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THE RELATION BETWEEN REAL AW*-FACTORS AND ANTI-AUTOMORPHISMS OF
INVOLUTIVE (I.E. WITH PERIOD 2) *-(COMPLEX) AW*-FACTORS
(1) Kh. Kh. Bol ae , (2) F. B. Rasulo a
(1,2) Na ional Pedagogical Uni e si y o Uzbekis an named a e Nizami and (1) Tashken
In e na ional Uni e si y, Tashken , Uzbekis an
[email p o ec ed]
asulo a e uza1[email p o ec ed]
ABSTRACT. The pape o he is o ini ia e he s udy o eal AW*-algeb as in he amewo k o he
heo y o eal C*-algeb as and W*-algeb as. I happens ha in some aspec s eal AW*-algeb as
beha e unlike complex AW*-algeb as and some imes hei p ope ies a e comple ely diffe en also
om co esponding p ope ies o eal W*-algeb as. We p o e ha i he complexifica ion o
a eal C*-algeb a A is a (complex) AW*-algeb a hen A i sel is a eal AW∗-algeb a. By modi ying
he Takenouchi’s examples o complex non-W*, AW*- ac o s we show ha he e exis eal non-W*,
AW*- ac o s. The co espondence be ween eal AW*- ac o s and in olu i e (i.e. wi h pe iod 2) *-
an i-au omo phisms o (complex) AW*- ac o s is es ablished. We gi e he decomposi ion o eal
AW*-algeb as in o ypes I, II and III simila o he case o complex AW*-algeb as o W*-algeb as. I
is p o ed ha i A is a eal AW*- ac o and i s complexifica ion is also an AW*-algeb a
(and he e o e an AW*- ac o ) hen he ypeso A and M coincide.
KEYWORDS: AW*-algeb a, C*-algeb a, ac o , in olu i e *-an iau omo phism, complex Hilbe
space, commu an , complexifica ion, linea *-au omo phism, conjuga e, bicommu an , qua e nions
algeb a, p ojec ion, isomo phic.
1. INTRODUCTION
The heo y o ope a o algeb as was ini ia ed in a se ies o pape s by Mu ay and on Neumann in
hi ies. La e such algeb as we e called on Neumann algeb as o W*-algeb as. These algeb as a e
sel -adjoin uni al subalgeb as M o he algeb a B(H) o bounded linea ope a o s on a complex
Hilbe space H, which is closed in he weak ope a o opology. Equi alen ly M is a on Neumann
algeb a in B(H) i i is equal o he commu an o i s commu an ( on Neumann’s bicommu an
heo em). A ac o (o W*- ac o ) is a on Neumann algeb a wi h i ial cen e and in es iga ion o
gene al W*-algeb as can be educed o he case o W*- ac o s, which a e classified in o ypes I, II
and III.
Real ope a o algeb a is a *-algeb a consis ing o bounded ( eal) linea ope a o s on a eal Hilbe
space H. I i is closed in he weak ope a o opology we ha e eal W*-algeb a, and i i is uni o mly
closed (i.e. in he no m opology) hen we come o he no ion o he eal C*-algeb a.
In his monog aph [7] Li Bing-Ren has se up he undamen als o eal ope a o algeb as and ga e a
sys ema ic discussion o he eal coun e pa o he heo y o W*- and C*-algeb as.
A sligh ly diffe en (bu almos he same up o *-isomo phism) defini ion o eal W*-algeb as was
gi en by E.S ø me [13,14]: A eal on Neumann algeb a (o eal W*-algeb a) is a eal *-algeb a
o bounded linea ope a o s on a complex Hilbe space con aining he iden i y ope a o 1, which is
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closed in he weak ope a o opology and sa isfies he condi ion The smalles
(complex) on Neumann algeb a con aining coincides wi h i s complexifica ion
, i.e. . Mo eo e gene a es a na u al in olu i e (i.e. o o de 2) *-
an iau omo phism o , namely , whe e
I is clea ha . Con e sely, gi en a (complex) on Neumann algeb a
U and any in olu i e *-an iau omo phism α on U, he se is a eal on
Neumann algeb a in he abo e sense.
I is no diffucul o see ha wo eal on Neumann algeb as gene a ing he same (complex) on
Neumann algeb a a e isomo phic i and only i he co esponding in olu i e *-an iau omo phisms
a e conjuga e. Thus he s udy o he abo e eal on Neumann algeb a can be educed o he s udy o
pai s (U, α), whe e U is a (complex) on Neumann algeb a and α - i s in olu i e *-an iau omo phism.
2. PRELIMINARIES
Le H be a complex Hilbe space, deno e he algeb a o all bounded linea ope a o s on H.
The weak (ope a o ) opology on is he locally con ex opology, gene a ed by semi no ms
o he o m: W*-algeb a is a weakly closed complex *-
algeb a o ope a o s on a Hilbe space H con aining he iden i y ope a o 1. Recall ha W∗-algeb as
a e also called on Neumann algeb as.
Le u he M be a W*-algeb a. The se o all elemen s om commu ing wi h each elemen
om M is called he commu an o he algeb a M. The cen e o a W*-algeb a M is he se o
elemen s o M, commu ing wi h each elemen om M. I is easy o see ha .
Elemen s o a e called cen al elemen s. A W*-algeb a M is called ac o , i consis s
o he complex mul iples o 1, i.e i We say ha a W*-algeb a M is injec i e
i he eexis s a p ojec ion P in on o M such ha and 1. This isequi alen o
he condi ion ha M is hype fini e, i.e., ha he e exis s an inc easing sequence o ma ix
subalgeb as o he algeb a M con aining 1 and such ha he union is weakly dense in M.
Le be p ojec ions om M. We say ha is equi alen o , and w i e , i
o some pa ial isome y om M. A p ojec ion is called: fini e, i
implies ; infini e - o he wise; pu ely infini e, i doesn’ ha e any nonze o fini e
subp ojec ion; abelian, i he algeb a is an abelian W∗-algeb a. A W∗-algeb a M is called
fini e, infini e, pu ely infini e, i 1 is a fini e, infini e, pu ely infini e espec i ely; M is σ-fini e, i any
amily o pai wise o hogonal p ojec ions om M is a mos coun able; semifini e, i each p ojec ion
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in M con ains a nonze o fini e subp ojec ion; p ope ly infini e, i e e y nonze o p ojec ion om
is infini e; disc e e, o o ype I, i i con ains a ai h ul abelian p ojec ion (i.e. an abelian
p ojec ion wi h he cen al suppo 1); con inuous, i he e is no abelian p ojec ion in M excep ze o;
M is o ype II, i M is semifini e and con inuous; ype ( espec i ely ), i M is o ype I and
fini e ( espec i ely p ope ly infini e); ype ( espec i ely ype ), i M is o ype II and fini e
( espec i ely p ope ly infini e); ype III, i M is pu ely infini e. I is known ha any W*-algeb a has a
unique decomposi ion along i s cen e in o he di ec sum o W*-algeb as o he
and III ypes.
A linea mapping is called a *-au omo phism ( espec i ely a *-an iau omo phism) i
and ( espec i ely , o all . A
mapping α is called in olu i e i . A *-au omo phism α is called inne i he e exis s a uni a y
in M, such ha , o all . A *-au omo phism is called cen ally i ial i
*-s ongly as o any cen al sequence . We shall deno e by
he g oup o all *-au omo phisms, by he g oup o all *an iau omo phisms, by
he g oup o all inne *-au omo phisms, and by C (R) he subg oup o i s cen ally i ial
*-au omo phisms o M. Two *-au omo phisms o *-an iau omo phisms α and β a e said o be
conjuga e (o ou e conjuga e), i ( espec i ely ) o some *-
au omo phism θ (and an inne *-au omo phism Adu). A linea unc ional ω on M is called posi i e, i
o all . A posi i e linea unc ional ω on M wi h is called a s a e. Le
be he posi i e pa o M. A weigh on M is a homogeneous addi i e unc ion
(we suppose ha ). A weigh (o a s a e) ω is called: ai h ul, i o
any implies ; no mal, i o any ne in M, inc easing o an elemen
x, we ha e ; fini e, i o all ; semifini e, i o any
he e exis s a ne o elemen s , such ha , and in σ - weak
opology; ω is a ace, i o all and each uni a y .
The ype o a W*-algeb a is igh ly connec wi h he exis ence o aces on i . Namely a W*-algeb a
M is a fini e i and only i i possesses a sepa a ing amily o fini e no mal aces; i is semifini e i
and only i i possesses a ai h ul semifini e no mal ace; M is pu ely infini e i and only i he e is
no nonze o semifini e no mal ace on M (see [15]).
Defini ion. [4]. By a eal C*-algeb a we mean a eal Banach *-algeb a R such ha he ela ion
holds and he elemen is in e ible o any
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Defini ion*. [8,9]. A eal C*-algeb a R such ha R+iR is a complex W*-algeb a is e e ed o as a
eal W*-algeb a.
We p oceed wi h ano he defini ion o a eal W*-algeb a, which can be ound in pape s o S ø me .
Defini ion∗∗. [2,13]. A uni al weakly closed eal *-algeb a R in such ha is
called a eal W*-algeb a.
A eal W*-algeb a R is called a ( eal) ac o i i s cen e Z(R) consis s o elemen s λ1, We say
ha a eal W*-algeb a R is o ype and III, i he en eloping W*-algeb a
(i.e., he leas W*-algeb a con aining R) is o he co esponding ype wi h espec
o he usual classifica ion o W*-algeb as.
3. MAIR RESULTS
Le A be a eal C*-algeb a, wi h he complexifica ion . Then M is a complex C*-algeb a
and, as we ha e seen in he p e ious sec ion, i A is a eal AW*-algeb a M may no be a (complex)
AW*-algeb a. Now le us conside he con e se p oblem i is an AW*-algeb a is A
necessa ily a eal AW∗-algeb a? The ollowing esul gi es a posi i e answe o his p oblem.
P oposi ion 1. Le A be a eal C*-algeb a and le be i s complexifica ion. Suppose ha
M is an AW*-algeb a. Then A is a eal AW*-algeb a.
P oo . As we ha e men ioned in he fi s sec ion, A coincides wi h he fixed poin se unde he
conjuga e linea *-au omo phism o M, whe e , i.e.
I S is a nonemp y subse in A hen o i s igh -annihila o (wi h espec o M)
we ha e
and
because This means ha i and only i .
Now suppose ha M is an AW*-algeb a, hen o a sui able p ojec ion . Since
om abo e i ollows ha . The e o e is a p ojec ion and , i.e.
. Thus i.e.
This means ha . Bu hen
i.e. A is a eal AW*-algeb a.
P oposi ion 2. The e exis eal AW*- ac o s which a e no eal W*- ac o s.
Theo em 1. A eal AW
∗
-algeb a A is a eal W*-algeb a i and only i
(i) A possesses a sepa a ing amily o no mal s a es;
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(ii) i s complexifica ion is an AW*-algeb a.
P oo . Necessi y is ob ious, since i A is a eal W*-algeb a, hen is a (complex) W*-
algeb a (see [7, Chap.5]). The e o e, M is an AW*-algeb a and i possesses a sepa a ing amily o
no mal s a es, he es ic ions o which on A gi e a sepa a ing amily o no mal s a es on A.
Sufficiency. Le be an AW*-algeb a and le A possess a sepa a ing amily o no mal
s a es, which we deno e by , i.e. o any exis s wi h
Fo we pu A s aigh o wa d calcula ion shows ha
α is an in olu i e (i.e. wi h pe iod 2) *-an i-au omo phism o M, and
The ex ension o by linea i y on M we deno e by , and we shall show, ha he amily
is a sepa a ing amily o no mal s a es on M.
Fo we ha e , and since is he mici ian
we ob ain since . Thus, o we ha e
and since
I , hen (because α is a *-an i-au omo phism), and hence i.e.
. The e o e , i.e. all unc ionals a e posi i e on M.
Mo eo e , we ha e , i.e. is a amily o s a es on M.
Now le us show, ha each s a e is no mal. I is an a bi a y ne wi h hen
since α is an o de isomo phism o M, we ha e The e o e and
. Since is a no mal we ob ain
i.e. all unc ionals a e no mal on M.
Finally, le and o all γ. Then , and since is a
sepa a ing amily o s a es, . Hence we ha e i.e.
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. Thus, he AW*-algeb a M possesses a sepa a ing amily o no mal s a es . By he
heo em o Pede sen [10] M is a W∗-algeb a. The e o e, by [7] A is a eal W∗-algeb a.
Now, le M be a (complex) AW*- ac o , α i s in olu i e *-an i-au omo phism. Then as i was
men ioned abo e he se is a eal C*-algeb a such ha
(ac ually in e ms o ope a ion ”-”) and om P oposi ion 4.3.1 i ollows ha A is a eal
AW*- ac o . I is known ha wo eal W*-algeb as gene a ing he same (complex) W*-algeb a, a e
isomo phic i and only i he co esponding in olu i e *-an i-au omo phisms a e conjuga e [2,13,14].
A simila esul is also alid o eal AW*-algeb as:
P oposi ion 3. Le α and β be in olu i e *-an i-au omo phisms o a (complex) AW*- ac o M. Then
he eal AW*- ac o s
and
a e eal *-isomo phic i and only i he in olu i e *-an i-au omo phisms α and β a e conjuga e, i.e.
o a sui able *-au omo phism o he AW*- ac o M.
P oo . Le A and B be eal *-isomo phic wi h a *-isomo phism . Then can be
na u ally ex ended o a (complex) *-isomo phism θ o hei complexifica ions and
bo h coincide wi h M. The e o e θ is a *-au omo phism o M and , i.e. i and
only i . Thus o we ha e
, i.e. o all
Since is a *-au omo phism on M which is iden ical on A and any eal *au omo phism o A
can be uniquely ex ended o a complex *-au omo phism o M, i ollows ha on
whole M, i.e. and , i.e. α and β a e conjuga e.
Con e sely, i α and β a e conjuga e, i.e. o a sui able complex *-au omo phism θ o M,
hen and i and only i , i.e. . The e o e,
θ es ic ed on A gi es he needed *-isomo phism be ween eal AW∗- ac o s A and B.
Now we conside one o he main esul s o his sec ion.
Theo em 2. Le A be a eal AW*-algeb a and i s complexifica ion is a (complex) AW*-
algeb a. Then A is o ype I i and only i M o ype I.
Co olla y. Le A be a eal AW*-algeb a o ype I, and i s complexifica ion is a
(complex) AW*-algeb a. Then A is a eal W*-algeb a i and only i i s cen e is a eal W*-
algeb a.
P oo . I A is a eal W*-algeb a hen, ob iously, i s cen e is a eal W*-algeb a. Con e sely, le
A be an AW*-algeb a o ype I and i s cen e is a W*-algeb a. Then by Theo em 4.5.2,
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is an AW*-algeb a o ype I, and i s cen e is a W*-algeb a. F om Kaplanskys
heo em [6, Theo em 2] i ollows ha M is an W*-algeb a. The e o e, A is a eal W*-algeb a.
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