On one class of topological *-algebras with standard identities
Let A be a unital semisimple topological nuclear *-algebra over C and let Z be its center. Then A is topologically isomorphic to M n (Z) if and only if A satisfies the standart identity and the maximality condition.
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| author | Tishchenko, S. V. Тищенко, С. В. |
| author_facet | Tishchenko, S. V. Тищенко, С. В. |
| author_sort | Tishchenko, S. V. |
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| description | Let A be a unital semisimple topological nuclear *-algebra over C and let Z be its center.
Then A is topologically isomorphic to M n (Z) if and only if A satisfies the standart identity and the maximality condition. |
| first_indexed | 2026-03-24T02:39:54Z |
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| fulltext |
UDK 513.88:�517.98
S.�V.�Tywenko (Ky]v. nac. un-t im. T.�Íevçenka)
PRO DEQKYJ KLAS TOPOLOHIÇNYX ∗∗∗∗ -ALHEBR
IZ STANDARTNYMY TOTOÛNOSTQMY
Let A be a unital semisimple topological nuclear ∗-algebra over C and let Z be its center. Then A is
topologically isomorphic to Mn ( Z ) if and only if A satisfies the standart identity and the maximality
condition.
Pust\ A — unytal\naq poluprostaq topolohyçeskaq qdernaq �∗ -alhebra nad C y Z — ee
centr. A topolohyçesky yzomorfna Mn ( Z ) tohda y tol\ko tohda, kohda A udovletvorqet
standartnomu toΩdestvu y uslovyg maksymal\nosty.
Vstup. Korysnym uzahal\nennqm komutatyvnyx alhebr [ matryçni alhebry nad
komutatyvnymy topolohiçnymy alhebramy. Tak, u roboti [1] dovedeno, wo bud\-
qka napivprosta banaxova alhebra A nad polem kompleksnyx çysel C z odyny-
ceg e i centrom Z izomorfna matryçnij alhebri M n ( Z ) todi i til\ky todi,
koly A [ alhebrog iz standartnymy totoΩnostqmy ( F2n -alhebrog) i mistyt\
pidalhebru A0 , qka izomorfna Mn ( C ) i mistyt\ odynycg e . U�roboti [2] cej
rezul\tat uzahal\neno na unital\ni alhebry nad deqkym polem skalqriv G .
TakoΩ u [2] (naslidok�3) dovedeno, wo u vypadku, koly A — normovana alhebra
nad C , ma[ misce topolohiçnyj izomorfizm miΩ A ta Mn ( Z ) .
Metog dano] roboty [ dovedennq rezul\tativ, podibnyx do otrymanyx u [1, 2],
dlq klasu topolohiçnyx qdernyx ∗ -alhebr.
Osnovni oznaçennq i ponqttq. Nexaj ( )H Tτ τ∈ ( T — dovil\na mnoΩyna in-
deksiv) — sim’q kompleksnyx hil\bertovyx prostoriv iz skalqrnymy dobutkamy
( , )⋅ ⋅ τ : = ( , )⋅ ⋅ Hτ
i normamy || ⋅ ||τ : = || ⋅ ||Hτ
. Topolohiçnyj prostir A =
= pr limτ τ∈T H nazyva[t\sq qdernym [3, s. 21], qkwo dlq koΩnoho τ ∈ T znaj-
det\sq τ′ ∈ T take, wo operator vkladennq H ′τ → Hτ [ operatorom Hil\berta –
Ímidta.
Topolohiçnyj qdernyj prostir A budemo nazyvaty qdernog alhebrog, qkwo
A [ asociatyvnog alhebrog, pryçomu operaciq mnoΩennq A × A � ( f , g ) � f ⋅ g ∈
∈ A [ sumisno neperervnog u topolohi] proektyvno] hranyci, tobto neperervnog
za sukupnistg zminnyx. Pid qdernog ∗ -alhebrog budemo rozumity qdernu alheb-
ru z involgci[g ∗, qka zadovol\nq[ umovu || || || ||∗ =f fτ τ dlq vsix f ∈ A , τ ∈ T .
Dlq dovil\no vybranyx elementiv a1 , a2 , … , an ∈ A vyznaçymo standartnyj
polinom stepenq n za dopomohog formuly
Fn ( a1 , a2 , … , an ) = ( ) ( ) ( )−
∈
∑ 1 1
σ
σ σ
σ
a a n
Sn
… ,
de Sn i ( –1 )
σ — vidpovidno symetryçna hrupa i znak pidstanovky σ . Alhebra A
nazyva[t\sq Fn -alhebrog, qkwo Fn ( a1 , a2 , … , an ) = 0 dlq dovil\nyx fiksova-
nyx a1 , a2 , … , an ∈ A [4].
DopomiΩni rezul\taty.
Lema�1. Nexaj U — topolohiçnyj qdernyj prostir. Todi mnoΩyna A =
= Mn ( U ) takoΩ bude topolohiçnym qdernym prostorom.
Dovedennq. Na mnoΩyni Mn ( U ) vvedemo strukturu topolohiçnoho prosto-
ru, qkyj [ proektyvnog hranyceg hil\bertovyx. Oskil\ky prostir U [ topolo-
hiçnym qdernym, to U = pr limτ τ∈T H , pryçomu vykonu[t\sq umova qdernosti
© S.�V.�TYWENKO, 2007
140 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
PRO DEQKYJ KLAS TOPOLOHIÇNYX ∗ -ALHEBR IZ STANDARTNYMY … 141
dlq proektyvno] hranyci. Dali, pry koΩnomu α = ( ) ,τij i j
n
=1 = ( ) ,τ i j
n
=1 ∈ T
n
×
n : = Γ
(tobto τ i j = τ ∈ T dlq vsix i , j = 1, 2, …, n ) vyznaçymo hil\bertovyj prostir
Hα : = Mn ( Hτ ) matryçnoznaçnyx funkcij F = ( ) ,fij i j
n
=1, G = ( ) ,gij i j
n
=1 ( f i j , gi j ∈
∈ Hτ ) , v qkomu skalqrnyj dobutok i normu vyznaçymo za dopomohog formul
( F , G )α : = ( ),
,
f gij ij
i j
τ∑ , || || || ||= ∑F fij
i j
α τ
2 2:
,
tut( �i dali budemo vykorystovuvaty skoroçene poznaçennq ( )
,
⋅∑i j
zamist\
( )
,
⋅ )=∑i j
n
1
. Ma[mo sim’g hil\bertovyx prostoriv ( )Hα α∈Γ . MnoΩyna ∩α α∈Γ H =
= ∩ ∩τ τ τ τ∈ ∈= ( )T n n TM H M H( ) = Mn ( U ) [ wil\nog u koΩnomu prostori Hα ,
oskil\ky prostir U [ wil\nym u koΩnomu Hτ . Sim’q hil\bertovyx prostoriv
( )Hα α∈Γ takoΩ [ napravlenog po vkladenng: dlq dovil\nyx α1 = ( ) ,τ1 1i j
n
= ,
α2 = ( ) ,τ2 1i j
n
= ∈ Γ znajdet\sq take α3 = ( ) ,τ3 1i j
n
= ∈ Γ, wo H Hα α3 1
⊂ , H Hα α3 2
⊂ ,
pryçomu vkladennq [ topolohiçnymy.
Perevirymo, wo linijnyj topolohiçnyj prostir A = Mn ( U ) = pr limα α∈Γ H [
qdernym, tobto dlq n\oho vykonu[t\sq umova: dlq dovil\noho α = ( ) ,τ i j
n
=1 ∈ Γ
znajdet\sq α′ = ( ) ,′ =τ i j
n
1 ∈ Γ take, wo vkladennq H ′α → H α [ kvaziqdernym
(operator vkladennq [ operatorom Hil\berta – Ímidta). Spravdi, rozhlqnemo
lokal\no opuklyj topolohiçnyj prostir A = pr limα α∈Γ H , qkyj, qk mnoΩyna,
zbiha[t\sq z peretynom ∩α α∈Γ H hil\bertovyx prostoriv Hα . Bazys okoliv
nulq v A utvorggt\ mnoΩyny W ( 0 ; α , δ ) = { F ∈ A : || F ||α < δ } pry dovil\nyx
α ∈ Γ i δ > 0. Oskil\ky prostir U = pr limτ τ∈T H [ qdernym, to dlq dovil\noho
τ ∈ T znajdet\sq τ′ ∈ T take, wo operator vkladennq O ′τ
τ
: H ′τ → Hτ [ kvazi-
qdernym, tobto norma Hil\berta – Ímidta || ||′Oτ
τ operatora vkladennq O ′τ
τ [
skinçennog. Poznaçymo α′ = ( ) ,′ =τ i j
n
1 ∈ Γ. Todi norma Hil\berta – Ímidta ope-
ratora vkladennq O ′α
α
: H ′α → Hα takoΩ bude skinçennog:
|| ||′Oα
α 2 = || || || ||′ ′∑ =O n O
i j
τ
τ
τ
τ2 2 2
,
< ∞ ,
wo j dovodyt\ kvaziqdernist\ vkladennq H ′α → Hα .
Lemu dovedeno.
Lema 2. Nexaj U — unital\na komutatyvna topolohiçna qderna ∗ -alheb-
ra. Todi A = Mn ( U ) takoΩ bude unital\nog topolohiçnog qdernog ∗ -alheb-
rog.
Dovedennq. Zhidno z lemog�1, Mn ( U ) [ topolohiçnym qdernym prostorom.
Oçevydno takoΩ, wo A = Mn ( U ) [ unital\nog alhebrog nad polem C iz zvy-
çajnymy linijnymy operaciqmy nad matrycqmy F = ( ) ,fij i j
n
=1, G = ( ) ,gij i j
n
=1, mat-
ryçnym mnoΩennqm ta involgci[g ( ) ,fij i j
n
=1 = F � F∗ = ( ) ,f ji j i
n∗
=1.
Dovedemo, wo A [ topolohiçnog ∗ -alhebrog, tobto mnoΩennq j involgciq [
neperervnymy operaciqmy v A . Spravdi, vnaslidok sumisno] neperervnosti ope-
raci] mnoΩennq i neperervnosti involgci] v topolohi] proektyvno] hranyci vy-
xidno] alhebry dlq dovil\noho τ ∈ T znajdut\sq τ1 , τ2 ∈ T taki, wo dlq dovil\-
nyx f , g ∈ U = pr limτ τ∈T H magt\ misce nastupni ocinky dlq cyx operacij:
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
142 S.�V.�TYWENKO
|| || || || || ||⋅ ≤f g f gτ τ τ1 2
i || || || ||∗ =f fτ τ .
Teper, poznaçagçy α : = ( ) ,τ i j
n
=1, α1 : = ( ) ,τ1 1i k
n
= , α 2 : = ( ) ,τ2 1k j
n
= , dlq vidpo-
vidnyx matryçnoznaçnyx funkcij F = ( ) ( ),f H M Hij i j
n
n= ∈ =1 1 1α τ , G = ( ) ,gij i j
n
=1 ∈
∈ Hα2
= M Hn( )τ2
i F ⋅ G ∈ Hα = Mn ( Hτ ) na pidstavi nerivnosti Bunqkovs\ko-
ho – Ívarca oderΩu[mo
|| ||⋅F G α
2 = f g f g f gik kj
ki j
ik kj
ki j
ik kj
ki j
∑∑ ∑∑ ∑∑≤
≤
|| || || ||
τ
τ τ τ
2 2 2
1 2
, , ,
≤
≤ || || || || || || || ||∑ ∑
=f g F Gik
i k
kj
k j
τ τ α α1 2 1 2
2 2 2 2
, ,
.
Zvidsy
|| || || || || ||⋅ ≤F G F Gα α α1 2
.
Ostannq nerivnist\ dovodyt\, wo mnoΩennq v A [ sumisno neperervnym.
Dali, oskil\ky pry α = ( ) ,τ i j
n
=1
|| || || || || || || || || ||∗ ∗= = = =∑ ∑ ∑F f f f Fji
j i
ji
j i
ij
i j
α τ τ τ α
2 2 2 2 2
, , ,
,
to involgciq ∗ [ unitarnym operatorom u koΩnomu prostori Hα , a otΩe, [
neperervnym operatorom v alhebri A .
Lema 3. Nexaj A — unital\na napivprosta qderna ∗ - alhebra nad C i
Z ⊂ A — ]] centr. Todi Mn ( Z ) takoΩ bude unital\nog napivprostog qdernog
∗ -alhebrog.
Dovedennq. ZauvaΩymo spoçatku, wo centr Z qderno] ∗ -alhebry A , budu-
çy zamknenog pidalhebrog v A [5, s. 203], [ komutatyvnog qdernog ∗ -alhebrog
v indukovanij z A topolohi]. Todi za lemamy 1,�2 matryçna alhebra Mn ( Z )
takoΩ bude qdernog ∗ -alhebrog.
Dali vidmitymo, wo alhebra Z , a otΩe i Mn ( Z ) , bude napivprostog. Sprav-
di, peretyn IZ = I ∩ Z bud\-qkoho maksymal\noho livoho idealu I ⊂ A z centrom
Z [ maksymal\nym livym idealom u Z . OtΩe, qkwo a naleΩyt\ radykalu
R ( Z ) , to a naleΩyt\ koΩnomu maksymal\nomu livomu idealu, a ce j oznaça[,
wo a ∈ R ( Z ) , tobto a = 0.
Osnovnyj rezul\tat.
Teorema. Nexaj A — unital\na napivprosta topolohiçna qderna ∗ -alheb-
ra nad C i Z — ]] centr. A topolohiçno izomorfna M n ( Z ) todi i til\ky
todi, koly:
a) A [ F2n -alhebrog;
b) A mistyt\ pidalhebru A0 , qka izomorfna Mn ( C ) i mistyt\ odynycg e .
Dovedennq. Zhidno z osnovnog teoremog roboty [2], umovy a)� i b) [ neob-
xidnymy ta dostatnimy dlq alhebra]çnoho izomorfizmu alhebr A i M n ( Z ) .
Zaznaçymo, wo dovedennq neobxidnosti teoremy [ naslidkom teoremy Amicura –
Levyc\koho pro te, wo matryçna alhebra Mn ( C ) [ F2n -alhebrog (dyv., napryk-
lad, [6], §�6). Dlq povnoty dovedennq teoremy nahada[mo, qk budu[t\sq alhebra-
]çnyj izomorfizm alhebr A i Mn ( Z ) . Nexaj ( ) ,ejk j k
n
=1 ∈ Mn ( C ) [ matryçnymy
odynycqmy. Qkwo ϕ — izomorfizm miΩ A0 ta Mn ( C ) , to, zhidno z umovog�b),
isnugt\ odnoznaçno vyznaçeni elementy ajk ∈ A0 taki, wo ϕ( )ajk = ejk . Dali,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
PRO DEQKYJ KLAS TOPOLOHIÇNYX ∗ -ALHEBR IZ STANDARTNYMY … 143
dlq bud\-qkoho z ∈ A vyznaçymo matryçni elementy w zjk ( ) = a z as j ks
s
∑ , j , k =
= 1, 2, … , n , a takoΩ matryçnoznaçnu funkcig σ ( z ) = w zjk j k
n( )
,( ) =1
∈ Mn ( A ) .
Nastupnymy krokamy u roboti [2] [ dovedennq toho, wo: 1)�vidobraΩennq σ [
linijnym mul\typlikatyvnym; 2)�qdro σ [ tryvial\nym; 3) w zjk ( ) ∈ Z dlq vsix
j , k = 1, 2, … , n ta z ∈ A ; 4) σ vidobraΩa[ alhebru A na vsg alhebru Mn ( Z ) .
Z ohlqdu na lemy 1 – 3 zalyßa[t\sq dovesty, wo alhebra A topolohiçno izo-
morfna Mn ( Z ) , tobto obydva vidobraΩennq σ ta σ–1
[ neperervnymy.
Spravdi, z odnoho boku, vnaslidok neperervnosti operaci] mnoΩennq u vy-
xidnij alhebri U dlq dovil\noho α = ( ) ,τ i j
n
=1 znajdut\sq α1 = ( ) ,τ1 1i j
n
= , α 2 =
= ( ) ,τ2 1i j
n
= , α3 = ( ) ,τ3 1i j
n
= taki, dlq qkyx vykonu[t\sq ocinka
σ α τ
τ
τ
( ) ( )
, , ,
z w z a z a a z ajk
j k
m j km
mj k
m j km
mj k
2 2
2 2
= = ≤
∑ ∑∑ ∑∑ ≤
≤ 2
1 2 3 1 3 2 2
2 2
2
1
2|| || || || || ||
= || || || ||
|| || = || ||∑∑ ∑∑a z a a a z K zmj km
mj k
mj km
mj k
τ τ τ τ τ τ τ
, ,
,
de K1 = || || || ||( )∑∑ a amj kmmj k τ τ1 3
2
,
.
Z inßoho boku, takoΩ vnaslidok neperervnosti operaci] mnoΩennq v U dlq
dovil\noho τ ∈ T znajdut\sq τ1 , τ2 , τ3 ∈ T taki, wo vykonu[t\sq ocinka
|| || = ≤
∑ ∑z a w z a a w z aj jk k
j k
j jk k
j k
τ
τ
τ
2
1 1
2
1 1
2
( ) ( )
, ,
≤
≤ a w z a K w z K zj jk k
j k
jk
j k
1 1
2
2
2
2
2
1 2 3 2 2τ τ τ τ ασ( ) ( ) ( )
, ,
( ) ≤ =∑ ∑ ,
de K2 = 2 1 1
2
1 3
max maxj j k ka a|| || || ||( )τ τ .
Z oderΩanyx ocinok vyplyva[, wo obydva vidobraΩennq σ ta σ–1
[ nepe-
rervnymy, a ce i dovodyt\ topolohiçnist\ izomorfizmu.
Teoremu dovedeno.
1. Krupnik N., Silbermann B. The structure of some Banach algebras fulfilling a standart identity //
Math. Nachr. – 1989. – 142. – P. 175 – 180.
2. Roch S., Silbermann B. On algebras with standart identities // Linear Algebra and its Appl. – 1990.
– 137/138. – P. 239 – 247.
3. Berezanskyj.G..M. SamosoprqΩenn¥e operator¥ v prostranstvax funkcyj beskoneçnoho
çysla peremenn¥x. – Kyev: Nauk. dumka, 1978. – 360�s.
4. Rabanovich S. V., Samoilenko Yu. S. Representations of Fn -algebras and applications // Meth.
Funct. Anal. and Top. – 1998. – 4, # 4. – P. 86 – 96.
5. Najmark.M..A. Normyrovann¥e kol\ca. – M.: Nauka, 1968. – 664�s.
6. Herstein I. N. Noncommutative rings. – New York: Wiley, 1968.
OderΩano 14.04.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
|
| id | umjimathkievua-article-3298 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:39:54Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/41/3273042986b6ca7b859dd90551bd5c41.pdf |
| spelling | umjimathkievua-article-32982020-03-18T19:50:22Z On one class of topological *-algebras with standard identities Про деякий клас топологічних *-алгебр із стандартними тотожностями Tishchenko, S. V. Тищенко, С. В. Let A be a unital semisimple topological nuclear *-algebra over C and let Z be its center. Then A is topologically isomorphic to M n (Z) if and only if A satisfies the standart identity and the maximality condition. Пусть A — унитальная полупростая топологическая ядерная * -алгебра над C и Z — ее центр. A топологически изоморфна M n (Z) тогда и только тогда, когда A удовлетворяет стандартному тождеству и условию максимальности. Institute of Mathematics, NAS of Ukraine 2007-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3298 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 1 (2007); 140–143 Український математичний журнал; Том 59 № 1 (2007); 140–143 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3298/3341 https://umj.imath.kiev.ua/index.php/umj/article/view/3298/3342 Copyright (c) 2007 Tishchenko S. V. |
| spellingShingle | Tishchenko, S. V. Тищенко, С. В. On one class of topological *-algebras with standard identities |
| title | On one class of topological *-algebras with standard identities |
| title_alt | Про деякий клас топологічних *-алгебр із стандартними тотожностями |
| title_full | On one class of topological *-algebras with standard identities |
| title_fullStr | On one class of topological *-algebras with standard identities |
| title_full_unstemmed | On one class of topological *-algebras with standard identities |
| title_short | On one class of topological *-algebras with standard identities |
| title_sort | on one class of topological *-algebras with standard identities |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3298 |
| work_keys_str_mv | AT tishchenkosv ononeclassoftopologicalalgebraswithstandardidentities AT tiŝenkosv ononeclassoftopologicalalgebraswithstandardidentities AT tishchenkosv prodeâkijklastopologíčnihalgebrízstandartnimitotožnostâmi AT tiŝenkosv prodeâkijklastopologíčnihalgebrízstandartnimitotožnostâmi |