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Weakly ordered a-commutative partial groups of linear operators densely defined on Hilbert space Jiˇr´ı Janda Department of Mathematics and Statistics Masaryk University, Brno, Czech Republic The 50th Summer School on Algebra and Ordered Sets

4. 9. 2012

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An Effect algebra

Definition (Foulis, Bennett, 1994) 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 : (Ei) x + y = y + x if x + y is defined, (Eii) (x + y ) + z = x + (y + z) if one side is defined, (Eiii) for every x ∈ E there exists a unique y ∈ E such that x + y = 1 (we put x 0 = y ), (Eiv) if 1 + x is defined then x = 0. On every effect algebra E a partial order ≤ can be introduced as follows: x ≤ y iff x + z is defined and x + z = y .

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A Generalized effect algebra Definition A partial algebra (E ; +, 0) is called a generalized effect algebra if 0 ∈ E is a distinguished element and + is a partially defined binary operation on E which satisfies the following conditions for any x, y , z ∈ E : (GEi)

x + y = y + x, if one side is defined,

(GEii) (x + y ) + z = x + (y + z), if one side is defined, (GEiii) x + 0 = x, (GEiv) x + y = x + z implies y = z (cancellation law), (GEv) x + y = 0 implies x = y = 0. A binary relation ≤ (being a partial order) on E can be defined by: x ≤ y iff x + z is defined and x + z = y . 3 / 23

A Weakly ordered a-commutative partial group Definition A partial algebra (G ; +, 0) is called an 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 + 0 is defined and x + 0 = x, (Giii) for every x ∈ E there exists a unique y ∈ E such that x + y = 0 (we put −x = y ), (Giv) If (x + y ) + z and (y + z) are defined, then x + (y + z) is defined and (x + y ) + z = x + (y + z). An a-commutative partial group (G ; +, 0) is called weakly ordered (shortly woa-group) with respect to a reflexive and antisymmetric relation (we call it a weak order) ≤ on G if (Ri) x ≤ y iff 0 ≤ z and x + z is defined and x + z = y , (Rii) 0 ≤ x, y and x + y defined, then 0 ≤ x + y , (Riii) 0 ≤ x, 0 ≤ z, x ≤ y and y + z defined implies x + z defined. 4 / 23

Woa-subgroup Definition Let (G , +, 0) be an a-commutative partial group and let S be 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 a-commutative partial subgroup of G . Let G be a woa-group with respect to a weak order ≤G and let ≤S be a weak order on an 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 woa-subgroup of G . Lemma Let (G , +, 0) be a woa-group w.r.t ≤ and S ⊆ G its woa-subgroup w.r.t. ≤S . Then (S, +/S , 0) w.r.t ≤S is a woa-group. 5 / 23

A total operation +

Lemma Let (G , +, 0) w.r.t. to ≤ be a woa-group. Whenever + is a total operation, then (G , +, ≤, 0) is a partially ordered commutative group.

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Basic properties Lemma Let (G , +, 0) be an a-commutative partial group. Then for any a, b, c, x, y , z ∈ G the following holds: 1

a + c = b iff c = b + (−a)

2

a + x = (a + y ) + z implies x = (y + z)

3

whenever a + b is defined, then (−a) + (−b) is defined and (−a) + (−b) = −(a + b)

Lemma Let (G , +, 0) w.r.t. ≤ be a woa-group. Then for any a, b, c, x, y , z ∈ G the following holds: 1

a ≤ b iff b + (−a) ≥ 0

2

a ≤ b iff −b ≤ −a 7 / 23

A positive cone forms a GEA

Theorem Let (G , +, 0) w.r.t. ≤ be a woa-group. Then the set Pos(G ) = {x ∈ G | 0 ≤ x} with the restriction of the partial operation + on Pos(G ), i.e., (Pos(G ), +/Pos(G ) , 0) forms a generalized effect algebra.

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An example of a woa-group

Example An interval [−1, 1] with +G defined for 0 ≤ x, y by x +G y = x + y iff x + y ≤ 1, x +G (−y ) = x − y , (−x) +G (−y ) = −(x +G y ) whenever (x +G y ) is defined and relation ≤G defined by x ≤G y iff x ≤ y and x − y ≤ 1 for all x, y ∈ [−1, 1] forms woa-group. Pos([−1, 1], +G , 0) forms with the restriction of operation +G on [0, 1] an effect algebra ([0, 1], +E , 0, 1).

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Positive elements Lemma Let (G , +, 0) be a woa-group w.r.t ≤. Let a, b ∈ G , a ≤ b and ⊕ be a partial operation on interval [a, b] such that for x, y ∈ [a, b], x ⊕ y exists if and only if (x − a) + (y − a) ∈ [0, b − a] ⊆ Pos(G ) and is given by x ⊕ y = ((x − a) + (y − a)) + a. Then ([a, b], ⊕, a, b) forms an effect algebra isomorphic to ([0, b − a], +/[0,b−a] , 0, b − a).

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Positive elements Lemma Let (G , +, 0) be a woa-group w.r.t ≤. Let a, b ∈ G , a ≤ b and ⊕ be a partial operation on interval [a, b] such that for x, y ∈ [a, b], x ⊕ y exists if and only if (x − a) + (y − a) ∈ [0, b − a] ⊆ Pos(G ) and is given by x ⊕ y = ((x − a) + (y − a)) + a. Then ([a, b], ⊕, a, b) forms an effect algebra isomorphic to ([0, b − a], +/[0,b−a] , 0, b − a). Lemma Let (G , +, 0) be an a-commutative partial group and E ⊆ G a subset closed under the +, i.e., x, y ∈ E , x + y ∈ G implies x + y ∈ E , such that 0 ∈ E and (E , +/E , 0) forms a generalized effect algebra. Define a relation ≤ by x ≤ y iff (−x) + y is defined and ((−x) + y ) ∈ E . Then (G , +, 0) is a woa-group w.r.t. ≤, Pos(G ) = E and ≤ on Pos(G ) coincides with induced partial order ≤E from (E , +/E , 0). 11 / 23

Woa-group as an extension of a GEA

Theorem Let (E , +, 0) be a generalized effect algebra with induced order ≤. Then there exists a woa-group (G , ⊕, 0) w.r.t. relation ≤G such that (Pos(G ), ⊕/Pos(G ) , 0) = (E , +, 0) and ≤G /Pos(G ) =≤.

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A construction of the negative cone Let (E , +, 0) be a generalized effect algebra with induced order ≤. For any a, b ∈ E , a ≤ b, the symbol b − a denotes such element that a + (b − a) = b. Let E − be a set with the same cardinality disjoint from E . Consider a bijectionSϕ : E → E − . We set a− = ϕ(a) for a ∈ E r {0} and 0− = 0. Let G = E ˙ (E − r {ϕ(0)}) be a disjoint union of E and E − r {ϕ(0)}. Let us define define a partial binary operation ⊕ on G by a ⊕ b exists iff a + b exists and then a ⊕ b = a + b for all a, b ∈ E a− ⊕ b − exist iff a + b exists and then a− ⊕ b − = (a + b)− for all a, b ∈ E a ⊕ b − = b − ⊕ a is defined iff 1 2

b ≤ a (a − b exists) then a ⊕ b − = a − b or a ≤ b (b − a exists) then a ⊕ b − = (b − a)− for any nonzero a, b ∈ E .

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Hilbert space

Infinite-dimensional complex Hilbert space H - a linear space over C with inner product h· , ·i which is complete in the induced metric. Linear operator A - linear map A : D(A) → H where D(A) is dense subspace in H w. r. t. induced metric. To every linear operator A : D(A) → H with D(A) = H there exists the adjoint operator A∗ of A such that D(A∗ ) = {y ∈ H | there exists y ∗ ∈ H such that hy ∗ , xi = hy , Axi for every x ∈ D(A)} and A∗ y = y ∗ for every y ∈ D(A∗ ).

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Linear operators

We call linear operator A: bounded - if there is a real number c ≥ 0 Such, that kAxk ≤ ckxk for all x ∈ D(A), B(H) denotes the set of all bounded operators with domain H, unbounded - if it is not bounded, positive - hAx, xi ≥ 0 for all x ∈ D(A), self-adjoint - A = A∗ , By 0, we denote null operator. For an arbitrary A, B, the usual sum of operators : (A + cB)x = Ax + cBx, for c ∈ C.

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Sets of linear oparators

For an infinite-dimensional complex Hilbert space H, let us define Gr (H) = {A : D(A) → H | D(A) = H and D(A) = H if A is bounded}

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Sets of linear oparators

For an infinite-dimensional complex Hilbert space H, let us define Gr (H) = {A : D(A) → H | D(A) = H and D(A) = H if A is bounded} and its subsets GrD (H) = {A ∈ Gr (H) | D(A) = D or A is bounded} SaGr (H) = {A ∈ Gr (H) | A = A∗ } SaGrD (H) = {A ∈ SaGr (H) | D(A) = D or A is bounded} V(H) = {A ∈ Gr (H) | A ≥ 0}.

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Operation ⊕D Theorem 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 = Ab . Definition 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  A + B (the usual sum) if A + B is unbounded     and (D(A) = D(B) or     one out of A, B  is bounded), A ⊕D B =  b  (A + B) if A + B is bounded and      D(A) = D(B),   undefined otherwise. 18 / 23

Operation ⊕u Definition Let H be an infinite-dimensional complex Hilbert space. Let ⊕u be a partial operation on Gr (H) defined as follows. A ⊕u B = A ⊕D B iff at least one of A, B ∈ Gr (H) is bounded or A, B ∈ Gr (H) are both unbounded, D(A) = D(B) and there exists a real number λA B 6= 0 such that A − λA B B is bounded. Definition X For an arbitrary subset X ⊆ Gr (H) let us define a relation ≤X D (resp. ≤u ) X X such that for any A, B ∈ X , A ≤D B (resp. A ≤u B) iff there exists a positive C ∈ X such that A ⊕D C = B (resp. A ⊕u C = B).

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Woa-groups Theorem Let H be an infinite-dimensional complex Hilbert space. Then Gr (H) (Gr (H), ⊕D , 0) w.r.t. relation ≤D forms a woa-group. Moreover, Gr (H) (GrD (H), ⊕D/GrD (H) , 0) w.r.t. relation ≤D D forms its woa-subgroup. Theorem Let H be an infinite-dimensional complex Hilbert space. Then Gr (H) (Gr (H), ⊕u , 0) w.r.t relation ≤u forms a woa-group. Moreover, Gr (H) (GrD (H), ⊕u /GrD (H) , 0) w.r.t. relation ≤u D , SaGr (H)

(SaGr (H), ⊕u /SaGr (H) , 0) w.r.t. relation ≤u (SaGrD (H), ⊕u /SaGrD (H) , 0) w.r.t. relation woa-subgroups.

and

SaGr (H) ≤u D

form its

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GEA’s and Woa-Groups

Woa-Groups: (B(H), +, 0),

Generalized Effect Algebras: B(H) ≤+

Pos(B(H), +, 0) = (B + (H), +, 0)

(Gr (H), ⊕, 0), ≤

Pos(Gr (H), ⊕, 0) = (V(H), ⊕, 0)

(Gr (H), ⊕D , 0), ≤D

Pos(Gr (H), ⊕D , 0) = (V(H), ⊕D , 0)

(Gr (H), ⊕u , 0), ≤u

Pos(Gr (H), ⊕u , 0) = (V(H), ⊕v , 0)

(SaGr (H), ⊕u , 0),

SaGr (H) ≤u

Pos(SaGr (H), ⊕u , 0) = (Sa, ⊕v , 0)

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Intervals on Woa-groups Gr (H)

Gr D (H)

SaGrD(H)

B A

A

[B]u B-C

B-A

C A-C B

0 GrD2(H)

B(H)

0 GrD1(H)

[A]u B(H)

0 C

Sa

[B]u [A]uSa HGr(H)

Corollary Let H be an infinite-dimensional complex Hilbert space and let Gr (H) Gr (H) A, B ∈ Gr (H), A ≤D B (resp. A ≤u B). Then the interval Gr (H) Gr (H) SaGr (H) [A, B]D (resp. [A, B]u or [A, B]u if A, B ∈ SaGr (H)) can be organised into an effect algebra which has nonempty set of states.

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Thank you for your attention!

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