Acta Polytechnica


Acta Polytechnica 53(3):289–294, 2013 © Czech Technical University in Prague, 2013
available online at http://ctn.cvut.cz/ap/

WEAKLY ORDERED A-COMMUTATIVE PARTIAL GROUPS OF
LINEAR OPERATORS DENSELY DEFINED ON HILBERT SPACE

Jiří Janda∗

Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, CZ-611 37 Brno,
Czech Republic

∗ corresponding author: 98599@mail.muni.cz

Abstract. The notion of a generalized effect algebra is presented as a generalization of effect algebra
for an algebraic description of the structure of the set of all positive linear operators densely defined on a
Hilbert space with the usual sum of operators. The structure of the set of not only positive linear
operators can be described with the notion of a weakly ordered partial commutative group (wop-group).
Due to the non-constructive algebraic nature of the wop-group we introduce its stronger version called a
weakly ordered partial a-commutative group (woa-group). We show that it also describes the structure of
not only positive linear operators.

Keywords: (generalized) effect algebra, partial group, weakly ordered partial group, Hilbert space,
unbounded linear operator, self-adjoint linear operator.

AMS Mathematics Subject Classification: 06F05, (03G25, 81P10, 08A55).

1. Introduction
The notion of an effect algebra was presented by Foulis
and Bennett in [3]. The definition was motivated by
giving an algebraic description of positive self-adjoint
linear operators between the zero and the identity
operator in a complex Hilbert space H. The notion of
a generalized effect algebra extends these ideas on
unbounded sets of positive linear operators. To answer
the natural question concerning the structure of sets
of not only positive linear operators Paseka started to
investigate a partially ordered commutative group of
operators with a fixed domain [5]. In [6] Paseka and
Janda introduced the structure of a weakly ordered
partial commutative group (shortly a wop-group).
They also showed that the set of all linear operators on
complex Hilbert space H with the usual sum, which is
restricted to the same domain for unbounded operators
(partial operation ⊕D), possesses this structure. In [4]
we considered the structure on the important subset
of self-adjoint operators, showing that it is also a
wop-group.

Wop-groups have only a non-constructive associa-
tivity (the equation holds if and only if both sides
are defined). It has been shown [4] that the set of
all linear operators has generally stronger algebraic
properties. This was a motivation for introducing
the notion of a weakly ordered partial a-commutative
group (woa-group) where the associative law is more
constructive. Also a weak order is more strongly
related to the partial operation. Moreover, every
positive cone of a woa-group is a generalized effect
algebra. On the other hand, we present a construction
showing that every generalized effect algebra is a
positive cone of some woa-group.

2. Preliminaries
We review some basic terminology, definitions and
statements. The basic reference for this text is the
book by Dvurečenskij and Pulmannová [2].

Definition 1. 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.

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. A generalized effect algebra with
the top element 1 ∈ E is called an effect algebra and
we usually write (E, +, 0, 1).
A subset S of E is called a sub-generalized effect

algebra (sub-effect algebra) of E iff (i) 0 ∈ S (1 ∈ S),
(ii) if out of elements x,y,z ∈ E such that x + y = z
at least two are in S, then all x,y,z ∈ S.

Definition 2. [6] A partial algebra (G, +, 0) is called
a commutative partial group if 0 ∈ E is a distinguished

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Jiří Janda Acta Polytechnica

element and + is a partially defined binary operation
on E which satisfies the following conditions for any
x,y,z ∈ E:

(GPi) x + y = y + x if x + y is defined,
(GPii) (x + y) + z = x + (y + z) if both sides are
defined,

(GPiii) x + 0 is defined and x + 0 = x,
(GPiv) for every x ∈ E there exists a unique y ∈ E
such that x + y = 0 (we put −x = y),

(GPv) x + y = x + z implies y = z (cancellation
law).

We say that a commutative partial group (G, +, 0)
is 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.

Due to the non-constructive algebraic nature of
wop-groups, we will introduce a stronger structure
with the notion of a woa-group.

Definition 3. A partial algebra (G, +, 0) is called an
a-commutative partial group if 0 ∈ G is a distinguished
element and + is a partially defined binary operation
on G which satisfies the following conditions for any
x,y,z ∈ G:

(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 a woa-group) with respect to a
reflexive and antisymmetric relation ≤ on G (we call
it a weak order) if

(Ri) x ≤ y iff there exists 0 ≤ z, 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.

Note that in the case of generalized effect algebras
the partial order is induced from a partial operation.
On the other hand, the weak order for woa-groups
(wop-groups) can be chosen in various ways. We will
show that the weak order is determined by the partial
operation and the set of positive elements.

Remark 1. Using the commutativity (Gi) with the
axiom (Giv), we can obtain similar formulas to (Giv)
for any permutation of variables. That is, let us
have an a-commutative group (G, 0, +). Then for
any x,y,z ∈ G, from the existence of (x + y) + z
and x + z we have (x + y) + z = (x + z) + y. Or

similarly, the existence of x + (y + z) and x + y implies
x + (y + z) = (x + y) + z and so on. In the following,
we will be using (Giv) in this more general sense and
we omit mentioning the commutativity.

Definition 4. Let (G, +, 0) be an (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 that x + y is defined
also x + y ∈ S.

Then we call S an (a-)commutative partial subgroup
of G.
Let G be a wop-group (woa-group) with respect

to a relation ≤G and let ≤S be a relation on a
(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 (woa-subgroup) of G.

Lemma 1. 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).

Proof. (1.) We have c = c + (a + (−a)) = (c + a) +
(−a) = b + (−a).

(2.) Let us have a + x = (a + y) + z. Then x =
((a + y) + z) + (−a). Since y + a and y + (a + (−a))
are defined then (y + a) + (−a) is defined. We have
x = ((a+y)+z)+(−a) = (((a+y)+(−a))+z) = y+z.

(3.) Let us have a,b ∈ G such that a + b ∈ G. We
have a = a + (b + (−b)) = (a + b) + (−b) defined.
Then by (1.) (−b) = a + (−(a + b)) and also by (1.)
(−a) + (−b) = (−(a + b)).

Lemma 2. 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.

Proof. (1.) Let a ≤ b. Then there exists c ≥ 0 such
that a + c = b and from Lemma 1 (1.) b + (−a) ≥ 0.
On the other hand, let b + (−a) ≥ 0. Then b =
b + (a + (−a)) = a + (b + (−a)) hence by (Ri) a ≤ b.

(2.) Let a ≤ b for some a,b ∈ G. Then (b + (−a)) ≥ 0
and −a = ((−b) + b) + (−a) = (b + (−a)) + (−b) ie.
−b ≤−a.

Lemma 3. Every a-commutative partial group
(G, +, 0) is a partial commutative group.

Proof. The cancellation law follows from Lemma 1
(2.) choosing z = 0.

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Lemma 4. Every woa-group (G, +, 0) w.r.t. ≤ is a
wop-group w.r.t. ≤.

Proof. By the previous lemma, we have (G, +, 0) is a
partial commutative group. Let us have x,y,z ∈ G,
x ≤ y, x + z,y + z defined. Then by Lemma 2
(1.) 0 ≤ (y + (−x)) and x + (y + (−x)) = y hence
y + z = (x + (y + (−x)) + z = (x + z) + (y + (−x))
and according to (Rii) x + z ≤ y + z.

Example 1. Let G = {0,a,b,c,−a,−b,−c} be a set
with partial operation + defined for a + b = c and
0 + x = x, x + (−x) = 0 for all x ∈ G. Then
(G, +, 0) is a commutative partial group, but it is
not an a-commutative partial group. Note that even
c + (−b) = (a + b) + (−b) is not defined.

Lemma 5. Let (G, +, 0) be an a-commutative partial
group and S ⊆ G its a-commutative partial subgroup.
Then (S, +/S, 0) is an a-commutative partial group.

Proof. Immediately (Gi) and (Giv) follows from (Siii),
(Gii) from (Si), (Giii) from (Sii).

Lemma 6. 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.

Proof. By the previous lemma (S, +/S, 0) is an a-
commutative partial group. For (Ri) let x,y ∈ S,
x ≤S y. Then by Lemma 2 (1.) 0 ≤ y + (−x) is
defined and by (Siii) y + (−x) ∈ S. The other direction
is straightforward. (Rii) is clear and (Riii) follows
from (Siii).

Corollary 1. Let (G, +, 0) be a wop-group w.r.t. ≤
and S ⊆ G its wop-subgroup w.r.t. ≤S. Whenever
(G, +, 0) w.r.t. ≤ is also a woa-group, then S is its
woa-subgroup.

Theorem 1. 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.

Proof. (GEi) and (GEiii) hold from definition and
the cancellation law follows from Lemma 4. For the
axiom (GEv) let x,y ∈ Pos(G) such that x + y = 0.
Then x = (−y) and with Lemma 2 (2.) we have
−y ≤ 0 hence x = y = 0. We will verify (GEii).
Let us have x,y,z ∈ Pos(G) such that (x + y) + z is
defined. Therefore y ≤ x + y, (x + y) + z exists and
(Riii) implies that y + z exists. Using (Giv) we have
(x + y) + z = x + (y + z).

Lemma 7. 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).

Proof. Reflexivity is clear since 0 ∈ E. Let x ≤ y and
y ≤ x, then −x + y,x + (−y) ∈ E and with (Giv)
(−x + x) + (−y + y) = (−x + y) + (x + (−y)) = 0.
E is a generalized effect algebra hence by (GEv)
x + (−y) = −x + y = 0 that is x = y. Clearly
Pos(G) = E. Then (Ri) is straightforward using
the definition of ≤ and Lemma 1 (1.). (Rii) holds
because we want E to be closed under the +. For
(Riii), let x,z ∈ E, x ≤ y and y + z be defined. Then
y + z = (x + ((−x) + y)) + z = (x + z) + ((−x) + y))
using the associativity of generalized effect algebra E
since x,z, ((−x) + y) ∈ E.

We show the coincidence. Let us have −x + y,x,y ∈
E. Because E is closed on + we have x + (−x + y)
defined and x + (−x + y) = y, i.e., x ≤E y. On the
other hand let x ≤E y, x,y ∈ E. Then there exists
z ∈ E such that x + z = y and by Lemma 1 (1.)
z = y + (−x).

The previous lemma formalizes the idea of determin-
ing a weak order by the set of positive elements. Let
us have an a-commutative partial group and choose
some elements to be positive. Consider the smallest
set closed under partial addition containing zero and
the chosen elements. If the set has the form of a
generalized effect algebra, then there exists such a
weak order that our set is exactly the set of positive
elements. On the other hand, it is not hard to show
that if the set is not a generalized effect algebra, then
there is no such weak order that all of our chosen
elements are positive.

Theorem 2. 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)=≤.

Proof. Let (E, +, 0) be a generalized effect algebra
with induced order ≤. For any a,b ∈ E, a ≤ b, the
symbol b−a denotes such an element that a+(b−a) = b.
Let E− be a set with the same cardinality disjoint
from E.

Consider a bijection ϕ : 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.) b ≤ a (a− b exists) then a⊕ b− = a− b or
(2.) a ≤ b (b−a exists) then a⊕ b− = (b−a)− for
any nonzero a,b ∈ E.

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Jiří Janda Acta Polytechnica

It is not hard to show that the definition is correct.
For any a,b ∈ E it holds that
(α1)(a⊕ b)− = (a + b)− = (a− ⊕ b−).
Let a⊕ b− be defined and
(β1)b ≤ a (a−b exists), then (a⊕b−)− = (a−b)− =
a− ⊕ b.
We show that (G,⊕, 0) forms an a-commutative

partial group. Commutativity (Gi) is clear from the
definition. Since 0 ∈ E it follows a⊕ 0 = (a + 0) = a
and (a− ⊕ 0) = (a− 0)− = a− for all a ∈ E, that is
(Gii). Clearly for any a ∈ E there exists a− ∈ E−
where a⊕a− = a−a = 0. This defines also inverse
elements for any a− ∈ E−.
Let us verify the associativity case by case. We

assume that x,y,z ∈ E.

Case i. First, let us have (x⊕y)⊕z defined and y⊕z
defined. Then (x⊕y)⊕z = (x+y) +z = x+ (y +z) =
x⊕(y⊕z) where the existence and the equation follow
from the associativity of the generalized effect algebra
E.

Case ii. Let (x⊕y) ⊕z− and y⊕z− be defined. And
let

(α2) z ≤ y (i.e., y−z is defined). Hence z ≤ (x + y)
and (x⊕y) ⊕ z− = (x + y) − z. Since y − z exists,
we have y = (y − z) + z and then ((x + y) − z) +
z = x + y = x + ((y − z) + z) = (x + (y − z)) + z,
where we used the associativity of the generalized
effect algebra E. By the cancellation law we have
(x⊕y)⊕z− = (x + y)−z = x + (y−z) = x⊕(y⊕z−).

(β2) y ≤ z and z ≤ (x + y) (that is z − y and
(x + y) − z exist). We have z = (z − y) + y and
also ((x + y) − z) + z = x + y. Putting together
((z − y) + y) + ((x + y) − z) = x + y from which
(z−y)+((x+y)−z) = x hence (x+y)−z = x−(z−y).
So we have (x⊕y)⊕z− = (x + y)−z = x−(z−y) =
x⊕(z⊕y−)− = x⊕(z−⊕y). The last equation holds
by β1.

(γ2) y ≤ z and (x+y) ≤ z (hence z−y and z−(x+y)
exist). From (z−y)+y = (z−(x+y))+(x+y) we have
z−y = (z−(x+y)) +x hence (z−y)−x = z−(x+y).
Therefore (x⊕y) ⊕z− = (z − (x + y))− = ((z −y) −
x)− = (x⊕ (z −y)−) = x⊕ (y ⊕z−).

Case iii. Let (x− ⊕y) ⊕z and y ⊕z be defined. Let
(α3) x ≤ y (i.e., y −x is defined and also x ≤ y ≤

y+z). Then y = (y−x)+x and y+z = ((y−x)+x)+z
hence (y+z)−x = (y−x)+z. Therefore (x−⊕y)⊕z =
(y −x) + z = (y + z) −x = x− ⊕ (y ⊕z).

(β3) y ≤ x and (x − y) ≤ z (that is x − y and
z−(x−y) are defined). Since z + y is defined, we have
from z = (z − (x−y)) + (x−y) with x = (x−y) + y
the equation z +y = (z−(x−y)) +x and (z +y)−x =
z − (x − y). Hence (x− ⊕ y) ⊕ z = (x − y)− ⊕ z =
z − (x−y) = (z + y) −x = x− ⊕ (y ⊕z).

(γ3) y ≤ x and z ≤ (x − y) (hence x − y and
(x − y) − z are defined). We have (x − y) − z =
(x−y)−z from which x = ((x−y)−z) + y) + z hence

x− (y + z) = (x−y) −z. And then (x− ⊕y) ⊕z =
(x − y)− ⊕ z = ((x − y) − z)− = (x − (y + z))− =
(x ⊕ (y ⊕ z)−)− = x− ⊕ (y ⊕ z) (the last equation
follows from β1).

Case iv. Let (x⊕y−) ⊕z and y− ⊕z be defined. Let
(α4) y ≤ x and y ≤ z (i.e., x − y and z − y are

defined). Using y = x−(x−y) and z = (z−y)+y we get
(x−y) +z = (x−y) + ((z−y) +y) = (z−y) +x. Hence
(x⊕y−)⊕z = (x−y) + z = (z−y) + x = x⊕(y−⊕z).

(β4) y ≤ x and z ≤ y (hence x−y and y − z are
defined). Similarly y = (y−z) + z and x = (x−y) + y
and then x = (x − y) + ((y − z) + z) from which
x − (y − z) = (x − y) + z. Hence (x ⊕ y−) ⊕ z =
(x−y)+z = x−(y−z) = x−(y⊕z−) = x⊕(y⊕z−)− =
x⊕ (y− ⊕z) (the last equation follows from β1).

(γ4) x ≤ y and y ≤ z which implies (y − x) ≤ z
(hence y−x, z− (y−x) and z−y are defined). Then
z = (z −y) + y = (z −y) + ((y −x) + x) from which
z − (y −x) = (z −y) + x and hence (x⊕y−) ⊕z =
(y−x)−⊕z = z−(y−x) = (z−y) +x = x⊕(y−⊕z).

(δ4) x ≤ y, (y − x) ≤ z and z ≤ y (i.e., y − x,
z−(y−x) and y−z are defined). Then y = (y−z)+z =
(y−z) + ((z− (y−x)) + (y−x)) = (y−x) + x hence
(y−z) + (z− (y−x)) = x from which (x⊕y−) ⊕z =
(y−x)−⊕z = z−(y−x) = x−(y−z) = x−(y⊕z−) =
x⊕ (y− ⊕z) (using β1).

(�4) x ≤ y, z ≤ (y −x) and z ≤ y (that is y −x,
(y−x)−z and y−z are defined). Then y = (y−x)+x =
((y−x)−z)+z)+x = (y−z)+z. Hence (y−x)−z)+x =
(y−z) and (x⊕y−)⊕z = (y−x)−⊕z = ((y−x)−z)− =
((y −z) −x)− = (x⊕ (y −z)−) = x⊕ (y− ⊕z).

Until now, the map − : E → E− ∪ 0 has been
defined only for elements of E. We can extend it on
G by (a−)− = a, which gives us an involution on G.
Then for any a,b ∈ E whenever b − a is defined it
holds (a ⊕ b−)− = ((b − a)−)− = (b − a) = a− ⊕ b.
Together with (α1) and (β1) we have

(γ1)(a⊕ b−)− = a− ⊕ b for all a,b ∈ E,
(δ1)(a−⊕b−)− = ((a⊕b)−)− = a⊕b for all a,b ∈ E

Case v. Let (x−⊕y−)⊕z and y−⊕z be defined. Then
also y ⊕z− is defined and using (α1), (γ1) and (ii) we
get (x−⊕y−)⊕z = ((x⊕y)⊕z−)− = (x⊕(y⊕z−))− =
(x⊕ (y− ⊕z)).

Case vi. Let (x⊕y−)⊕z− and y−⊕z− be defined. Then
y ⊕z is defined and using (α1), (γ1) and (iii) we have
((x⊕y−)⊕z−) = ((x−⊕y)⊕z)− = (x−⊕(y⊕z))− =
(x⊕ (y− ⊕z−)).

Case vii. Let (x− ⊕y) ⊕z− and y ⊕z− be defined.
Then y− ⊕z is defined and with (α1), (γ1) and (iv)
we can see that ((x− ⊕y) ⊕z−) = ((x⊕y−) ⊕z)− =
(x⊕ (y− ⊕z))− = (x− ⊕ (y ⊕z−)).

Case viii. Let (x−⊕y−)⊕z− and y−⊕z− be defined.
Then y⊕z is defined and (x−⊕y−)⊕z− = (x⊕y)−⊕
z− = ((x⊕y)⊕z)− = (x⊕(y⊕z))− = x−⊕(y−⊕z−).

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vol. 53 no. 3/2013 Weakly Ordered A-Commutative Partial Groups of Linear Operators

Hence (G,⊕, 0) forms an a-commutative partial
group. Since E is a generalized effect algebra, by
Lemma 7 there exists a relation ≤G such that (G,⊕, 0)
w.r.t. ≤G forms a woa-group.

Example 2. An interval [−1, 1] with ⊕ defined for
0 ≤ x,y by
• x⊕y = x + y iff x + y ≤ 1,
• x⊕ (−y) = x−y,
• (−x) ⊕ (−y) = −(x⊕y) iff (x⊕y) exists
and relation ≤G defined by x ≤G y iff x ≤ y and
y −x ≤ 1 for all x,y ∈ [−1, 1] forms a woa-group. A
positive cone

(
[0, 1],⊕/[0,1], 0

)
forms a well-known unit

interval effect algebra.

3. Hilbert spaces
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. The
term dimension of H in the following always means
the Hilbertian dimension defined as the cardinality of
any orthonormal basis of H (see [1]).

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.
By positive linear operators A, (denoted by A ≥ 0) it
means that 〈Ax,x〉≥ 0 for all x ∈ D(A).
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 〈y∗,x〉 = 〈y,Ax〉 for every x ∈ D(A)
}
and

A∗y = y∗ for every y ∈ D(A∗). When A∗ = A, A is
called self-adjoint (for more details see [1]).
Recall that a linear operator A : D(A) → H

is called a bounded operator if there exists a real
constant C ≥ 0 such that ‖Ax‖ ≤ C‖x‖ for all
x ∈ D(A) and hence A is an unbounded operator
if to every C ∈ R, C ≥ 0 there exists xC ∈ D(A)
with ‖AxC‖ > C‖xC‖. By symbol 0 we mean a
null operator and it is a bounded operator. 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 exten-
sion B such as D(B) = H and Ax = Bx for every
x ∈ D(A). We will denote this extension B = Ab (see
again [1]).

Definition 5. For an infinite-dimensional complex
Hilbert space H, let us define these sets of operators
(in order to [4, 6]):

• Gr(H) = {A : D(A) →H| D(A) = H and D(A) =
H if A is bounded}

• 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}.

We also define an operation ⊕D on Gr(H) by

A⊕DB =




A + B if A + B is unbounded and(
D(A) = D(B) or
one out of A,B is bounded

)
,

(A + B)b if A + B is bounded
and D(A) = D(B),

undefined otherwise,

and operation ⊕u by 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 λAB 6= 0 such that A−λ

A
BB is bounded[4].

For an arbitrary subset X ⊆Gr(H) let us define a
relation ≤XD (resp. ≤

X
u ) such that for any A,B ∈ X,

A ≤XD B (resp. A ≤
X
u B) iff there exists a positive

C ∈ X such that A⊕D C = B (resp. A⊕u C = B).

Theorem 3. Let H be an infinite-dimensional com-
plex Hilbert space. Then

(
Gr(H),⊕D, 0

)
w.r.t.

relation ≤Gr(H)D forms a woa-group. Moreover,(
GrD (H),⊕D/GrD (H), 0

)
w.r.t. relation ≤GrD (H)D forms

its woa-subgroup.

Proof. It has been shown [6] that
(
Gr(H),⊕D, 0

)
w.r.t. relation ≤Gr(H)D forms a wop-group. Moreover,
in [4, Lemma 4] the axiom (Giv) is proved. Axiom
(Ri) holds by definition. Since

(
V(H),⊕D/V(H), 0

)
=(

Pos(Gr(H)),⊕D/P os(Gr(H)), 0
)
is a generalized ef-

fect algebra [8], (Rii) and (Riii) hold. According to
[6]
(
GrD (H), ⊕D/GrD (H), 0

)
w.r.t. ≤Gr(H)D is a wop-

subgroup hence by Corollary 1 it is also a woa-
subgroup.

Note that since the operation ⊕D/GrD (H) is total on
GrD (H), it is also a partially ordered commutative
group.

Theorem 4. Let H be an infinite-dimensional com-
plex Hilbert space. Then

(
Gr(H),⊕u, 0

)
w.r.t.

relation ≤Gr(H)u forms a woa-group. Moreover,(
GrD (H), ⊕u/GrD(H), 0

)
w.r.t. relation ≤GrD (H)u ,(

SaGr(H),⊕u/SaGr(H), 0
)
w.r.t. relation ≤SaGr(H)u

and
(
SaGrD (H),⊕u/SaGrD(H), 0

)
w.r.t. relation

≤SaGrD (H)u form its woa-subgroups.

Proof. We have shown [6] that
(
Gr(H),⊕u, 0

)
w.r.t. relation ≤Gr(H)u forms a wop-group. In [4,
Lemma 6]. is proved the axiom (Giv). (Ri)
holds by definition. Because

(
V(H)⊕u/V(H), 0

)
=(

Pos(Gr(H)),⊕u/P os(Gr(H)), 0
)
is a generalized effect

algebra [4, 7], we have (Rii) and (Riii). According to
[4]
(
GrD (H),⊕u/GrD(H), 0

)
w.r.t. relation ≤GrD (H)u ,

293



Jiří Janda Acta Polytechnica

(
SaGr(H),⊕u/SaGr(H), 0

)
w.r.t. relation ≤SaGr(H)u

and
(
SaGrD (H),⊕u/SaGrD(H), 0

)
w.r.t. ≤SaGrD (H)u

are wop-subgroups hence by using Corollary 1 they
are woa-subgroups.

Acknowledgements
The author gratefully acknowledges financial support
from Masaryk University under grant 0964/2009 and
financial support from ESF Project CZ.1.07/2.3.00/20.0051
Algebraic methods in Quantum Logic of the Masaryk
University.

References
[1] Blank J., Exner P., Havlíček M., Hilbert Space Operators

in Quantum Physics, 2nd edn. Springer, Berlin (2008).
[2] Dvurečenskij A., Pulmannová S., New Trends in

Quantum Structures, Kluwer Acad. Publ.,
Dordrecht/Ister Science, Bratislava, 2000.

[3] Foulis D. J., Bennett M. K., Effect algebras and unsharp
quantum logics, Found. Phys. 24 (1994), 1331–1352.

[4] Janda J., Weakly ordered partial commutative group of
self-adjoint operators densely defined on Hilbert space,
Tatra Mt. Math. Publ., 50 (2011), 1-16.

[5] Paseka J., PT -Symmetry in (Generalized) Effect
Algebras, Internat. J. Theoret. Phys., 50 (2011),
1198–1205.

[6] Paseka J., Janda J., More on PT -Symmetry in
(Generalized) Effect Algebras and Partial Groups, Acta
Polytechnica, 51 (2011), No. 4, 65–72.

[7] Paseka J., Riečanová Z., Considerable Sets of Linear
Operators in Hilbert Spaces as Generalized Effect
Algebras. Found. Phys. 41 (2011), 1634–1647.

[8] Riečanová Z., Zajac M. and Pulmannová S., Effect
Algebras of Positive Linear Operators Densely Defined
on Hilbert Spaces, Reports on Mathematical Physics 68,
(2011), 261–270.

294


	Acta Polytechnica 53(3):289–294, 2013
	1 Introduction
	2 Preliminaries
	3 Hilbert spaces
	Acknowledgements
	References