Acta Polytechnica Vol. 51 No. 4/2011

Extensions of Effect Algebra Operations

Z. Riečanová, M. Zajac

Abstract

We study the set of all positive linear operators densely defined in an infinite-dimensional complex Hilbert space. We
equip this set with various effect algebraic operations making it a generalized effect algebra. Further, sub-generalized
effect algebras and interval effect algebras with respect of these operations are investigated.

Keywords: generalized effect algebra, effect algebra, Hilbert space, densely defined linear operators, extension of
operations.

1 Introduction and some
basic definitions and facts

The aim of this paper is to show that (generalized)
effect algebras may be suitable and natural alge-
braic structures for sets of linear operators (includ-
ing unbounded ones) densely defined on an infinite-
dimensional complex Hilbert space H. In all cases, if
the effect algebraic sum of operators A, B is defined
then it coincides with the usual sum of operators in
H.

Effect algebras were introduced by D. Foulis and
M. K. Bennet in 1994 [2]. The prototype for the
abstract definition of an effect algebra was the set
E(H) (Hilbert space effects) of all selfadjoint opera-
tors between null and identity operators in a complex
Hilbert space H. If a quantum mechanical system is
represented in the usual way by a complex Hilbert
space H, then self-adjoint operators from E(H) rep-
resent yes-no measurements that may be unsharp.
The subset P(H) of E(H) consisting of orthogonal
projections represents yes-no measurements that are
sharp.

The abstract definition of an effect algebra follows
the properties of the usual sum of operators in the in-
terval [0, I] (i.e. between null and identity operators
in H) and it is the following.
Definition 1 (Foulis, Bennet [2]) A partial algebra
(E; ⊕, 0, 1) is called an effect algebra if 0,1 are two
distinguished elements and ⊕ is a partially defined
binary operation on E which satisfy the following
conditions for any x, y, z ∈ E:
(E1) x ⊕ y = y ⊕ x if x ⊕ y is defined,
(E2) (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) if one side is defined,
(E3) for every x ∈ E there exists a unique y ∈ E

such that x ⊕ y = 1 (we put x′ = y),
(E4) If 1 ⊕ x is defined then x = 0.

Immediately in 1994 the study of generalizations
of effect algebras (without the top element 1) was

started by several authors (Foulis and Bennet [2],
Kalmbach and Riečanová [4], Hedĺıková and Pulman-
nová [3], Kôpka and Chovanec [5]). It was found that
all these generalizations coincide, and their common
definition is the following:

Definition 2 A generalized effect algebra (E; ⊕, 0)
is a set E with element 0 ∈ E and partial binary op-
eration ⊕ satisfying for any x, y, z ∈ E the conditions
(GE1) x ⊕ y = y ⊕ x if one side is defined,
(GE2) (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) if one side is defined,
(GE3) If x ⊕ y = x ⊕ z then y = z,
(GE4) If x ⊕ y = 0 then x = y = 0,
(GE5) x ⊕ 0 = x for all x ∈ E.

In every (generalized) effect algebra E a partial or-
der ≤ and a binary operation � can be introduced
as follows: for any a, b ∈ E, a ≤ b and b � a = c iff
a ⊕ c is defined and a ⊕ c = b.

If the elements of a (generalized) effect algebra
E are positive linear operators in a given infinite-
dimensional complex Hilbert space, then E is called
an operator (generalized) effect algebra.

Throughout the paper we assume that H is an
infinite-dimensional complex Hilbert space, i.e., a
linear space with inner product (·, ·) which is com-
plete in the induced metric. Recall that here for any
x, y ∈ H we have (x, y) ∈ C (the set of all 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 (x, x) ≥ 0 with (x, x) = 0 iff x = 0. The term
dimension of H in the following always means the
Hilbertian dimension, i.e. the cardinality of any or-
thonormal basis of H (see [1, p. 44]).

For notions and results on Hilbert space operators
we refer the reader to [1]. We will assume that the
domains D(A) of all considered linear operators A are
dense linear subspaces of H (in the metric topology
induced by inner product). We say that operators A
are densely defined on H. The set of all densely de-
fined linear operators on H will be denoted by L(H).

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Acta Polytechnica Vol. 51 No. 4/2011

Recall that A : D(A) → H is a bounded opera-
tor if there exists a real constant C > 0 such that
‖Ax‖ ≤ C‖x‖ for all x ∈ D(A). If A is not bounded
then it is called unbounded.

Let T ∈ L(H). Since D(T ) is dense in H, for
any y ∈ H there is at most one y∗ ∈ H satisfy-
ing (y, T x) = (y∗, x) for all x ∈ D(T ). This al-
lows us to define the adjoint T ∗ of T by putting
D(T ∗) = {y ∈ H | there exists y∗ ∈ H such that
(y, T x) = (y∗, x) for all x ∈ D(T )} and T ∗y = y∗.
Operator T is said to be self-adjoint if T = T ∗. An
operator T ∈ L(H) is called symmetric, if T (x, y) =
(x, T y) for all x, y ∈ D(A). It is well-known that
this is equivalent with (T x, x) ∈ R for all x ∈ D(T ).
Clearly every self-adjoint operator is symmetric but
the converse need not hold for unbounded operators
(see [1], p. 98).

Since every (generalized) effect algebra includes
the zero element 0 as the least element of E, we
will assume that all considered operators are posi-
tive (written A ≥ 0). This means that (Ax, x) ≥ 0
for all x ∈ D(A) and hence A is also symmetric, i.e.
(Ax, y) = (x, Ay) for all x, y ∈ D(A) (see [1, pp. 68
and 94]). For two operators A : D(A) → H and
B : D(B) → H we write A ⊂ B iff D(A) ⊂ D(B)
and Ax = Bx for all x ∈ D(A). Then B is called an
extension of A.

We show some examples of partial binary oper-
ations (sums) on the set V(H) of all positive lin-
ear operators densely defined on infinite-dimensional
Hilbert space H. Our main goal is to study the prop-
erties of sub-generalized effect algebras and effect al-
gebras being intervals in V(H) if V(H) is equipped
with two different partial sums such that one of them
is an extension of the other.

2 Some properties of
unbounded operators in
complex Hilbert spaces

Before defining (generalized) effect algebras consist-
ing of operators A ∈ L(H), we review some necessary
results on Hilbert space operators [1, Chapter 4].
Theorem 3 Let D1 ⊂ D2 ⊂ H be linear subspaces,
D1 = D2 = H. Let A ∈ L(H), D(A) = D2 and its
restriction A1 = A|D1 = 0 and (Ax, x) ∈ R for all
x ∈ D2. Then A = 0.
Proof. If A �= 0, then ∃e ∈ D2 \ D1 for which
Ae �= 0. For all d ∈ D1 and all λ ∈ C

Ad = 0, d + λe ∈ D2 , (1)(
A(d + λe), d + λe

)
= λ(Ae, d) + |λ|2(Ae, e) ∈ R .

Since D⊥1 = {0} and Ae �= 0, we can choose d1 ∈ D1

for which (Ae, d1) �= 0. Then, by (1),

∀ λ ∈ C , λ(Ae, d1) + |λ|2(Ae, e) ∈ R .

Since the second summand is real, the first one must
be real for all λ ∈ C. However, this is possible only
if (Ae, d1) = 0, a contradiction.

Corollary 4 If A ∈ L(H), D = D(A) �= H. is a
symmetric bounded operator and B ∈ L(H) is its
symmetric extension, then B is also bounded.

Proof. Let B be a proper symmetric extension
of A and let à be the unique bounded extension
of A. Then (B − Ã)|D = 0 and, by Theorem 3,
(B−Ã)|D(B) = 0. It follows that B = B−Ã|D(B)+
Ã|D(B) = Ã|D(B) is a bounded linear operator.
Theorem 5 Let A ∈ L(H) and there exists

k > 0 such that ∀ x ∈ D(A) |(Ax, x)| ≤ k‖x‖2

Then A is bounded.

Proof. Let there exist k > 0 such that

∀x ∈ D(A) |(Ax, x)| ≤ k‖x‖2

Using the polarization identity:

(Ax, y) =
1
4

{
(A(x + y), x + y) − (A(x − y), x − y) +

i[(A(ix + y), ix + y) − (A(ix − y), ix − y)]
}

,

we obtain for any x, y ∈ D(A), ‖x‖ = ‖y‖ = 1, that

|(Ax, y)| ≤
1
4

{
|(A(x + y), x + y)| + |(A(x − y), x − y)| +

|(A(ix + y), ix + y)| + |(A(ix − y), ix − y)|
}
≤ (2)

k

4

{
‖x + y‖2 + ‖x − y‖2 + ‖ix + y‖2 + ‖ix − y‖2

}
=

k

4

{
(x + y, x + y) + (x − y, x − y) +

(ix + y, ix + y) + (ix − y, ix − y)
}

=

k

4

{
2‖x‖2 + 2‖y‖2 + 2‖ix‖2 + 2‖y‖2

}
=

k(‖x‖2 + ‖y‖2) = 2k .

This means that the quadratic form (Ax, y) is
bounded, i.e.

∀ x, y ∈ D(A) |(Ax, y)| ≤ 2k‖x‖ ‖y‖ . (3)

It follows that for any fixed x ∈ D(A) the linear
functional ϕ(y) = (Ax, y) is bounded and defined on
D(A) therefore ϕ̃ : D(A) = H → C, ϕ̃(y) = (Ax, y)
for all y ∈ H is its unique bounded linear extension
and ‖ϕ̃‖ = ‖ϕ‖ ≤ 2k‖x‖. Now putting y = Ax we
obtain, by (3),

‖Ax‖2 = (Ax, Ax) = ϕ̃(Ax) ≤ 2k‖x‖‖Ax‖ =⇒
‖Ax‖ ≤ 2k‖x‖ .

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Acta Polytechnica Vol. 51 No. 4/2011

Corollary 6 Let A, B be nonnegative densely de-
fined linear operators having the same domain D. If
A + B is bounded then both A, B are bounded.

Proof. It suffices to observe that

0 ≤ (Ax, x) ≤ (Ax, x) + (Bx, x) =
((A + B)x, x) ≤ ‖A + B||‖x‖2

and then, by Theorem 5, A is bounded. By the same
reasoning we obtain that B is bounded.

3 Extensions of effect algebra
operations

It is well known that if a set E includes two distin-
guished elements 0, 1 then there may exist more than
one partial binary operations ⊕ on E making E an
effect algebra.
Example 7 Let E = {0, a, b, 1} and let us define op-
erations ⊕1, ⊕2 on E as follows:
⊕1: a ⊕1 b = b ⊕1 a = 1 and 0 ⊕1 x = x ⊕1 0 = x for
all x ∈ E,
⊕2: a ⊕2 a = b ⊕2 b = 1 and 0 ⊕2 x = x ⊕2 0 = x for
all x ∈ E.
Then (E; ⊕1, 0, 1), (E; ⊕2, 0, 1) are effect algebras
with the same set of elements. On the other hand
(E; ⊕1, 0, 1) is a Boolean algebra, while (E; ⊕2, 0, 1)
is a horizontal sum of two chains {0, a, a ⊕2 a = 1}
and {0, b, b ⊕2 b = 1}, hence in this case elements a, b
are noncompatible.

We obtain special cases of two operations ⊕1 �= ⊕2
on the same underlying set E, if ⊕2 extends ⊕1:

Definition 8 Let (E; ⊕1, 0) and (E; ⊕2, 0) be gen-
eralized effect algebras. We say that the operation
⊕2 extends ⊕1 (written ⊕1 ⊂ ⊕2) if for any a, b ∈ E
the existence of a ⊕1 b implies that a ⊕2 b exists and
a ⊕2 b = a ⊕1 b.

Lemma 9 Let E1 = (E; ⊕1, 0), E2 = (E; ⊕2, 0) be
generalized effect algebras and ⊕1 ⊂ ⊕2. Then
(i) If G ⊆ E is a sub-generalized effect algebra of

E2, then G is also a sub-generalized effect alge-
bra of E1.

(ii) If ≤1 and ≤2 are the partial orders on E derived
from ⊕1 and ⊕2, respectively, then, for a, b ∈ E,
if a ≤1 b then a ≤2 b.

(iii) For intervals in E1, E2 the following inclusion
holds:
[0, q]E1 ⊆ [0, q]E2 for any nonzero q ∈ E.

Proof. The proof obviously follows from the fact
that, for any a, b ∈ E, the existence of a ⊕1 b im-
plies a ⊕2 b = a ⊕1 b. Let us prove, e.g., (i): Let
a, b, c ∈ E with a ⊕1 b = c and assume that at least

two out of elements a, b, c are in G. Since ⊕1 ⊂ ⊕2
implies a ⊕1 b = c = a ⊕2 b = c and G is a sub-effect
algebra of E2 we obtain a, b, c ∈ G. Hence G is a
sub-generalized effect algebra of E1.

The following example shows that the converses
of assertions (i)–(iii) do not hold.

Example 10 Let E = {0, 1, 2, . . .} and G =
{0, 1, 2, 4, 6, . . .}. Define the partial binary operations
⊕1 and ⊕2 for a, b ∈ E

a ⊕1 b =
{

a + b, if a = 0 or both a, b are even,

not defined, otherwise,
(4)

a ⊕2 b = a + b for all a, b ∈ E . (5)

and let ≤1 and ≤2 be the corresponding derived par-
tial orders. Then obviously ⊕1 ⊂ ⊕2 and
(i) E1 = (E, ⊕1, 0) and E2 = (E, ⊕2, 0) are gener-

alized effect algebras.
(ii) G is a sub-generalized effect algebra of E1.
(iii) G is not a sub-generalized effect algebra of E2.
(iv) There exist a, b ∈ E for which a ≤2 b but a �≤1 b.
(v) There exist q ∈ E for which [0, q]E2 �⊆ [0, q]E1.
Let us prove conditions (i)–(v).

(i) Let us show that ⊕1 is associative. To show
this, suppose that for a, b, c ∈ E the sum (a⊕1 b) ⊕1 c
exists. First, if c = 0 then (a ⊕1 b) ⊕1 c = a ⊕1 b =
a ⊕1 (b ⊕1 c). Next, if c �= 0 then either a = b = 0
or both c and a + b are even. If a = b = 0 then
(a ⊕1 b) ⊕1 c = a ⊕1 (b ⊕1 c) is obvious. a + b is even
if both a, b are odd, but this is impossible because
a ⊕1 b exists. So, both a, b are even and, then, again,
(a ⊕1 b) ⊕1 c = a ⊕1 (b ⊕1 c) is obvious. The rest of
the proof of (i) is obvious.

(ii) Suppose that a, b, c ∈ E satisfy a ⊕1 b = c. If
a, b ∈ G then clearly c ∈ G as well. If a, c ∈ G and
a = 0, then b = c, hence b ∈ G. a, c ∈ G are nonzero
then both are even and then b is even, as well. So, if
two elements out of a, b, c are in G, then all three are
in G. This proves (ii).

(iii) Put a = 1, b = 2. Now a, b ∈ G and
a ⊕2 b = 3 /∈ G shows that G is not a sub-generalized
effect algebra of E2.

(v) Clearly, [0, q]E1 = {0, q} for any odd q. So,
e.g. [0, 3]E2 = {0, 1, 2, 3} �⊆ [0, 3]E1.

(iv) obviously (v) implies (iv).

4 Extensions of operator
effect algebra operations

We introduce examples of operator generalized effect
algebra with the same set of elements and different
operations. Moreover, we consider intervals in these
algebras.

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Acta Polytechnica Vol. 51 No. 4/2011

A generalized effect algebra whose elements are
positive linear operators on a complex Hilbert space
H is called an operator generalized effect algebra.
Definition 11 Assume that H is an infinite-
dimensional Hilbert space and that A ∈ L(H) is pos-
itive. Let D denote the set of all dense linear sub-
spaces of H and
(i) V(H) = {A ∈ L(H) | A ≥ 0, D(A) = H if A is

bounded and D(A) ∈ D if A is unbounded}.
(ii) GD (H) = {A ∈ V(H) | A is bounded or D(A) =

D if A is unbounded}, D ∈ D.
(iii) Let ⊕ be a partial binary operation on V(H) de-

fined by:
For A, B ∈ V(H), A ⊕ B is defined and A ⊕ B =
A + B (the usual sum) iff at least one out of op-
erators A, B is bounded. The triple (V(H); ⊕, 0)
will be denoted by V(H) for short.

(iv) Let ⊕D be a partial binary operation on V(H)
defined by:
For A, B ∈ V(H), A ⊕D B is defined and A ⊕D
B = A + B (the usual sum) iff either at least one
out of A, B is bounded, or D(A) = D(B) ∈ D if
both are unbounded. The triple (V(H); ⊕D, 0)
will be denoted by VD(H) for short.

From the above definition it is clear that V(H) =⋃
{GD(H) | D ∈ D} and for D1 �= D2 it holds

GD1 (H) ∩ GD1(H) = B
+(H) = {A ∈ V(H) | A is

bounded with D(A) = H}.
Moreover, for any A, B ∈ V(H) if A⊕B is defined

then A⊕D B is defined and A⊕D B = A⊕B = A+B.
Hence ⊕ ⊂ ⊕D.

Recently in [6, 8] the following theorems were
proved.

Theorem 12 [6] For V(H) from Definition 11 it
holds:
(i) (V(H); ⊕, 0) is a generalized effect algebra.

(ii) If Sp(H) = {A ∈ V(H) | A = A∗} is equipped
with ⊕|Sp(H), i.e., for A, B ∈ Sp(H) there ex-
ists A ⊕|Sp(H) B = A ⊕ B iff there exists A ⊕
B in V(H), then (Sp(H); ⊕|Sp(H) , 0) is a sub-
generalized effect algebra of (V(H); ⊕, 0).

Theorem 13 [8] Using the notation from Defini-
tion 11 we obtain:
(i) (V(H); ⊕D, 0) is a generalized effect algebra.
(ii) Let D ∈ D be a fixed dense subspace of H.

Let GD,D(H) = {A ∈ V(H) | either A is
bounded with D(A) = H or A is unbounded with
D(A) = D}. Let for A, B ∈ GD,D(H) the sum
A ⊕D|GD,D(H) B = A ⊕D B if there exists A ⊕D
B ∈ GD,D(H), otherwise A ⊕D|GD,D(H) B is not
defined. Then (GD,D(H); A ⊕D|GD,D(H) , 0) is a
sub-generalized effect algebra of (V(H), ⊕D, 0).

Now, because ⊕ ⊂ ⊕D on V(H) and, for every fixed
D ∈ D, GD,D(H) is a sub-generalized effect algebra of

(V(H); ⊕D, 0) we obtain that GD,D(H) is also a sub-
generalized effect algebra of (V(H); ⊕, 0). Clearly,
the intersection of two sub-generalized effect algebras
of (V(H); ⊕, 0) is again its sub-generalized effect al-
gebra. Thus we obtain the following corollary of The-
orems 12 and 13.

Theorem 14 Let for every fixed D ∈ D, Sp,D(H) =
{Sp(H) ∩ GD,D(H) = {A ∈ V(H) | A = A∗ and ei-
ther D(A) = H, if A is bounded, or D(A) = D if A
is unbounded}. Let ⊕|Sp,D(H) on Sp,D(H) be defined
as follows: For A, B ∈ Sp,D(H), A ⊕|Sp,D(H) B =
A ⊕ B iff A ⊕ B is defined in V(H). Then
(Sp,D(H); ⊕|Sp,D(H) , 0) is a sub-generalized effect al-
gebra of (Sp(H); ⊕, 0).

The above observations show that generalized ef-
fect algebra VD(H) is a pasting of its sub-generalized
effect algebras GD,D(H) through the sub-generalized
effect algebra B+(H) = {A ∈ V(H) | A is bounded}.
Consequently, the generalized effect algebra Sp(H)
is a pasting of its sub-generalized effect algebras
Sp,D(H) through the sub-generalized effect algebra
B+(H). More precisely:

Theorem 15 (Pasting Theorem)
(i) For generalized effect algebra

VD(H) = {V(H); ⊕D, 0) and its sub-generalized
effect algebras GD,D(H), D ∈ D, and B+(H)
(1) GD,D1(H) ∩ GD,D2(H) = B

+(H) for every
D1, D2 ∈ D, D1 �= D2.

(2) VD(H) =
⋃

{GD,D(H) | D ∈ D}.
(ii) For generalized effect algebra Sp(H) and its sub-

generalized effect algebras Sp,D(H), D ∈ D
(1) SpD1(H) ∩ SpD2 (H) = B

+(H) for every
D1, D2 ∈ D, D1 �= D2.

(2) Sp(H) =
⋃

{Sp,D(H) | D ∈ D}.

Remark 16 It is worth noting that the operations
⊕, ⊕D are the usual sum of operators in H, but
only partially applied on pairs A, B ∈ V(H). There-
fore all effect algebra operations ⊕, ⊕D, ⊕|Sp(H),
⊕D|GD,D(H), ⊕|Sp,D(H) coincide with the usual sum
of operators A + B if the corresponding effect algebra
sum of A and B exists.

5 Intervals in generalized
effect algebras of
self-adjoint operators

Assume that (E; ⊕, 0) is a generalized effect algebra.
For any q ∈ E, q �= 0, let [0, q]E = {a ∈ E | there ex-
ists b ∈ E with a ⊕ b = q} be an interval in (E; ⊕, 0].
We will denote by ⊕|[0,q]E the partial binary opera-
tion on [0, q]E defined as follows:

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Acta Polytechnica Vol. 51 No. 4/2011

For a, b ∈ [0, q]E the sum a ⊕|[0,q]E b = a ⊕ b iff a ⊕ b
is defined in E and a ⊕ b ∈ [0, q]E . It is known that
[0, q]E equipped with ⊕|[0,q]E is an effect algebra (see,
e.g. [7]).

In this way, for all nonzero Q ∈ V(H), we obtain
the operator effect algebras(

[0, Q]V(H); ⊕|[0,Q]V(H) , 0, Q
)

and (
[0, Q]VD(H); ⊕D|[0,Q]VD(H) , 0, Q

)
(see [8]). By definition of ⊕, ⊕D and the results
of Section 2, it is clear that for Q ∈ V(H) with
D(Q) = D ∈ D we have [0, Q]V(H) ⊂ GD (H) and
[0, Q]VD(H) ⊂ GD,D(H). The same is true for any Q ∈
Sp(H), Q �= 0, D(Q) = D. Namely, [0, Q]Sp(H) =
[0, Q]Sp,D(H) and

(
[0, Q]Sp,D(H); ⊕|[0,Q]Sp,D(H) , 0, Q

)
,

D ∈ D, are effect algebras of positive self-adjoint
operators A ≤D Q, where ≤D is the partial order
on Sp(H) derived from operation ⊕|Sp,D(H). Since
D(Q) = D is dense in H, the next Theorem 18 about
states on intervals is Sp(H) (hence in Sp,D (H)) can
be proved by the same argument as Theorem 7 in [8]
for states on intervals in GD,D(H).
Definition 17 Let E be an effect algebra.
(i) A map ω : E → [0, 1] is called a state on E if

1. ω(0) = 0, ω(1) = 1,
2. ω(a ⊕ b) = ω(a) + ω(b) for all a, b ∈ E with

a ⊕ b defined in E.
3. A state ω is faithful if ω(a) = 0 implies a = 0.
4. A set M of states is called an ordering set of

states on E if
a ≤ b iff ω(a) ≤ ω(b) for all ω ∈ M, a, b ∈ E.

Theorem 18 Let D ∈ D and Q ∈ Sp,D(H), Q �= 0.
Then
(i) There exists x̃ ∈ D(Q) such that cx̃ = (Qx̃, x̃) >

0.
(ii) The mapping ωx̃ : [0, Q]Sp,D(H) → [0, 1] ⊂ R

given by ωx̃(A) =
1
cx̃

(Ax̃, x̃) for every A ∈
[0, Q]Sp,D(H) is a state.

(iii) If D0 = {x ∈ D(Q) | cx = (Qx, x) > 0} then
M = {ωx | x ∈ D0} is an ordering set of states
on [0, Q]Sp,D(H).

(iv) If H is separable, then there exists a faithful
state ω : [0, Q]Sp,D(H) → [0, 1].

Acknowledgement

Supported by grants VEGA 1/0297/11 and VEGA
1/0021/10 of the Ministry of Education of the Slo-
vak Republic.

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[7] Riečanová, Z.: Subalgebras, intervals and central
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Zdenka Riečanová
E-mail: zdenka.riecanova@stuba.sk
M. Zajac
E-mail: zajacm@stuba.sk
Department of Mathematics
Faculty of Electrical Engineering
and Information Technology STU
Ilkovičova 3, SK-81219 Bratislava

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