Kokanduni uz

www.
kokanduni.uz 
yig‘indisi 
(
)


2
1
2
1
1
2
,
:
l
l
x y
y
k x
b
b
+ =


+ +
ko‘rinishdagi yarim tekislikdan iborat 
ekanligi kelib chiqadi. 
Agar 
1
2
k
k

bo‘lsa, u holda 
1
1
1
:
l y
k x b
=
+
va 
2
2
2
:
l
y
k x b
=
+
tog‘ri chizqlar o‘zaro 
parallel bo‘lma
ydi. [1, Pr
oposition 3.2.] ga ko‘ra bu to‘g‘ri chiziqlarning Minkovskiy yig‘indisi 
bu to‘g‘ri chiziqlar yotuvchi tekislik bo‘ladi, ya’ni 
2
.
Faraz qilaylik, 
2
1
2
l
l
+ 
bo‘lsin. U holda shunday
(
)
2
0
0
;
x y

nuqta topilib, 
(
)
0
0
1
2
;
x y
l
l
 +
bo‘ladi. 
(
)
0
0
1
2
;
x y
l
l
 +
munosabatning ma’nosi 
0
1
2
x
x
x
=
+
va 
0
1
2
y
y
y
=
+
tengliklarni 
qanoatlantiradigan 
(
)
1
1
1
;
x y
l

,
(
)
2
2
2
;
x y
l

nuqtalar 
topilmasligini bildiradi. Lekin 
1
x

va 
2
x

ekanligidan doimo 
0
1
2
x
x
x
=
+
tenglikni 
qanoatlantiradigan 
1
2
,
x x
lar topiladi. Bundan 
1
0
2
x
x
x
=
−
yoza olamiz.
1
1 1
1
y
k x
b

+
va 
2
2
2
2
y
k x
b

+
bo‘lgani uchun 
1
2
1 1
2
2
1
2
y
y
k x
k x
b
b
+

+
+ +
ifoda kelib chiqadi. Bu ifodaga 
1
0
2
x
x
x
=
−
ni qo‘ysak 
(
)
1
2
1 0
2
1
2
1
2
y
y
k x
k
k x
b
b
+ 
+
−
+ +
tengsizlikk
a ega bo‘lamiz. 
1
2
k
k

bo‘lgani 
uchun 
har 
qanday 
(
)
2
0
0
;
x y

nuqta 
olinmasin 
doim 
(
)
0
1 0
2
1
2
1
2
y
k x
k
k x
b
b

+
−
+ +
munosabat o‘rinli bo‘luvchi 
2
x

topiladi. Demak, 
(
)
0
0
1
2
;
x y
l
l
 +
va 
2
1
2
l
l
+ =
. Demak, bu to‘g‘ri chiziqlar hosil qilgan yarim tekisliklar 
Minkovskiy yig‘indisi ham butun tekislikdan iborat bo‘lar ekan.
2-teorema
. 
Aytaylik 
( )


2
1
1
1
,
:
l
x y
y
k x
b
=


+
va 
( )


2
2
2
2
,
:
l
x y
y
k x
b
=


+
to‘plamlar 
2
da aniqlangan yarim tekisliklar bo‘lsin. Agar 
1
2
k
k
=
tenglik 
bajarilsa, 
u 
holda 
ularning 
Minkowskiy 
ayirmasi 
( )


2
1
2
1
1
2
,
:
l l
x y
y
k x
b
b

=


+ −
yarim tekislik bo‘ladi. Agar 
1
2
k
k

munosabat 
bajarilsa 
1
2
l l

Minkovskiy ayirmasi bo‘sh to‘plamdan iborat bo‘ladi.
Isbot. 
Agar 
1
2
k
k
=
bo‘lsa 
1
l
va 
2
l
to‘plamlarning mos ravishda yasovchilari bo‘lgan,
1
1
1
:
l y
k x b
=
+
va 
2
2
2
:
l
y
k x b
=
+
to‘g‘ri chiziqlar o‘zaro parallel bo‘lib qoladi. [2, 1
-teorema] 
ga ko‘ra bu chiziqlarning Minkowskiy ayirmasi shu to‘g‘ri chiziqlarga parallel to‘g‘ri chiziq 
bo‘ladi.
Aytaylik, 
(
)
0
0
1
2
;
x y
l l


bo‘lsin. 
Minkovskiy 
ayirmasining 
ta’rifiga 
ko‘ra 
(
)
0
0
2
1
;
x y
l
l
+ 
munosabat o‘rinli bo‘ladi. 
(
)
0
0
2
;
x y
l
+
to‘plam 
2
l
to‘plamni 
(
)
0
0
;
x y
nuqtaga parallel ko‘chirishdan hosil bo‘lgan to‘plam bo‘lib, uni 
2
l

orqali belgilaymiz va 
( )
(
)


2
2
2
0
2
0
,
:
l
x y
y
k
x
x
b
y
 =


−
+ +
(3) 

---

40
www.
kokanduni.uz 
ko‘ri
nishda ifodalaymiz. 
2
1
l
l
 
bo‘lgani uchun 
( )


2
1
1
1
,
:
l
x y
y
k x
b
=


+
ekanligini bilgan holda
(
)
2
0
2
0
1
1
k x
x
b
y
k x b
−
+ + 
+
(4) 
tengsizlikni yoza olamiz. 
1
2
k
k
=
bo‘lgani uchun (4) munosabatni
0
1 0
1
2
y
k x
b
b

+ −
(5)
ko‘rinishda yoza olamiz. Demak, 
(
)
0
0
1
2
;
x y
l l


nuqtalar uchun (5) shart o‘rinli ekan. 
Bundan 
esa 
1
l
va 
2
l
yarim 
tekisliklarning 
Minkovskiy 
ayirmasi 
( )


2
1
2
1
1
2
,
:
l l
x y
y
k x
b
b

=


+ −
ko‘rinishda
gi yarim tekislikdan iborat ekanligi kelib 
chiqadi. 
1
2
k
k

bo‘lsin.
2
2
l
l

to‘g‘ri chiziqni olaylik. Minkovskiy ayirmasi aniqlanishiga ko‘ra 
1
2
l l

to‘plam shunday nuqtalardan iborat bo‘lishi kerakki, 
2
l
to‘g‘ri chiziqning bu nuqtalarga 
parallel ko‘chirganimizdagi obrazi 
2
l
yarim tekis
likda to‘la yotib qolishi lozim. 
1
1
1
:
l y
k x b
=
+
va 
2
2
2
:
l
y
k x b
=
+
to‘g‘ri chiziqlar
o‘zaro parallel bo‘lmagani uchun bunday nuqtalar 
topilmaydi. Bu esa 
1
2
l l

= 
ekanini anglatadi. 
2
2
l
l

bo‘lgani uchun 
1
2
l l

= 
munosabat 
ham o‘rinli bo‘ladi. 
Foydalanilgan adabiyo
tlar ro‘yxati:
 
1.
D. Velichova. Notes on properties and applications of Minkowski point set 
operations// South Bohemia Mathematical Letters. 2016. Volume 24. 

1. pp.57-71. 
2.
J.T. Nuritdinov. To‘g‘ri chiziq va tekisliklar Minkovskiy ayirmasi haqida
.// Differential 
equations and related problems of analysis. Republican Scientific Conference with the 
participation of foreign scientists Bukhara, Uzbekistan, November 04
–
05, 2021 

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41
www.
kokanduni.uz 
THE CONNECTION OF A RICKART REAL C*-ALGEBRA WITH ITS ENVELOPING RICKART 
(COMPLEX) C*-ALGEBRA 
Abduxamidova Dilshoda Bayram qizi 
Kokand University Student of MLS 
–
 1 
–
 22, 
Nilufarxon Raxmonova Vaxobjon qizi 
Kokand University teacher at Department 
of Digital technologies and mathematics 
Abstract: 
In the paper Rickart complex and real C*-algebra are consider. For Rickart real 
C*-algebra, its connection with the enveloping (complex) C*-algebra is studied. It is shown 
that the fact that A is a Rickart real C*-algebra does not imply that a complexification A +i A of 
A is a Rickart (complex) C*-algebra. Proved that if A is a real C*-algebra and A +i A is a Rickart 
C*-algebra, then A is a Rickart real C*-algebra. It is shown that there exists a Rickart real C*-
algebra whose projection lattice is not complete. 
Keywords
: C*-algebra; Rickart complex and real C*-algebras; complex and real AW*-
algebras. 
Introduction: 
The theory of operator algebras was initiated in a series of papers by 
Murray and von Neumann in thirties. Later such algebras were called von Neumann algebras 
or W*-algebras. These algebras are self-adjoint unital subalgebras M of the algebra B(H) of
 
bounded linear operators on a complex Hilbert space H, which are closed in the weak 
operator topology. Equivalently M is a von Neumann algebra in B(H) if it is equal to the 
commutant of its commutant (von Neumann's bicommutant theorem). A factor (or W*-factor) 
is a von Neumann algebra with trivial centre and investigation of general W*-algebras can be 
reduced to the case of W*-factors, which are classified into types I, II and III. 
 
Rings and algebras, which will be discussed below, first studied by C.E. Rickart [1]. These 
algebras were further developed in the works of I. Kaplansky [2-4]. Exactly, AW*-algebras 
were proposed by I. Kaplansky as an appropriate setting for certain parts of the algebraic 
theory of von Neumann algebras. In this article, we consider the real analogue of these 
algebras. 
Preliminaries:
Definition 2.1.
A C*-algebra is a (complex) Banach *-algebra whose norm satisfies the 
identity 
2
|| * || || ||
x x
x
=
. Now let A be a real Banach *-algebra. A is called a real C*-algebra, if 
c
A
A
iA
= +
can be normed to become a (complex) C*-algebra by extending the original norm 
on A. 
Note that a C*-norm on 
c
A
is unique, if it exists. It is known that [5, Corollary 5.2.11] 
A is real C*-algebra if and only if 
2
|| * || || ||
x x
x
=
and 
1
*
x x
+
is invertible, for any 
x
A

. 
Definition 2.2.
If A is a ring and S is a nonempty subset of A, we write 
( ) {
:
0,
}
R S
x
A
sx
s
S
= 
=
 
and call R(S) the right-annihilator of S. Similarly, 
( ) {
:
0,
}
L S
x
A
xs
s
S
= 
=
 

---

42