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
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
=
=
|