Tensor product systems of Hilbert spaces

Transcription

1 Tensor product systems of Hilbert spaces B. V. Rajarama Bhat, Indian Statistical Institute, Bangalore. January 14, 2016 Indo-French Program for Mathematics Matscience, Chennai, 11-24, January 2016

2 Acknowledgements Thanks to the Organisers.

3 Acknowledgements Thanks to the Organisers. Remembering Prof. P. A. Meyer.

4 E-semigroups: Semigroups of endomorphisms of B(H) Let H be a complex separable Hilbert space.

5 E-semigroups: Semigroups of endomorphisms of B(H) Let H be a complex separable Hilbert space. Let θ = {θ t : t 0} be a family of maps satisfying:

6 E-semigroups: Semigroups of endomorphisms of B(H) Let H be a complex separable Hilbert space. Let θ = {θ t : t 0} be a family of maps satisfying: θ t : B(H) B(H) is a -endomorphism (linear, multiplicative, -preserving).

7 E-semigroups: Semigroups of endomorphisms of B(H) Let H be a complex separable Hilbert space. Let θ = {θ t : t 0} be a family of maps satisfying: θ t : B(H) B(H) is a -endomorphism (linear, multiplicative, -preserving). θ s+t (X ) = θ s (θ t (X )) for all s, t 0 and X B(H).

8 E-semigroups: Semigroups of endomorphisms of B(H) Let H be a complex separable Hilbert space. Let θ = {θ t : t 0} be a family of maps satisfying: θ t : B(H) B(H) is a -endomorphism (linear, multiplicative, -preserving). θ s+t (X ) = θ s (θ t (X )) for all s, t 0 and X B(H). X θ t (X ) is normal for every t.

9 E-semigroups: Semigroups of endomorphisms of B(H) Let H be a complex separable Hilbert space. Let θ = {θ t : t 0} be a family of maps satisfying: θ t : B(H) B(H) is a -endomorphism (linear, multiplicative, -preserving). θ s+t (X ) = θ s (θ t (X )) for all s, t 0 and X B(H). X θ t (X ) is normal for every t. t θ t (X ) is continuous in weak operator topology for every X.

10 E-semigroups: Semigroups of endomorphisms of B(H) Let H be a complex separable Hilbert space. Let θ = {θ t : t 0} be a family of maps satisfying: θ t : B(H) B(H) is a -endomorphism (linear, multiplicative, -preserving). θ s+t (X ) = θ s (θ t (X )) for all s, t 0 and X B(H). X θ t (X ) is normal for every t. t θ t (X ) is continuous in weak operator topology for every X. θ having all these properties is called an E-semigroup.

11 E-semigroups: Semigroups of endomorphisms of B(H) Let H be a complex separable Hilbert space. Let θ = {θ t : t 0} be a family of maps satisfying: θ t : B(H) B(H) is a -endomorphism (linear, multiplicative, -preserving). θ s+t (X ) = θ s (θ t (X )) for all s, t 0 and X B(H). X θ t (X ) is normal for every t. t θ t (X ) is continuous in weak operator topology for every X. θ having all these properties is called an E-semigroup. An E-semigroup which is unital for every t, (θ t (I ) = I ), is said to be an E 0 -semigroup.

12 Automorphism semigroups: Wigner s theorem and Stone s theorem An E 0 -semigroup θ is an automorphism semigroup if each θ t is an automorphism.

13 Automorphism semigroups: Wigner s theorem and Stone s theorem An E 0 -semigroup θ is an automorphism semigroup if each θ t is an automorphism. Note that every automorphism of B(H) is of the form X UXU for some unitary U.

14 Automorphism semigroups: Wigner s theorem and Stone s theorem An E 0 -semigroup θ is an automorphism semigroup if each θ t is an automorphism. Note that every automorphism of B(H) is of the form X UXU for some unitary U. Wigner s theorem: Every automorphism semigroup of B(H) is of the form θ t (X ) = U t XU t, t 0, X B(H) for some unitary representation t U t of R.

15 Automorphism semigroups: Wigner s theorem and Stone s theorem An E 0 -semigroup θ is an automorphism semigroup if each θ t is an automorphism. Note that every automorphism of B(H) is of the form X UXU for some unitary U. Wigner s theorem: Every automorphism semigroup of B(H) is of the form θ t (X ) = U t XU t, t 0, X B(H) for some unitary representation t U t of R. Stone s theorem: U t = e ith t 0, where H is a (possibly unbounded) self-adjoint operator.

16 What are all the E-semigroups? R.T. Powers initiated a study of general E-semigroups. He initially gave several examples and suggested an invariant.

17 What are all the E-semigroups? R.T. Powers initiated a study of general E-semigroups. He initially gave several examples and suggested an invariant. W. Arveson associated a tensor product system of Hilbert spaces with an E 0 -semigroup and broadly classified E 0 -semigroups into three types.

18 What are all the E-semigroups? R.T. Powers initiated a study of general E-semigroups. He initially gave several examples and suggested an invariant. W. Arveson associated a tensor product system of Hilbert spaces with an E 0 -semigroup and broadly classified E 0 -semigroups into three types. Basic Reference: W. Arveson, Noncommutative Dynamics and E-Semigroups, Monographs in Mathematics, Springer, New York, 2003.

19 Basic structure of E 0 -semigroups Suppose θ is an E 0 -semigroup of B(H):

20 Basic structure of E 0 -semigroups Suppose θ is an E 0 -semigroup of B(H): There exists a Hilbert space P t with a unitary W t : H P t H such that θ t (X ) = W t (X 1 Pt )W t X.

21 Basic structure of E 0 -semigroups Suppose θ is an E 0 -semigroup of B(H): There exists a Hilbert space P t with a unitary W t : H P t H such that θ t (X ) = W t (X 1 Pt )W t X. P t form a tensor product system of Hilbert spaces.

22 Tensor product systems of Hilbert spaces A family {P t : t 0} of Hilbert spaces with a family {U s,t : s, t 0} of unitaries, U s,t : P s P t P s+t,

23 Tensor product systems of Hilbert spaces A family {P t : t 0} of Hilbert spaces with a family {U s,t : s, t 0} of unitaries, U s,t : P s P t P s+t, satisfying natural associativity condition: U r,s+t (I r U s,t ) = U r+s,t (U r,s 1 t ),

24 Tensor product systems of Hilbert spaces A family {P t : t 0} of Hilbert spaces with a family {U s,t : s, t 0} of unitaries, U s,t : P s P t P s+t, satisfying natural associativity condition: U r,s+t (I r U s,t ) = U r+s,t (U r,s 1 t ), and some technical conditions such as measurability.

25 Left Cocycles and Cocycle Conjugacy Let θ be an E 0 -semigroup on B(H). A family of unitaries {U t : t 0} on H, is said to be a left cocycle of θ if

26 Left Cocycles and Cocycle Conjugacy Let θ be an E 0 -semigroup on B(H). A family of unitaries {U t : t 0} on H, is said to be a left cocycle of θ if U 0 = I, U s+t = U s θ s (U t ) s, t,

27 Left Cocycles and Cocycle Conjugacy Let θ be an E 0 -semigroup on B(H). A family of unitaries {U t : t 0} on H, is said to be a left cocycle of θ if U 0 = I, U s+t = U s θ s (U t ) s, t, and t U t continuous in strong operator topology.

28 Left Cocycles and Cocycle Conjugacy Let θ be an E 0 -semigroup on B(H). A family of unitaries {U t : t 0} on H, is said to be a left cocycle of θ if U 0 = I, U s+t = U s θ s (U t ) s, t, and t U t continuous in strong operator topology. In such a case, ψ t (X ) = U t θ t (X )Ut defines an E 0 -semigroup.

29 Left Cocycles and Cocycle Conjugacy Let θ be an E 0 -semigroup on B(H). A family of unitaries {U t : t 0} on H, is said to be a left cocycle of θ if U 0 = I, U s+t = U s θ s (U t ) s, t, and t U t continuous in strong operator topology. In such a case, defines an E 0 -semigroup. ψ t (X ) = U t θ t (X )U t Two E 0 -semigroups θ, ψ on B(H) are said to be cocycle conjugate, if such a relation holds for some left cocycle.

30 Classification up to cocycle conjugacy Theorem (Arveson): Two E 0 -semigroups are cocycle conjugate if and only if their product systems are isomorphic.

31 Classification up to cocycle conjugacy Theorem (Arveson): Two E 0 -semigroups are cocycle conjugate if and only if their product systems are isomorphic. Isomorphism of product systems for cocycle conjugate E 0 -semigroups is obvious. The converse is also not very difficult.

32 E 0 -semigroups from product systems Does every product system arise from an E 0 -semigroup?

33 E 0 -semigroups from product systems Does every product system arise from an E 0 -semigroup? This question was answered in the affirmative, by W. Arveson through his theory of spectral C -algebras. It is a very indirect proof.

34 E 0 -semigroups from product systems Does every product system arise from an E 0 -semigroup? This question was answered in the affirmative, by W. Arveson through his theory of spectral C -algebras. It is a very indirect proof. More direct proofs were found by V. Liebscher and M. Skeide.

35 E 0 -semigroups from product systems Does every product system arise from an E 0 -semigroup? This question was answered in the affirmative, by W. Arveson through his theory of spectral C -algebras. It is a very indirect proof. More direct proofs were found by V. Liebscher and M. Skeide. Inspired by the work of Skeide, Arveson got a simpler proof. Further, Skeide has combined his and Arveson s methods.

36 E 0 -semigroups from product systems Does every product system arise from an E 0 -semigroup? This question was answered in the affirmative, by W. Arveson through his theory of spectral C -algebras. It is a very indirect proof. More direct proofs were found by V. Liebscher and M. Skeide. Inspired by the work of Skeide, Arveson got a simpler proof. Further, Skeide has combined his and Arveson s methods. Conclusion: Tensor product systems completely classify E 0 -semigroups up to cocycle conjugacy.

37 Exponentiation of Hilbert spaces For any Hilbert space G let Γ(G) denote the Boson or symmetric Fockspace over G.

38 Exponentiation of Hilbert spaces For any Hilbert space G let Γ(G) denote the Boson or symmetric Fockspace over G. Γ(G) = C G G s2 G s3.

39 Exponentiation of Hilbert spaces For any Hilbert space G let Γ(G) denote the Boson or symmetric Fockspace over G. Γ(G) = C G G s2 G s3. For g G, define the exponential vector: e(g) = 1 g g 2 2 g n n!.

40 Exponentiation of Hilbert spaces For any Hilbert space G let Γ(G) denote the Boson or symmetric Fockspace over G. Γ(G) = C G G s2 G s3. For g G, define the exponential vector: e(g) = 1 g g 2 2 g n n!. e(g), e(f ) = e g,f.

41 Exponentiation of Hilbert spaces For any Hilbert space G let Γ(G) denote the Boson or symmetric Fockspace over G. Γ(G) = C G G s2 G s3. For g G, define the exponential vector: e(g) = 1 g g 2 2 g n n!. e(g), e(f ) = e g,f. {e(f ) : f G} are linearly independent and total in Γ(G).

42 Exponentiation of Hilbert spaces For any Hilbert space G let Γ(G) denote the Boson or symmetric Fockspace over G. Γ(G) = C G G s2 G s3. For g G, define the exponential vector: e(g) = 1 g g 2 2 g n n!. e(g), e(f ) = e g,f. {e(f ) : f G} are linearly independent and total in Γ(G). Γ(G H) = Γ(G) Γ(H), by e(g h) e(g) e(h).

43 Exponential product systems Take P t = Γ(L 2 ([0, t)))

44 Exponential product systems Take P t = Γ(L 2 ([0, t))) Define U s,t : P s P t P s+t by

45 Exponential product systems Take P t = Γ(L 2 ([0, t))) Define U s,t : P s P t P s+t by U s,t (e(g) e(h)) = e(f ), where f (r) = g(r) for 0 r < s and f (r) = h(r s) for s r < s + t.

46 Exponential product systems Take P t = Γ(L 2 ([0, t))) Define U s,t : P s P t P s+t by U s,t (e(g) e(h)) = e(f ), where f (r) = g(r) for 0 r < s and f (r) = h(r s) for s r < s + t. Then (P t, U s,t ) is a tensor product system.

47 Exponential product systems Take P t = Γ(L 2 ([0, t))) Define U s,t : P s P t P s+t by U s,t (e(g) e(h)) = e(f ), where f (r) = g(r) for 0 r < s and f (r) = h(r s) for s r < s + t. Then (P t, U s,t ) is a tensor product system. By considering L 2 ([0, t), K) we can get more examples. These are called exponential tensor product systems.

48 Units of product systems Let (P t, U s,t ) be a product system.

49 Units of product systems Let (P t, U s,t ) be a product system. A measurable, non-trivial family {u t }, with U s,t (u s u t ) = u s+t is called a unit of the product system.

50 Units of product systems Let (P t, U s,t ) be a product system. A measurable, non-trivial family {u t }, with U s,t (u s u t ) = u s+t is called a unit of the product system. In terms of E 0 -semigroup, a unit is a one parameter semigroup of operators {V t }, satisfying θ t (X )V t = XV t for all X.

51 Units of product systems Let (P t, U s,t ) be a product system. A measurable, non-trivial family {u t }, with U s,t (u s u t ) = u s+t is called a unit of the product system. In terms of E 0 -semigroup, a unit is a one parameter semigroup of operators {V t }, satisfying θ t (X )V t = XV t for all X. A product system may or may not have a unit.

52 Classification of product systems A product system is said to be type I, if it has units and units generate the product system.

53 Classification of product systems A product system is said to be type I, if it has units and units generate the product system. A product system is said to be type II, if it has units and they don t generate the product system.

54 Classification of product systems A product system is said to be type I, if it has units and units generate the product system. A product system is said to be type II, if it has units and they don t generate the product system. A product system is said to be type III, if it has no units.

55 Classification of product systems A product system is said to be type I, if it has units and units generate the product system. A product system is said to be type II, if it has units and they don t generate the product system. A product system is said to be type III, if it has no units. A numerical invariant called index can be defined and computed if the product system has units.

56 Classification of product systems A product system is said to be type I, if it has units and units generate the product system. A product system is said to be type II, if it has units and they don t generate the product system. A product system is said to be type III, if it has no units. A numerical invariant called index can be defined and computed if the product system has units. Exponential product systems are type I. They are all the type I product systems.

57 Exotic product systems: type II R. T. Powers was the first to construct type II and type III product systems.

58 Exotic product systems: type II R. T. Powers was the first to construct type II and type III product systems. B. Tsirelson constructed whole families of type II product systems using random sets such as zeros of a Brownian motion.

59 Exotic product systems: type II R. T. Powers was the first to construct type II and type III product systems. B. Tsirelson constructed whole families of type II product systems using random sets such as zeros of a Brownian motion. In the converse direction, V. Liebscher showed that any pair of a product system and a sub-system in it gives raise to such random sets. He showed that type II class is much richer than previously thought.

60 Exotic product systems: type II R. T. Powers was the first to construct type II and type III product systems. B. Tsirelson constructed whole families of type II product systems using random sets such as zeros of a Brownian motion. In the converse direction, V. Liebscher showed that any pair of a product system and a sub-system in it gives raise to such random sets. He showed that type II class is much richer than previously thought. R. T. Powers and his collaborators obtained large families of type II product systems using dilation theory for quantum dynamical semigroups.

61 Exotic product systems: type III R. T. Powers constructed a type III product system for the first time using CAR flows.

62 Exotic product systems: type III R. T. Powers constructed a type III product system for the first time using CAR flows. B. Tsirelson constructed families of type III product systems using gaussian measures and sum systems (almost direct sum systems).

63 Exotic product systems: type III R. T. Powers constructed a type III product system for the first time using CAR flows. B. Tsirelson constructed families of type III product systems using gaussian measures and sum systems (almost direct sum systems). Bh. and Srinivasan proved a generalization of Shale s theorem and got a functional analytic approach to Tsirelson s construction.

64 Exotic product systems: type III R. T. Powers constructed a type III product system for the first time using CAR flows. B. Tsirelson constructed families of type III product systems using gaussian measures and sum systems (almost direct sum systems). Bh. and Srinivasan proved a generalization of Shale s theorem and got a functional analytic approach to Tsirelson s construction. Izumi and Srinivasan introduced generalized CCR/CAR flows and showed that we can only get type I or type III examples through these methods.

65 Further work and Future directions Dilation theory of quantum dynamical semigroups.

66 Further work and Future directions Dilation theory of quantum dynamical semigroups. More general algebras.

67 Further work and Future directions Dilation theory of quantum dynamical semigroups. More general algebras. More general semigroups.

68 Further work and Future directions Dilation theory of quantum dynamical semigroups. More general algebras. More general semigroups. Quantum Stochastic Calculus.

69 Further work and Future directions Dilation theory of quantum dynamical semigroups. More general algebras. More general semigroups. Quantum Stochastic Calculus. Open problems.

70 THANKS AND HAPPY PONGAL!!