure arises. 20 / 27

Dilations and superproduct systems Dilation ⇒ what? Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. Following a construction from [Skeide02], we see what structure arises. Define a family ( E s ) s ∈ S of B -correspondences as follows: E := A p 20 / 27

Dilations and superproduct systems Dilation ⇒ what? Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. Following a construction from [Skeide02], we see what structure arises. Define a family ( E s ) s ∈ S of B -correspondences as follows: E := A p E s := ϑ s ( p ) E . , 20 / 27

Dilations and superproduct systems Dilation ⇒ what? Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. Following a construction from [Skeide02], we see what structure arises. Define a family ( E s ) s ∈ S of B -correspondences as follows: E := A p E s := ϑ s ( p ) E . , W*-correspondence structure: b · x s := ϑ s ( b ) x s x s · b := xb , x s ∈ E s , b ∈ B . , � x s , y s � := x ∗ s y s ∈ p A p = B . 20 / 27

Dilations and superproduct systems Dilation ⇒ what? Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. Following a construction from [Skeide02], we see what structure arises. Define a family ( E s ) s ∈ S of B -correspondences as follows: E := A p E s := ϑ s ( p ) E . , W*-correspondence structure: b · x s := ϑ s ( b ) x s x s · b := xb , x s ∈ E s , b ∈ B . , � x s , y s � := x ∗ s y s ∈ p A p = B . Unit: η s := ϑ s ( p ) p ∈ E s . 20 / 27

Dilations and superproduct systems Dilation ⇒ what? Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. Following a construction from [Skeide02], we see what structure arises. Define a family ( E s ) s ∈ S of B -correspondences as follows: E := A p E s := ϑ s ( p ) E . , W*-correspondence structure: b · x s := ϑ s ( b ) x s x s · b := xb , x s ∈ E s , b ∈ B . , � x s , y s � := x ∗ s y s ∈ p A p = B . Unit: η s := ϑ s ( p ) p ∈ E s . ( E s , η s ) represents T � η s , b · η s � 20 / 27

Dilations and superproduct systems Dilation ⇒ what? Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. Following a construction from [Skeide02], we see what structure arises. Define a family ( E s ) s ∈ S of B -correspondences as follows: E := A p E s := ϑ s ( p ) E . , W*-correspondence structure: b · x s := ϑ s ( b ) x s x s · b := xb , x s ∈ E s , b ∈ B . , � x s , y s � := x ∗ s y s ∈ p A p = B . Unit: η s := ϑ s ( p ) p ∈ E s . ( E s , η s ) represents T � η s , b · η s � = p ϑ s ( p ) ϑ s ( b ) ϑ s ( p ) p = p ϑ s ( b ) p 20 / 27

Dilations and superproduct systems Dilation ⇒ what? Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. Following a construction from [Skeide02], we see what structure arises. Define a family ( E s ) s ∈ S of B -correspondences as follows: E := A p E s := ϑ s ( p ) E . , W*-correspondence structure: b · x s := ϑ s ( b ) x s x s · b := xb , x s ∈ E s , b ∈ B . , � x s , y s � := x ∗ s y s ∈ p A p = B . Unit: η s := ϑ s ( p ) p ∈ E s . ( E s , η s ) represents T � η s , b · η s � = p ϑ s ( p ) ϑ s ( b ) ϑ s ( p ) p = p ϑ s ( b ) p = T s ( b ) . 20 / 27

Dilations and superproduct systems Dilation ⇒ what? II Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. We constructed a family ( E s ) s ∈ S of B -corresopndences, and a family ( η s ) s ∈ S of unit vectors ( η s ∈ E s ) that represent T : � η s , b · η s � = p ϑ s ( b ) p = T s ( b ) . 21 / 27

Dilations and superproduct systems Dilation ⇒ what? II Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. We constructed a family ( E s ) s ∈ S of B -corresopndences, and a family ( η s ) s ∈ S of unit vectors ( η s ∈ E s ) that represent T : � η s , b · η s � = p ϑ s ( b ) p = T s ( b ) . Hence ( E s , η s ) "contains" the GNS representation ( E s , ξ s ) of T s . 21 / 27

Dilations and superproduct systems Dilation ⇒ what? II Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. We constructed a family ( E s ) s ∈ S of B -corresopndences, and a family ( η s ) s ∈ S of unit vectors ( η s ∈ E s ) that represent T : � η s , b · η s � = p ϑ s ( b ) p = T s ( b ) . Hence ( E s , η s ) "contains" the GNS representation ( E s , ξ s ) of T s . Q: 21 / 27

Dilations and superproduct systems Dilation ⇒ what? II Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. We constructed a family ( E s ) s ∈ S of B -corresopndences, and a family ( η s ) s ∈ S of unit vectors ( η s ∈ E s ) that represent T : � η s , b · η s � = p ϑ s ( b ) p = T s ( b ) . Hence ( E s , η s ) "contains" the GNS representation ( E s , ξ s ) of T s . Q: is ( E s ) s ∈ S a PRODUCT system? 21 / 27

Dilations and superproduct systems Dilation ⇒ what? III Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. Let (( E s ) s ∈ S , ( η s ) s ∈ S ) be as above, � η s , b · η s � = T s ( b ) . 22 / 27

Dilations and superproduct systems Dilation ⇒ what? III Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. Let (( E s ) s ∈ S , ( η s ) s ∈ S ) be as above, � η s , b · η s � = T s ( b ) . Define v s , t : E s ⊙ E t → E s + t ( really E s ⊙ s E t ) v s , t : x s ⊙ y t �→ ϑ t ( x s ) y t . 22 / 27

Dilations and superproduct systems Dilation ⇒ what? III Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. Let (( E s ) s ∈ S , ( η s ) s ∈ S ) be as above, � η s , b · η s � = T s ( b ) . Define v s , t : E s ⊙ E t → E s + t ( really E s ⊙ s E t ) v s , t : x s ⊙ y t �→ ϑ t ( x s ) y t . A direct calculation shows: � x s ⊙ y t , x ′ s ⊙ y ′ t � = . . . = � ϑ t ( x s ) y t , ϑ t ( x ′ s ) y ′ t � . Hence v s , t : E s ⊙ E t → E s + t is an isometry: 22 / 27

Dilations and superproduct systems Dilation ⇒ what? III Let T = ( T s ) s ∈ S be a CP-semigroup on B , and ( A , ϑ, p ) a dilation. Let (( E s ) s ∈ S , ( η s ) s ∈ S ) be as above, � η s , b · η s � = T s ( b ) . Define v s , t : E s ⊙ E t → E s + t ( really E s ⊙ s E t ) v s , t : x s ⊙ y t �→ ϑ t ( x s ) y t . A direct calculation shows: � x s ⊙ y t , x ′ s ⊙ y ′ t � = . . . = � ϑ t ( x s ) y t , ϑ t ( x ′ s ) y ′ t � . Hence v s , t : E s ⊙ E t → E s + t is an isometry: E s ⊙ E t ⊆ E s + t . ( E s ) s ∈ S is a superproduct system (but not always a product system). 22 / 27

Dilations and superproduct systems Superproduct systems Definition A superproduct system is a family E � = ( E s ) s ∈ S of B -correspondences, together with a family { v s , t : E s ⊙ E t → E s + t } of isometric bimodule maps, which iterate associatively 23 / 27

� � � Dilations and superproduct systems Superproduct systems Definition A superproduct system is a family E � = ( E s ) s ∈ S of B -correspondences, together with a family { v s , t : E s ⊙ E t → E s + t } of isometric bimodule maps, which iterate associatively, i.e., the following diagram is commutative ( ∀ r , s , t ): E r ⊙ E s ⊙ E t E r ⊙ E s + t � E r + s + t E r + s ⊙ E t 23 / 27

� � � Dilations and superproduct systems Superproduct systems Definition A superproduct system is a family E � = ( E s ) s ∈ S of B -correspondences, together with a family { v s , t : E s ⊙ E t → E s + t } of isometric bimodule maps, which iterate associatively, i.e., the following diagram is commutative ( ∀ r , s , t ): E r ⊙ E s ⊙ E t E r ⊙ E s + t � E r + s + t E r + s ⊙ E t A product system is a superproduct system in which v s , t are all unitaries. 23 / 27

Dilations and superproduct systems Recap Subproduct system: E s ⊙ E t ⊇ E s + t Product system: E s ⊙ E t = E s + t Unit: ξ s ⊙ ξ t = ξ s + t 24 / 27

Dilations and superproduct systems Recap Subproduct system: E s ⊙ E t ⊇ E s + t Product system: E s ⊙ E t = E s + t Unit: ξ s ⊙ ξ t = ξ s + t Superproduct system: E s ⊙ E t ⊆ E s + t 24 / 27

Dilations and superproduct systems Recap Subproduct system: E s ⊙ E t ⊇ E s + t Product system: E s ⊙ E t = E s + t Unit: ξ s ⊙ ξ t = ξ s + t Superproduct system: E s ⊙ E t ⊆ E s + t For every CP-semigroup T on B , there exists a subproduct system E � = ( E s ) s ∈ S of B -correspondences (the GNS subproduct system ) and a unit ( ξ s ) s ∈ S such that T s ( b ) = � ξ s , b ξ s � for all s ∈ S , b ∈ B . 24 / 27

Dilations and superproduct systems Recap Subproduct system: E s ⊙ E t ⊇ E s + t Product system: E s ⊙ E t = E s + t Unit: ξ s ⊙ ξ t = ξ s + t Superproduct system: E s ⊙ E t ⊆ E s + t For every CP-semigroup T on B , there exists a subproduct system E � = ( E s ) s ∈ S of B -correspondences (the GNS subproduct system ) and a unit ( ξ s ) s ∈ S such that T s ( b ) = � ξ s , b ξ s � for all s ∈ S , b ∈ B . If T unital, and if the GNS subproduct system can be embedded into a product system , then T has a dilation ( A , ϑ, p ) (with A = B a ( E ) ). 24 / 27

Dilations and superproduct systems Recap Subproduct system: E s ⊙ E t ⊇ E s + t Product system: E s ⊙ E t = E s + t Unit: ξ s ⊙ ξ t = ξ s + t Superproduct system: E s ⊙ E t ⊆ E s + t For every CP-semigroup T on B , there exists a subproduct system E � = ( E s ) s ∈ S of B -correspondences (the GNS subproduct system ) and a unit ( ξ s ) s ∈ S such that T s ( b ) = � ξ s , b ξ s � for all s ∈ S , b ∈ B . If T unital, and if the GNS subproduct system can be embedded into a product system , then T has a dilation ( A , ϑ, p ) (with A = B a ( E ) ). If T has a dilation ( A , ϑ, p ) , then the GNS subproduct system must embed into a superproduct system. 24 / 27

Dilations and superproduct systems Dilations and superproduct systems Theorem (S.-Skeide) Let T = ( T s ) s ∈ S be a Markov semigroup on a von Neumann algebra B . • A sufficient condition for T to a have a dilation, is that the GNS subproduct system of T embeds into a product system. • A necessary condition for T to have a dilation, is that the GNS subproduct system of T embeds into a superproduct system. 25 / 27

Dilations and superproduct systems Dilations and superproduct systems Theorem (S.-Skeide) Let T = ( T s ) s ∈ S be a Markov semigroup on a von Neumann algebra B . • A sufficient condition for T to a have a dilation, is that the GNS subproduct system of T embeds into a product system. • A necessary condition for T to have a dilation, is that the GNS subproduct system of T embeds into a superproduct system. Corollary (S.-Skeide) There exist CP and Markov semigroups over N 3 that have no dilation. 25 / 27

Dilations and superproduct systems Dilations and superproduct systems Theorem (S.-Skeide) Let T = ( T s ) s ∈ S be a Markov semigroup on a von Neumann algebra B . • A sufficient condition for T to a have a dilation, is that the GNS subproduct system of T embeds into a product system. • A necessary condition for T to have a dilation, is that the GNS subproduct system of T embeds into a superproduct system. Corollary (S.-Skeide) There exist CP and Markov semigroups over N 3 that have no dilation. "Proof" (not really...) We have an example of a subproduct system over N 3 that cannot be embedded into a superproduct system. 25 / 27

Dilations and superproduct systems Dilations and superproduct systems Theorem (S.-Skeide) Let T = ( T s ) s ∈ S be a Markov semigroup on a von Neumann algebra B . • A sufficient condition for T to a have a dilation, is that the GNS subproduct system of T embeds into a product system. • A necessary condition for T to have a dilation, is that the GNS subproduct system of T embeds into a superproduct system. Corollary (S.-Skeide) There exist CP and Markov semigroups over N 3 that have no dilation. "Proof" (not really...) We have an example of a subproduct system over N 3 that cannot be embedded into a superproduct system. The truth: the SPS is not the GNS subproduct system of a CP-semigroup, so the proof does not really go like that . . . 25 / 27

More subproduct systems Another way subproduct systems arise Let E be a full W*-correspondence over B , and B a ( E ) the adjointable operators on E . E is a Morita W* equivalence from B a ( E ) to B : B = E ∗ ⊙ s E B a ( E ) = E ⊙ s E ∗ . , 26 / 27