More on PT -Symmetry in (Generalized) Eﬀect Algebras and Partial Groups

We continue in the direction of our paper on PT -Symmetry in (Generalized) Eﬀect Algebras and Partial Groups. Namely we extend our considerations to the setting of weakly ordered partial groups. In this setting, any operator weakly ordered partial group is a pasting of its partially ordered commutative subgroups of linear operators with a ﬁxed dense domain over bounded operators. Moreover, applications of our approach for generalized eﬀect algebras are mentioned.


Introduction
It is a well known fact that unbounded linear operators play the role of the observable in the mathematical formulation of quantum mechanics.Examples of such observables corresponding to the momentum and position observables, respectively, are the following self-adjoint unbounded linear operators on the Hilbert space L 2 (R): (i) The differential operator A defined by where i is the imaginary unit and f is a differentiable function with compact support.Then D(A) = L 2 (R), since otherwise the derivative need not exist.(ii) (Bf )(x) = xf (x), multiplication by x and again D(B) = L 2 (R), since otherwise xf (x) need not be square integrable.Note that in both cases the possible domains are dense sub-spaces of L 2 (R), i.e., D(A) = D(B) = L 2 (R).
The same is true in general, since for any unbounded linear operator there is no standard way to extend it to the whole space H.By the Hellinger-Toeplitz theorem, every symmetric operator A with D(A) = H is bounded.
An important attempt at an alternative formulation of quantum mechanics started in the seminal paper [1] by Bender and Boettcher in 1998.Bender and others adopted all the axioms of quantum mechanics except the axiom that restricted the Hamiltonian to be Hermitian.They replaced this condition with the requirement that the Hamiltonian must have an exact PT -symmetry.Later, A. Mostafazadeh [6] showed that PT -symmetric quantum mechanics is an example of a more general class of theories, called Pseudo-Hermitian Quantum Mechanics.
In [4] Foulis and Bennett introduced the notion of effect algebras that generalized the algebraic structure of the set E(H) of Hilbert space effects.In such a case the set E(H) of effects is the set of all self-adjoint operators A on a Hilbert space H between the null operator 0 and the identity operator 1 and endowed with the partial operation + defined iff A + B is in E(H), where + is the usual operator sum.Recently, M. Polakovič and Z. Riečanová [8] established new examples of generalized effect algebras of positive operators on a Hilbert space.
In [7] we showed how the standard effect algebra E(H) and the latter generalized effect algebras of positive operators are related to the type of attempt mentioned above.As a by-product, we placed some of the results from [8] under the common roof of partially ordered commutative groups.
The aim of the present note is to continue in this direction.
The paper is organized in the following way.In Section 1 we recall the basic notions concerning the theory of (generalized) effect algebras and partially ordered commutative groups.In Section 2 we show that the linear operators on H and symmetric linear operators on H are equipped with the structure of a weakly ordered commutative partial group.In Section 3 we manifest the fact that each of these operator structures is a pasting of partially ordered commutative groups of the respective operators with a fixed dense domain.In the last section we show that our results concerning the application of renormalization due to the PT -symmetry of an operator from [7] remain true for weakly ordered commutative partial groups.

Basic definitions and some known facts
The basic reference for the present text is the classic book by A. Dvurečenskij and S. Pulmannová [3], where the interested reader can find unexplained terms and notation concerning the subject.We now review some terminology concerning (generalized) effect algebras and weakly ordered partial commutative groups.Definition 1 ( [4]) A partial algebra (E; +, 0, 1) is called an effect algebra if 0, 1 are two distinct elements and + is a partially defined binary operation on E which satisfy the following conditions for any x, y, z ∈ E: one side is defined, (Eiii) for every x ∈ E there exists a unique y ∈ E such that x + y = 1 (we put x = y), (Eiv) if 1 + x is defined then x = 0.
In every generalized effect algebra E the partial binary operation and relation ≤ can be defined by (ED) x ≤ y and y x = z iff x + z is defined and x + z = y.Then ≤ is a partial order on E under which 0 is the least element of E.
Note that every effect algebra satisfies the axioms of a generalized effect algebra, and if a generalized effect algebra has the greatest element then it is an effect algebra.
Definition 3 A partial algebra (G; +, 0) is called a commutative partial group if 0 ∈ E is a distinguished element and + is a partially defined binary operation on E which satisfy the following conditions for any x, y, z ∈ E: (Gi) x + y = y + x if x + y is defined, (Gii) (x + y) + z = x + (y + z) if both sides are defined, (Giii) x + 0 is defined and x + 0 = x, (Giv) for every x ∈ E there exists a unique y ∈ E such that x + y = 0 (we put −x = y), (Gv) x + y = x + z implies y = z.

We will put ⊥
A commutative partial group (G; +, 0) is called weakly ordered (shortly a wop-group) with respect to a reflexive and antisymmetric relation ≤ on G if ≤ is compatible w.r.t.partial addition, i.e., for all x, y, z ∈ G, x ≤ y and both x + z and y + z are defined implies x+z ≤ y +z.We will denote by P os(G) Recall that wop-groups equipped with a total operation + such that ≤ is an order are exactly partially ordered commutative groups.
Throughout the paper we assume that H is an infinite-dimensional complex Hilbert space, i.e., a linear space with inner product • , • which is complete in the induced metric.Recall that here for any x, y ∈ H we have x, y ∈ C (the set of complex numbers) such that x, αy + βz = α x, y + β x, z for all α, β ∈ C and x, y, z ∈ H.Moreover, x, y = y, x and finally x, x ≥ 0 at which x, x = 0 iff x = 0.The term dimension of H in the following always means the Hilbertian dimension defined as the cardinality of any orthonormal basis of H (see [2]).
Moreover, we will assume that all considered linear operators A (i.e., linear maps A : D(A) → H) have a domain D(A) a linear subspace dense in H with respect to the metric topology induced by the inner product, so D(A) = H (we say that A is densely defined ).We denote by D the set of all dense linear subspaces of H.Moreover, by positive linear operators A, (denoted by A ≥ 0) it means that Ax, x ≥ 0 for all x ∈ D(A), therefore operators A are also symmetric, i.e., y, Ax = Ay, x for all x, y ∈ D(A) (for more details see [2]).
To every linear operator A :

The set of all bounded operators on H is denoted by B(H). For every bounded operator A : D(A) → H densely defined on D(A) = D ⊂ H exists a unique extension B such as D(B) = H and Ax = Bx for every x ∈ D(A).
We will denote this extension B = A b (for more details see [2]).Bounded and symmetric operators are called Hermitian operators.
We also write, for linear operators A : 3 Operator wop-groups as a pasting of operator sub-groups equipped with the usual sum of operators Definition 4 Let H be an infinite-dimensional complex Hilbert space.Let us define the following set of linear operators densely defined in H: Theorem 1 Let H be an infinite-dimensional complex Hilbert space.Let ⊕ D be a partial operation on Gr(H) defined for A, B ∈ Gr(H) by and ≤ be a relation on Gr(H) defined for A, B ∈ Gr(H) by A ≤ B iff there is a positive linear operator Then Gr(H) = (Gr(H); ⊕ D , 0) is a wop-group with respect to ≤.
Clearly, (Giv) follows from the fact that for every A ∈ Gr(H) there exists a unique B ∈ Gr(H) such that A ⊕ D B = 0 (namely we put −A = B and evidently A + B = 0 /D(A) yields 0 b /D(A) = 0).It remains to check (Gii) and (Gv).This will be proved by cases.Assume that (A

∩D(B) ∈ {D(A), D(B)} and D((A+B)+C) = D(A + B) ∩ D(C) ∈ {D(A), D(B), D(C)}. Assume for the moment that D((A + B) + C) = D(A) = H (the other cases follow by a symmetric argument).
We have the following possibilities: (the other case where D(B) = H is symmetric).We will distinguish the following cases: (α 2 ): Let (A⊕ D B)⊕ D C be of the form ((A+B) b +C) b , hence (A + B) b + C is bounded A + B is bounded and then C is also bounded.We will verify each of the following cases: (α 3 ): And in the last case, let (A If B is unbounded then D(B) = D(A) and we will distinguish the following cases: (γ 1 ): A + B is bounded and hence also A + C is bounded.It follows that C is unbounded and hence In both cases we get as above from (Gii) that If B is bounded we have that A+B is unbounded.This implies that A+ C is unbounded as well.
The situation for (A+ C 1 ) b = B with B bounded and (B + C 2 ) = A with A unbounded is symmetric to the previous case with The last case is (A + C 1 ) b = B with B bounded and (B + C 2 ) b = A with A bounded too.Hence C 1 and C 2 are bounded as well and (C Hence by (Giii) and (Giv) we obtain that So, it remains to check that ≤ is compatible with addition, i.
For the case when E is bounded, clearly, (C ⊕ D A) ⊕ D E is always defined.Now assume that E is unbounded.In the case when A is unbounded and B is bounded we get that D(E) = D(A) and D((E + A) b ) = H.We have the following possibilities: (a): In the case when both A, B are unbounded we obtain that In the last case assume that A is bounded and B is unbounded, hence Recall that we have the following result of [9].
Theorem 2 [9, Theorem 1] Let H be an infinitedimensional complex Hilbert space.Let us define the following set of positive linear operators densely defined in H: either at least one out of A, B is bounded 2. or both A, B are unbounded and ) is a generalized effect algebra such that ⊕ D extends the operation ⊕.

Then P os(Gr(H)) = V(H), hence the generalized effect algebra V(H) is the positive cone of Gr(H) and, for all A, B, C ∈ V(H), (A ⊕
Definition 5 Let (G, +, 0) be a commutative partial group and let S be a subset of G such as: (Si) 0 ∈ S, (Sii) −x ∈ S for all x ∈ S, (Siii) for every x, y ∈ S such x + y is defined also x + y ∈ S. Then we call S a commutative partial subgroup of G.
Let G be a wop-group with respect to a partial order ≤ G and let ≤ S be a partial order on a commutative partial subgroup S ⊆ G.If for all x, y ∈ S holds: x ≤ S y if and only if x ≤ G y, we call S a wop-subgroup of G.
For a commutative partial group G = (G, +, 0) and a commutative partial subgroup S, we denote + S = + /S 2 .We will omit an index and we will write S = (S, +, 0) instead of (S, + S , 0) where no confusion can result.
Lemma 1 Let G = (G, +, 0) be a commutative partial group and let S be a commutative partial subgroup of G. Then (S, +, 0) is a commutative partial group.
Let G be a wop-group and let S be a wop-subgroup of G. Then S is a wop-group.
Assume now that G is a wop-group such that S is a wop-subgroup of G.If x, y, z ∈ S, x ≤ S y and x + z, y + z are defined, then x + z, y + z ∈ S and Lemma 2 Let G = (G, +, 0) be a commutative partial group and S 1 , S 2 commutative partial subgroups of G. Then S = S 1 ∩S 2 is also a commutative partial subgroup of G.
(Siii): Assume that x, y ∈ S such that x + y is defined.Then x, y ∈ S 1 and x, y ∈ S 2 .Hence x+y ∈ S 1 and x + y ∈ S 2 .This yields x + y ∈ S.
If ϕ is a bijection such that ϕ and ϕ −1 are morphisms we say that ϕ is an isomorphism of commutative partial groups and G 1 and G 2 are isomorphic.
Moreover, let ≤ 1 on G 1 and ≤ 2 on G 2 be partial orders such that G 1 and G 2 are wop-groups.Let ϕ : G 1 → G 2 be a morphism between commutative partial groups.If for every x, y ∈ G 1 : x ≤ 1 y implies ϕ(x) ≤ 2 ϕ(y), then ϕ is a morphism between wopgroups.If ϕ is a bijection, ϕ and ϕ −1 are morphisms we say that ϕ is an isomorphism of wop-groups and G 1 and G 2 are isomorphic as wop-groups.From the definition we can see that HGr(H) ⊆ SGr(H).It is a well known fact that every positive operator is symmetric and every positive bounded operator is both self-adjoint and Hermitian (see [2]).

Proof.
Conditions (Si) and (Sii) are clearly satisfied.We have to verify that SGr(H) is closed under addition.Let A, B ∈ SGr(H) be bounded, then they are Hermitian and it is well known that the sum of two Hermitian operators is also Hermitian.
Recall that A ⊂ A * iff for all x, y ∈ D(A) : x, Ay = Ax, y .If A is bounded and B is unbounded then D(A + B) = D(B) and, for all x, y ∈ D(B), it holds x, (A + B)y = x, Ay + x, By = Gr D (H) ∩ SGr(H).We have to check order preservation.For any A, B ∈ SGr D (H) such that A ≤ GrD(H) B we have that there exists positive C ∈ Gr D (H) such that A + C = B. Since every positive operator is symmetric we have that C ∈ SGr D (H).This yields that ≤ SGrD(H) = (≤ GrD(H) ) /SGrD(H) 2 .
Theorem 7 (The pasting theorem for Gr(H)) Let H be an infinite-dimensional complex Hilbert space.Then the wop-group Gr(H) pastes their partially ordered commutative subgroups Gr D (H)

PT -symmetry and related effect algebras
Let us repeat some of the notions concerning the basics of PT -symmetry from [7].Let H be a Hilbert space equipped with an inner product ψ, φ  Proposition 3] Let H be an infinite-dimensional complex Hilbert space.Assume moreover that Ω : H → H is an invertible linear operator and D ∈ D. Then 1. (A) D Ω is a positive operator on H Ω iff A is a positive operator on H.

(•) D
Ω is an isomorphism of partially ordered commutative groups.

(A) D
Ω is a Hermitian operator on H Ω iff A is a Hermitian operator on H.

(I) D
Ω = I.The preceding proposition immediately yields that Theorem 9 Let H be an infinite-dimensional complex Hilbert space and Ω : H → H an invertible linear operator.Then the map (•) Ω : Gr(H) → Gr(H Ω ) defined by (A) Ω := (A)

D(A) Ω
for all A ∈ Gr(H) is an isomorphism of wop-groups.

Proof.
It follows from Proposition 1 and Theorem 7.
Corollary 1 Let H be an infinite-dimensional complex Hilbert space and Ω : H → H an invertible linear operator.Then V(H) and V(H Ω ) are isomorphic generalized effect algebras.

Proof.
It follows immediately from the fact that (•) Ω preserves and reflects positive operators and from Theorem 9.
We say that an operator H : D → H defined on a dense linear subspace D of a Hilbert space H is η + -pseudo-Hermitian and η + is a metric operator if η + : H → H is a positive, Hermitian, invertible, linear operator such that H * = η + Hη −1  + (see also [6]).Theorem 10 Let H be an infinite-dimensional complex Hilbert space, let D ⊆ H be a linear subspace dense in H and let H : D → H be a η +pseudo-Hermitian operator for some metric operator η + : H → H such that η + = ρ 2 + .Then 1. Gr(H ρ+ ) and Gr(H) are mutually isomorphic wop-groups such that H ∈ Gr(H ρ+ ) and H is a self-adjoint operator with respect to the positivedefinite inner product −, − = ρ + −, ρ + − on H ρ+ .
2. V(H) and V(H ρ+ ) are mutually isomorphic generalized effect algebras.If moreover H is a positive operator with respect to −, − (i.e., its real spectrum will be contained in the interval [0, ∞)) then H ∈ V(H ρ+ ). Proof.
It follows from the above considerations and [7, Theorem 3].

Conclusion
In this paper we have shown that a η + -pseudo-Hermitian operator for some metric operator η + of a quantum system described by a Hilbert space H yields an isomorphism between the weakly ordered commutative partial group of linear maps on H and the weakly ordered commutative partial group of linear maps on H ρ+ .The same applies to the generalized effect algebras of positive operators introduced in [9].Hence, from the standpoint of (generalized) effect algebra theory the two representations of our quantum system coincide.
is bounded and C is unbounded.Then we prove: (α 4 ): D(A) = D(B) = H i.e.A and B are bounded.Then D((A + B) b + C) = D(C) and D(C) = D(B + C) = D(A + (B + C)).Hence ((A + B) b + C) = (A + (B + C)) = A ⊕ D (B ⊕ D C). (β 4 ): D(A) = D(B) = H, i.e.A and B are unbounded.Then if D Therefore D(A + B) = D(A) ⊆ D(B) and D(A) ⊆ D(C).We then have A + B = A + C. It follows again by (Gii) that C /D(A) is bounded and B /D(A) = C /D(A) .Therefore B = C.Let us check that ≤ is reflexive and antisymmetric.Let A, B ∈ Gr(H).Evidently, A ≤ A since A = A ⊕ D 0 and 0 is a positive bounded linear operator on H. Now, assume that B = A ⊕ D C 1 and A = B ⊕ D C 2 for some positive linear operators C 1 , C 2 on H.By cases we have that (A+C 1 = B with B unbounded or (A + C 1 ) b = B with B bounded) and (B+C 2 = A with A unbounded or (B+C 2 ) b = A with A bounded).Assume first that A+C 1 = B with B unbounded and B + C 2 = A with A unbounded.We have the following possibilities for C 1 and C 2 (see Tab. 1).Now assume that A + C 1 = B with B unbounded and (B + C 2 ) b = A with A bounded.Then C 1 has to be unbounded with D(C 1 ) = D(B) and C 2 can only also be unbounded with D e., for all A, B, C ∈ Gr(H) such that A ≤ B, C ⊕ D A and C ⊕ D B are defined we have that C ⊕ D A ≤ C ⊕ D B. Again by cases we have that (C ⊕ D A = C + A with C + A unbounded or C ⊕ D A = (C + A) b with C + A bounded) and (C ⊕ D B = C + B with C + B unbounded or

Definition 7
Let H be an infinite-dimensional complex Hilbert space.Let us define the following sets of linear operators densely defined in H:SGr(H) = {A ∈ Gr(H) | A ⊂ A * } HGr(H) = {A ∈ Gr(H) | A ⊂ A * , D(A) = H}.i.e.SGr(H) is the set of all symmetric operators and HGr(H) is the set of all Hermitian operators.

Theorem 3
Let H be an infinite-dimensional complex Hilbert space.Let ≤ S be a relation on SGr(H) defined for A, B ∈ SGr(H) by A ≤ S B if and only if there exists a positive operator C ∈ SGr(H) such as A ⊕ D C = B. Then (SGr(H); ⊕ D , 0) equipped with ≤ S forms a wop-subgroup of Gr(H).

Table 1 :
, D ⊆ H a dense linear subspace of H, together over B(H), i.e.Gr D1 (H) ∩ Gr D2 (H) = B(H) for every pair D 1 , D 2 of dense linear subspaces of H, D 1 = D 2 , and Gr(H) = {Gr D (H) | D ∈ D}.Proof.Straightforward from definition, for D ∈ D, every bounded A ∈ Gr(H) lies in Gr D (H).For any unbounded B ∈ Gr(H), B ∈ Gr D (H) if and only if D(B) = D hence there is unique Gr D (H) in which B lies.Hence Gr D1 (H) ∩ Gr D2 (H) = B(H) for all D 1 = D 2 , D 1 , D 2 ∈ D. And because Gr D (H) in which B lies exists for every B ∈ Gr(H), we have Gr(H) = {Gr D (H) | D ∈ D}.Theorem 8 (The pasting theorem for SGr(H)) Let H be an infinite-dimensional complex Hilbert space.Then the wop-group SGr(H) pastes their partially ordered commutative subgroups SGr D (H), D ∈ D, together over HGr(H), i.e., for every pair D 1 , D 2 of dense linear subspaces of H, D 1 = D 2 , SGr D1 (H) ∩ SGr D2 (H) = HGr(H) and SGr(H) = {SGr D (H) | D ∈ D}.Proof.Let D ∈ D. Since SGr D (H) = Gr D (H) ∩ SGr(H), with previous theorem D∈D SGr D . Let us denote H Ω the corresponding Hilbert space.Hence Ω : H Ω → H and Ω −1 : H → H Ω provide a realization of the unitaryequivalence of the Hilbert spaces H Ω and H.
. Let Ω : H → H be an invertible linear operator.Then