aa r X i v : . [ m a t h . OA ] J a n Opposite product system for the multiparameterCAR flows
Anbu ArjunanJanuary 5, 2021
Abstract
We consider the multiparameter CAR flows and describe its opposite. Wealso characterize the symmeticity of CAR flows in terms of associated isometricrepresentations.
AMS Classification No. :
Primary 46L55; Secondary 46L99.
Keywords : E -semigroups, CCR flow, CAR flow, opposite product system. Let P be a closed convex cone in R d . We assume that P − P = R d and P ∩ − P = { } .Let V be a pure isometric representation of P and let α be the CCR flow associated tothe isometric representation V . The author in [4] have shown that the CCR flow is notcocycle conjugate to the CAR flow when the isometric representation V is proper. Theproduct system associated with the CAR flow is not decomposable in general; see [1]. Itwas shown in [5] that α is cocycle conjugate to α op if and only if V is unitary equivalent toits opposite V op . This result uses the characterization of decomposable product systemwhich admits a unit; see [6]. It is natural to ask whether the analogous result holds truefor the multiparameter CAR flows. In this article we answer this question affirmatively;see Theorem 3.5. We will achieve this by identifying the opposite of the product systemfor a CAR flow with the product system for an appropriate CAR flow. Also we will alsouse this to study the symmetricity of the CAR flows.1 Preliminaries
Let H be a Hilbert space and let Γ a ( H ) be the antisymmetric Fock space over H . For f ∈ H , define a bounded operator a ( f ) ∗ on Γ a ( H ) as a ( f ) ∗ (Ω) = f and a ( f ) ∗ ( h ∧ h ∧ ... ∧ h n ) = f ∧ h ∧ h ∧ ... ∧ h n where Ω is the vacuum vector of Γ a ( H ) and h ∧ h ∧ ... ∧ h n is an arbitrary antisymmetricelementary tensor element with h , h , ..., h n ∈ H and n ≥
1. Let a ( f ) be the adjoint of a ( f ) ∗ . The operators a ( f ) ∗ and a ( f ) are called the creation and the annihilation operatorassociated to a vector f .By an isometric representation of P on a Hilbert space H , we mean a strongly con-tinuous map V : P → B ( H ) such that each V x is an isometry and V x V y = V x + y for each x, y ∈ P . For a given isometric representation V : P → B ( H ), there exists a unique E -semigroup, denoted by β V , on Γ a ( H ) satisfying β Vx ( a ( f )) = a ( V x f ) for each f ∈ H. This E -semigroup β V is called the CAR flow associated to the isometric representation V ; see [4].Let H and K be Hilbert spaces. For an isometry W : H → K , there exists a uniquebounded operator Γ a ( W ), called the second quantization of W , from Γ a ( H ) to Γ a ( K ),satisfying Γ a ( W )(Ω) = Ω , andΓ a ( W )( f ∧ f ∧ ... ∧ f n ) = W f ∧ W f ∧ ... ∧ W f n , where Ω is the vacuum vector in the appropriate antisymmetric Fock space and f ∧ f ∧ ... ∧ f n is any antisymmetric elementary tensor element with f , f , ..., f n ∈ H and n ≥ Let V be a pure isometric representation of P on a Hilbert space H . Let β V be theCAR flow associated to the isometric representation V and denote its concrete productsystem by E β V . Set E V ( x ) = Γ a (Ker( V ∗ x )). Consider the set E V as E V = { ( x, f ) : x ∈ Ω and f ∈ E V ( x ) } . E V is a Borel subset of Ω × Γ a ( H ), E V is a standard Borel space. Define amultiplication . on E V as ( x, f ) . ( y, f ) := ( x + y, f ⊗ Γ a ( V x ) g )for every ( x, f ) , ( y, f ) ∈ E V . E V equipped with the above multiplication defines aproduct system structure over Ω. We define another multiplication ◦ on E V as( x, f ) ◦ ( y, f ) := ( x + y, g ⊗ Γ a ( V y ) f ) . Then the pair ( E V , ◦ ) also has a structure of product system over Ω, called the oppositeproduct system for ( E V , . ), denoted by ( E V ) op .Let x ∈ Ω and let f ∈ E V ( x ) be given. Define a bounded operator T f on Γ a ( H ) as T f η = f ⊗ Γ a ( V x ) η, for every η ∈ Γ a ( H ) . Then we have the following lemma.
Lemma 3.1
The map θ : E V ∋ ( x, f ) ( x, T f ) ∈ E β V is an isomorphism as productsystems.Proof : Let ( x, f ) , ( y, g ) ∈ E V be given. Since T f T g = T f ⊗ Γ a ( V x ) g , it follows that θ ( x, f ) θ ( y, g ) = θ (( x, f )( y, g )). For each x ∈ Ω, the restriction of θ to E V ( x ), θ | E V ( x ) : E V ( x ) → E β V ( x ) is a unitary. For let f, g ∈ E V ( x ) be given. Note that T ∗ g T f = h f, g i E V ( x ) and T f ∈ E β V ( x ). This implies that the map E V ( x ) ∋ f T f ∈ E β V ( x )is an isometry. To prove that the map is a unitary it suffices to show that whenever T ∈ E β V ( x ) such that h T f , T i = 0 for all f , then T = 0. Since the linear span of the set { f ⊗ Γ a ( V x ) η : f ∈ E V ( x ) and η ∈ Γ a ( H ) } is dense in Γ a ( H ), we see that T = 0.Since E V and E β V are standard Borel spaces and the restriction of θ to each fibre is aunitary, it follows that the map θ is a Borel isomorphism and hence it is a isomorphismas product systems by [2]. ✷ Let us recall the opposite isometric representation V op for the given isometric repre-sentation V considered in [6]. Let U be a minimal unitary dilation of V . More precisely,there exists a Hilbert space e H containing H as a subspace and a unitary representation U of R d on a Hilbert e H such that the following conditions hold.1. For x ∈ P , U x ξ = V x ξ .2. The set ∪ x ∈ P U ∗ x H is dense in e H . 3ote that for x ∈ P , K = H ⊥ is invariant under U x . For x ∈ P , define V op x on K to bethe restriction of U − x to K i.e. V op x := U − x | K . Then V op := { V op x } x ∈ P is an isometricrepresentation of P , called the opposite isometric representation for V . This isometricrepresentation V op is pure [5, Proposition 3.2]. Proposition 3.2
The map φ : ( E V ) op ∋ ( x, f ) ( x, Γ a ( U − x ) f ) ∈ E V op is an isomor-phism as product systems.Proof : For each x ∈ Ω, the map Ker( V ∗ x ) h U − x h ∈ Ker(( V op x ) ∗ ) is a unitary; seethe proof of [5, Proposition 3.2]. Then it follows that the map φ : ( E V ) op ∋ ( x, f ) ( x, Γ a ( U − x ) f ) ∈ E V op is a continuous bijection and its inverse is given by E V op ∋ ( x, ξ ) ( x, Γ a ( U x ) ξ ) ∈ ( E V ) op . Hence it is a Borel isomorphism by [2]. Now it remains to showthat φ follows product system structure. Let ( x, f ) , ( y, g ) ∈ E V be given. Then we have φ (( x, f )( y, g )) = φ ( x + y, f ⊗ Γ a ( V x ) g )= ( x + y, Γ a ( U − ( x + y ) )( f ⊗ Γ a ( V x ) g ))= ( x + y, Γ a ( U − ( x + y ) )Γ a ( V x ) g ⊗ Γ a ( U − ( x + y ) ) f )= ( x + y, Γ a ( U − y ) g ⊗ Γ a ( U − y )Γ a ( U − x ) f )= ( x + y, Γ a ( U − y ) g ⊗ Γ a ( V op y )Γ a ( U − x ) f )= ( y, Γ a ( U − y ) g )( x, Γ a ( U − x ) f )= φ ( y, g ) φ ( x, f ) . Hence the map φ is an isomorphism as product systems. ✷ Let E β V be the concrete product system for β V and let E op β V be its opposite productsystem. By [3, Theorem 3.14], there exists an E -semigroup denoted by ( β V ) op such that E op β V is isomorphic to E ( β V ) op . Corollary 3.3 An E -semigroup ( β V ) op is cocycle conjugate to β V op .Proof : By Proposition 3.2 and Lemma 3.1, we conclude that ( E V ) op is isomorphic to E β V op . This implies that the product system E ( β V ) op is isomorphic to E β V op by Lemma3.1. Then by [3, Theorem 2.9], we have ( β V ) op is cocycle conjugate to β V op . ✷ Remark 3.4
The above corollary implies that the opposite of a CAR flow over P isagain a CAR flow over P . Theorem 3.5
Let β V be the CAR flow associated to an isometric representation V .Then the following are equivalent. . The CAR flow β V is cocycle conjugate to its opposite ( β V ) op
2. The isometric representation V is unitary equivalent to its opposite V op .Proof : Proof follows from [4, Proposition 4.7] and Corollary 3.3. ✷ By a P -module we mean a non-empty closed subset A of R d such that A + P ⊆ A . Let A be a P -module. For x ∈ P , define an operator V Ax on L ( A ) as( V Ax f )( y ) = f ( y − x ) if y − x ∈ A, y − x / ∈ A. for each f ∈ L ( A ). Then the family { V Ax } x ∈ P is an isometric representation of P . Proposition 4.1 (See [5, Proposition 3.4]) We have the following.1. The isometric representation ( V A ) op is unitary equivalent to V A .2. There exists an element z ∈ R d such that A = − ( int ( A ) c ) + z .Here int ( A ) is the interior of A and int ( A ) c is the complement of int ( A ) in R d . Let β A be the CAR flow associated to the isometric representation V A . It follows fromTheorem 3.5 and Proposition 4.1 that the CAR flow β A is cocycle conjugate to itsopposite ( β A ) op if and only if A = − (int( A ) c ) + z for some z ∈ R d . Remark 4.2
By considering the existence of such P -modules, we can see that there areuncountably many symmetric CAR flows as well as asymmetric CAR flows over P . Acknowledgment
The author would like to thank The Institute of Mathematical Sciences for the InstitutePostdoctoral fellowship. 5 eferences [1] Anbu Arjunan,
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