Publicacions Ma em`a iques, Vol 39 (1995), 5–21.
POLAR DECOMPOSITION
IN RICKART C∗-ALGEBRAS
Dmi y Golds ein
Abs ac
A new p oo is ob ained o he ollowing ac : a Ricka C∗-algeb a
sa isfies pola decomposi ion. Equi alen ly, ma ix algeb as o e
a Ricka C∗-algeb a a e also Ricka C∗-algeb as.
In oduc ion.
In his pape we gi e new p oo o he ollowing esul : all Ricka
C∗-algeb as sa is y pola decomposi ion. This ac was es ablished in [2]
by P. A a and au ho by using a sui able ac o iza ion o he elemen s
in he egula o e ing o a fini e Ricka C∗-algeb a.
New p oo also uses he cons uc ion o he egula o e ing, bu in
a diffe en way. In pa icula , we don’ need he esul o Goodea l,
Law ence and Handelman abou algeb as wi hou one-dimensional ep-
esen a ions.
The egula ing o measu able ope a o s o a fini e AW∗-algeb a was
cons uc ed by S. K. Be be ian in [3]. La e Sai o modified Be be ian’s
app oach o gene al AW∗-algeb as [11]. E. Ch is ensen cons uc ed and
in es iga ed a ∗-algeb a o measu able ope a o s, associa ed o MSC C∗-
algeb as [5].
Handelman ound a egula ex ension Q(T) o a fini e Ricka C∗-
algeb a T, using a echnique o he module-homomo phisms on he es-
sen ial coun ably gene a ed ideals (ins ead o Be be ian’s coo dina ed
sequences) [9]. I was es ablished ([1], [10]) ha a fini e Ricka C∗-
algeb a Tsa isfies pola decomposi ion iff he bounded elemen s o Q( )
belong o T. Goodea l, Handelman and Law ence ha e p o ed ha T
sa isfies pola decomosi ion in he case whe e Thas no one-dimensional
ep esen a ions (see [8]).
P. A a in [1], using his special cons uc ion, p o ed ha le and igh
p ojec ions o elemen in a Ricka C∗-algeb a a e equi alen and he
6D. Golds ein
pola decomposi ion p oblem in gene al Ricka C∗-algeb as can be e-
duced o he fini e case. In his wo k A a also p o ed an equi alence o
he ollowing condi ions o a Ricka C∗-algeb a T:
(i) Tsa isfies pola decomposi ion;
(ii) The ma ix algeb as Mn(T) o e Ta e he Ricka C∗-algeb as
o all n;
(iii) The pa ial isome ies o Ta e ℵ0-addable.
Finally, by de elopmen o he me hods o [1], [8], i was p o ed in [2]
ha condi ions (i)-(iii) a e always ulfiled in Ricka C∗-algeb as.
In [6], [7] was cons uc ed a ∗-algeb a o measu able ope a o s o a
fini e Ricka C∗-algeb a and we e p o ed some algeb aic p ope ies o
his ∗-algeb a. We con inue o de elope his app oach in o de o sol e
he pola decomposi ion p oblem.
1. P elimina ies.
A∗-algeb a Ais Ricka , i o all x∈A he e exis s a p ojec ion e∈A
such ha R(x)={a∈A|xa =0}is eA. Because o he in olu ion,
L(x)={a∈A|ax =0}=T o some p ojec ion . We shall w i e
e=RA(x), =LA(x), 1 −e=RP(x), 1 − =LP(x) and P(A) o
he se o all p ojec ions o A.
The p ojec ions eand a e equi alen (e∼ )ina∗-algeb a Ai
e=uu∗, =u∗u o some pa ial isome y u∈A.Ais fini e i p∼1
implies p= 1. A Ricka C∗-algeb a is a C∗-algeb a ha is also a Ricka
∗-algeb a. We ecall some p ope ies o he Ricka C∗-algeb as.
Theo em 2.1. A Ricka C∗-algeb a sa isfies he ollowing p ope -
ies:
(i) P(T)is ℵ0-comple e la ice pa ially o de ed by p≥qiff pq =q
(see [4]).
I in addi ion Tis fini e hen he la ice P(T)is ℵ0-con inuous
[9, Co . 1.1].
(ii) LP(x)∼RP(x) o all x∈T[1, Th. 2.5].
(iii) Fo gi en sequences (en)and ( n)o o ogonal p ojec ions such
ha en∼ n o all n∈N, we ha e nen∼n n[1].
The pa ial isome ies a e ℵ0-addable in a Ricka C∗-algeb a Ti o
e e y sequence o pa ial isome ies {wn}such ha {wnw∗
n}and {w∗
nwn}
a e he sequences o o ogonal p ojec ions he e exis s a pa ial isome y
wsuch ha ww∗
nwn=wnw∗
nw=wn.
Pola decomposi ion in Ricka C∗-algeb as 7
2. S ongly dense domains.
Th ough his pape Tdeno es (i he opposi e is no specified) a fini e
Ricka C∗-algeb a.
A sequence o p ojec ions (en)⊂P(T) is a s ongly dense domain
(SDD) in case en↑1. Le e∈P(T), x∈T. We define x−1(e)=
RA[(1 −e)x].
P oposi ion 2.1. Le (en)and ( n)a e SDD, xn∈Tsuch ha
m≤nimplies xnem=xmem. Then a sequence ( n=x−1
n( n)en)
is a SDD.
P oo : Le dn=x−1
n( n). I m≤n hen
(1 −en)xn m=(1−en)(1 −em)xnem m=(1−en)(1 −em)xm m=0,
so ha m≤ n. Le p∈P(T). We show ha he e exis s a numbe k
such ha kp= 0. Fo ha choose a numbe isuch ha q=pei=
0. I xiq= 0, hen q≤ i. Now le xiq= 0. The e exis s a∈Tsuch ha
h=xiqa is non-ze o p ojec ion [4, pa . 8]. Obse e ha xiq=xnq o
all n≥iand h= kh= 0 o sufficien ly la ge k.Fo k≥iwe ha e
(1 − k)xkqah=(1− k)xiqah=(1− k)hh=0.
The e o e g=LP(qah)≤dk. In addi ion, g≤q≤ei≤ek, hence
g≤ k.Thusp k≥g=0.
Co olla y 2.2. I (en)and ( n)a e SDD, hen (en n)is also SDD.
P oo : Pu in P oposi ion 2.1 xn= 1 o all n.
3. A ing o measu able ope a o s.
An essen ially measu able ope a o (EMO) is a pai o sequences
(xn,e
n) wi h xn∈T,(en) an SDD, and such ha m≤nimplies
xnem=xmemand x∗
nem=x∗
mem. Two (EMO) (xn,e
n) and (yn,
n)
a e equi alen , i he e exis s an SDD (gn) such ha xngn=yngn,
gnxn=gnyn o all n∈N. Clea ly ha his ela ion is indeed equi a-
lence ela ion (By Co olla y 2.2). I (xn,e
n) is (EMO), [xn,e
n] deno es
i s equi alence class. We call [xn,e
n] a measu able ope a o (MO) and
deno e by S(T) he se o all (MO), and use he le e s x,y,z,... o
he elemen s o S(T). Now we define he algeb aic ope a ions on S(T).
We pu
[xn,e
n]+[yn,
n]=xn+yn,e
n n
λ[xn,e
n]=[λxn,e
n]
[xn,e
n][yn,
n]=[xnyn,k
n],
[xn,e
n]=[x∗
n,e
n],
8D. Golds ein
whe e kn= ny−1
n(en)en(x∗
n)−1( n).
Summa izing,
Theo em 3.1. The se S(T)o all MO is a ∗-algeb a. The mapping
x→ [x, 1](x∈T)is a ∗-isomo phism o Tin o Q, and [1,1] is a uni y
elemen o S(T).
We w i e x=[x, 1], o x∈T. The image o Tin S(T)isT.
We ecall he cons uc ion by Handelman o he ∗- egula ing associ-
a ed o a fini e Ricka C∗-algeb a. Le Abe a uni al ing. A igh (le )
ideal E⊆Ais essen ial i Ehas non i ial in e sec ion wi h any nonze o
igh (le ) ideal o A. We say ha Eis essen ial coun ably gene a ed
(ecg) igh ideal i he e exis a sequence { n}n∈N⊆Asuch ha  iA
is essen ial in A. Simila ly, we define le ecg ideal. I was p o ed in [9]
ha e e y ecg ideal o a fini e Ricka C∗-algeb a is gene a ed by SDD.
Le Tbe a fini e Ricka C∗-algeb a. Conside he ollowing pai s
o mappings [ ,E; 1,E
1], whe e is igh T-module homomo phisms
om essen ial coun ably gene a ed igh ideal E, 1is le T-module
homomo phism om essen ial coun ably gene a ed le ideal E1, and
hey a e balanced by he ollowing condi ion: e1 (e)= 1(e1)e o all
e∈Eand all e1∈E1. Two pai s [ ,E; 1,E
1] and [g,J;g1,G
1] a e
equi alen i (x)=g(x) and 1(y)=g1(y) o all x∈EJand all
y∈E1J1. Le Qbe he se o equi alence classes o jus defined pai s.
I was shown in [9] ha Qis endowed wi h algeb aic ope a ions, and
wi h espec o hese ope a ion Qbecomes a ∗- egula algeb a.
Define mapping om S(T) oQ.I [xn,e
n] is MO hen E=∞
n=1 enT
(E1=∞
n=1 Ten) is an essen ial coun ably gene a ed igh (le ) ideal in
Tco esponden ly. Define a igh T-module homomo phism : (en )=
xnen , whe e en ∈E. Ob iously, (en x)= (en )x o all x∈T. Le
en =ems(m≤n). Then
(ems)= (em)s=xmems=xnems=xnen = (en ).
Thus his defini ion is co ec . Simila ly, we define a le T-module ho-
momo phism 1:E1→T, ( en)= enxn. Now le e∈E,e1∈E1,
e=em ,e1= 1en.I m≤n hen
e1 (e)= 1en (em )= 1enxmem = 1enxnem
= 1( 1en)em = 1(e1)e.
By a simila a gumen e1 (e)= 1(e1)e. The e o e [ ,E, 1,E
1]∈
Q. We shall deno e jus defined mapping by π. Then π([xn,e
n]) =
Pola decomposi ion in Ricka C∗-algeb as 9
[ ,E, 1,E
1]. Le [xn,e
n]=[x
n,e

n], π([x
n,e

n]) = [ ,E, 
1,E
1].
Choose an SDD (pn) such ha xnpn=x
npn,pnxn=pnx
n o all
n∈N.Pu qn=pnene
n. No e ha qn∈EE1EE
1.We
ha e (qn)=xnqn=xnpnqn=x
nqn= (qn). Thus = on
∞

n=1
qnT.
In he same way we ob ain 1= 
1on
∞

n=1
Tqn.
Theo em 3.2. The mapping πis a ∗-isomo phism om S(T)on o
Q.
P oo : Le [ ,E, 1,E
1]∈Q,E=∞
n=1 enT,E1=∞
n=1 Ten,(en)an
SDD,
(ex)= (e)x, 1(xe1)=x 1(e1)
o all e∈E,e1∈E1,x∈T.Pu (en)=yn, 1(en)=zn. Ob iously,
ynen=yn,enzn=zn. Se
xn=yn+zn−znen=yn+zn−enyn
so ha xnen=yn,enxn=zn o all n∈N. I is easy o see ha
[xn,e
n] is MO. Se π(xn,e
n])=[g,E,g1,E
1], whe e g(en)=xnen,
g1(en)=enxn. Then g(en)=yn= (en), g1(en)=zn= 1(en),
hence [ ,E, 1,E
1]=[g,E,g1,E
1]. Thus πis su jec i e. Now we show
ha he mapping πp ese es he algeb aic ope a ions. Le [xn,e
n],
[yn,k
n]∈S(T). Pu
π([xn,e
n]) = [ ,E, 1,E
1],π([yn,k
n]) = [g,J,g1,J
1],
whe e
E=
∞

n=1
enT, E1=
∞

n=1
Ten,
J=
∞

n=1
knT, J1=
∞

n=1
Tkn.
We ha e (see [9, Sec ion 2])
[ ,E, 1,E
1]+[g,J,g1,J
1]= +g,E J, 1+g1,E
1J1,
[xn,e
n]+[yn,k
n]=xn+yn,e
nkn.

10 D. Golds ein
Le pn=enknand π([xn+yn,e
nkn]) = [ , L, 1,L
1]. We can ega d
ha
L=
∞

n=1
pnT, L1=
∞

n=1
Tpn,
(pn)=(xn+yn)pn,
1(pn)=pn(xn+yn).
Since (enkn)T=(enT)(knT) ( see [9]) i ollows L=JE,L1=
J1E1. In addi ion
(pn)=(xn+yn)pn=( +g)(pn),
1(pn)=pn(xn+yn)=( 1+g1)(pn).
Consequen ly
[ , L, 1,L
1]= +g,E J, 1+g1,E
1J1.
Fu he , [xn,e
n][yn,k
n]=[xnyn,
n], whe e nis a sui able SDD. On he
o he hand,
[ ,E, 1,E
1][g,J,g1,J
1]=[ g,g−1E,g1 1, −1
1J1].
We shall es ablish ha
∞

n=1
nTis an essen ial subideal in g−1E. By he
defini ion, n=hngn, whe e
hn=en(x∗
n)−1(kn),g
n=kny−1
n(en),g
−1E={x∈J:g(x)∈E}.
We ha e n≤gn≤kn, he e o e n∈J o all n∈N. I emains o
p o e ha g( n)∈E o all n. Really,
g( n)=g(gn) n=g(gnkny−1
n(en)) n=g(kn)y−1
n(en)gn n
=ynkny−1
n(en)gn n=yny−1
n(en)kngn n
=enyny−1
n(en)kngn n=enyny−1
n(en) n,
he e o e g( n)∈E.So n∈g−1E o all n.
Simila ly, we ob ain
∞

n=1
T n⊂ −1J1.Thus
[ g,g−1E, 1g1, −1
1J1]= g,
∞

n=1
nT,g1 1,
∞

n=1
T n.
I implies
π([xn,e
n][yn,k
n]) = π([xn,e
n])π([yn,k
n]).
Ob iously,
π(λ[xn,e
n]) = λπ([xn,e
n]).
I is s h igh o wa d o check ha π([xn,e
n]∗)=π([xn,e
n])∗.
Pola decomposi ion in Ricka C∗-algeb as 11
Co olla y 3.3. S(T)is a Ricka ∗-algeb a and ℵ0-con inuous ing.
P oo : I ollows om Theo em 3.2 and [9, Th. 2.1].
4. Some algeb aic p ope ies o S(T). Cayley ans o m.
Lemma 4.1. I x=[xn,e
n]∈Qand he xna e all in e ible hen x
is in e ible and x−1=[x−1
n,h
n] o a sui able SDD (hn).
P oo : Se n=LP(xnen). We show ha ( n) is a SDD. I m≤n
hen n(xmem)= nxnem= nxnenem=xnenem=xmem, m≤ n.
Since xnis in e ible hen RP(xnen)=en. We ha e n∼en[1, Th. 2.5].
As p∼qimplies 1 −p∼1−q o he p ojec ions pand qin a fini e
Ricka C∗-algeb a, so en+1 −en∼ n+1 − n. Then by ℵ0-addi i i y,
1 = [sup
n
(en+1 −en)] e1∼sup
n
( n+1 − n) 1= .
Se yn=x−1
n.I m≤n, hen yn m=ym m. Really,
ynxmem=ynxnem=em=ymxmem,
hence (yn−ym)xmem= 0 and (yn−ym) m= 0. Simila y on se ing
gn=LP(x∗
nen), we ha e ha (yn) is a SDD and y∗
ngm=y∗
mgmwhen
m≤n.Pu hn= ngn, hen y=[yn,h
n] is MO, and xy =yx =1.
Co olla y 4.2. Fo any x∈S(T)an elemen 1+x∗xis in e ible.
P oo : I ollows immedia ely om Lemma 4.1.
Lemma 4.3. I x=x∗, one can w i e x=[xn,e
n]wi h x∗
n=xn.
P oo : I x=[yn,
n], hen x=1/2(x∗+x)=[1/2(y∗
n+yn),
n].
Co olla y 4.4. I x=x∗, hen x+iis in e ible.
P oo : Le x=[xn,e
n], x∗
n=xn; hen x+i=[xn+i, en] and each
xn+iis in e ible.
Theo em 4.5. The o mulas
u=(x−i)(x+i)−1,x=i(1 + u)(1 −u)−1
define mu ually in e se one-one co espondences be ween he sel -adjoin
elemen s x∈Q, and he uni a y elemen s u o which 1−uis in e ible.
P oo : I ollows om Co olla y 4.4.
We call his u he Cayley ans o m o x.
12 D. Golds ein
Lemma 4.6. Le x=[xn,e
n]∈S(T)and xn−→ xin no m, hen
x=x.
P oo : E iden ly, xen−xnen=xen−xken o all k≥n. Then
xen−xnen≤x−xk o all k≥n,xen−xnen=0,xen=xnen.
In jus he same way, enx=enxn.
Lemma 4.7. Le x=[xn,e
n]∈S(T). Then xen=xnen.
P oo : Ob ious.
5. The bounded measu able ope a o s.
Le Tbe a fini e Ricka C∗-algeb a, Q=S(T) deno es a ∗-algeb a o
measu able ope a o s o T.
An elemen x=[xn,e
n]∈Qis bounded, i supnxn≤∞. Le Bbe
a se o all bounded MO. I is clea ha Bis ∗-algeb a. Since P(Q)⊂B
hence Bis Ricka ∗-algeb a. We define he mapping ·1:Bx→
x1= in supn{xn|(xn,e
n)∈x}∈R.
The bounded elemen s o S(T) play a c ucial ole in he ollowing
discussion o he pola decomposi ion p oblem (o ℵ0-addabili y o he
pa ial isome ies, see In oduc ion) in a fini e Ricka C∗-algeb a. I is
easy o see ha he pa ial isome ies o Ba e ℵ0-addable (Co olla y 7.3).
On he o he hand, i is well known ha he algeb as Band Tcoincide
i Tis AW∗-algeb a [3]. We shall p o e a simila esul o a gene al
Ricka C∗-algeb a.
Theo em 5.1. The mapping ·1is a C∗-no m.
P oo : Le x=[xn,e
n]∈B. Clea ly, x1≥0. I x1= 0 hen
o any ε≥0 he e exis s EMO (xn,e
n)∈xsuch ha supnxn≤ε.
Le y=[yn,e
n]∈B, supnyn=α. We can choose (x
n,e

n)∈xwi h
x
n≤ε/α o all n∈N. The e o e xnyn≤ε,xy1=0.
Now assume ha he e exis s x∈Bsuch ha x= 0 and x1=0.
Choose numbe nsuch ha xen= 0. By Lemma 4.7 xen=xnen.As
i was shown abo e xen1=xnen1= 0. Le a=xnen. By he
defini ion o he no m ·1we ha e a1= in supn{an|(an,k
n)∈a}.
Fo any (an,k
n)∈a he e exis s an SDD (pn) such ha apn=anpn.
No e apn=anpn≤an. Choose bsuch ha ba =eis a non-ze o
p ojec ion [4, pa . 8]. Since P(T)isℵ0-con inuous, he e exis s k∈N
such ha q=epk=0.
Consequen ly
1=q≤epk=bapk≤apkb,
Pola decomposi ion in Ricka C∗-algeb as 13
hence apk≥1/b. I ollows ak≥1/b, hence a1=0,a
con adic ion. Thus x1= 0 implies x=0.
Ob iously λx1=λx1 o each x∈B.
Fu he , le x,y∈B,x=[xn,e
n], y=[yn,
n]. Then
x+y1= in sup
n{cn|(cn,g
n)∈x+y}
≤in sup
n{x
n+y
n|(x
n,e

n)∈x(y
n, 
n)∈y}
≤in sup
n{x
n+y|(x
n,e

n)∈x,(y
n, 
n)∈y}
=x1+y1.
In jus he same way, we ge
xy1≤x1y1.
F om p e ious p ope y we ha e x∗x1≤x2
1. On he o he hand,
le (bn,q
n)∈x∗x,(sn,k
n)∈x o a sui able SDD kn.
Hence he e exis s SDD (pn) such ha
npn=s∗
nsnpn,p
nbn=pns∗
nsn.
Le n=pnknqn. Then ( ns∗
n)(sn n)= nbn nand so  ns∗
nsn n≤
bn. In addi ion, [ ns∗
n,
n]=[s∗
n,k
n] o sui able SDD ( n), he e-
o e ( ns∗
n,
n)∈x∗. Hence o any EMO (bn,q
n)∈x∗x he e ex-
is s EMO (zn,
n)∈x(zn=sn n) such ha zn2≤bn.Thus
x2
1≤x∗x1.
Co olla y 5.2. The no ms ·and ·1coincide on T.
P oo : Le xbe a posi i e elemen o T. By he defini ion, we ha e
x1= in sup
n{xn|(xn,e
n)∈x}.
Ob iously x1≤x. Se (xn,e
n)∈x. Then he e exis s SDD pnsuch
ha xnpn=xpn o all n. The e o e xpn=xnpn≤xn. Choose
a sequence o he posi i e numbe s εnwi h εn↑x. Se {x} =C(K),
o some Hausdo ff space K.Pu Un={a∈K:x(a)>ε
n},
bn(a)=


1
x(a),a∈U
0,o he wise.
20 D. Golds ein
Co olla y 7.4. All Ricka C∗-algeb as sa is y pola decomposi ion.
P oo : By [1, Th. 3.4], his asse ion is educed o a fini e case. Now
combine Co olla y 7.3 and [1, P op. 2.1] and he Co olla y ollows.
Co olla y 7.5. Le Tbe a Ricka C∗-algeb a, hen he ma ix alge-
b as Mn(T)o e Ta e also Ricka C∗-algeb as o all n∈N.
P oo : See [1, Th. 3.5].
8. Axiom (PSR) in Q.
Using Theo em 7.1 and he me hods o [3], [11]o [6], we can desc ibe
he sel -adjoin elemen s in Q.
Theo em 8.1. Le x=x∗∈Q,u=u(u∈T) i s Cayley ans o m.
One can w i e x=[xn,e
n]wi h xn,en∈{u},x∗
n=xn,xnen=xn,
x2
n↑.
P oo : See [3, Th. 4.2].
An elemen x∈Qis posi i e, w i en x≥0, i x=y∗y o some
y∈Q.
Theo em 8.2. Le x=x∗∈B,u=ui s Cayley ans o m. The
ollowing condi ions a e equi alen :
a) x≥0;
b) one can w i e x=[yn,
n]wi h yn≥0;
c) he spec um o ucon ained in {eiΘ:−π≤Θ≤0};
d) one can w i e x=[xn,e
n]wi h xn,en∈{u},xn≥0,xnen=
xn.
P oo : See [3, Th. 6.1].
Co olla y 8.2. Qsa isfies axiom (PSR).
P oo : See [3, Co . 6.2].
Acknowledgmen s. The au ho exp ess his g a i ude o V. I. Chilin
o s a ing he p oblem and o his conce n abou he wo k, and also o
P. A a and D. Handelman o aluable discussions.
I am e y g a e ul o Ben-Gu ion Uni e si y o he help in p epa ing
his pape .

Pola decomposi ion in Ricka C∗-algeb as 21
Re e ences
1. P. A a, Le and igh p ojec ions a e equi alen in Ricka C∗-al-
geb as, J. Algeb a 120 (1989), 433–448.
2. P. A a and D. Golds ein, A solu ion o he ma ix p oblem o
Ricka C∗-algeb as, Ma h. Nach . 164 (1993), 259–270.
3. S. K. Be be ian, The egula ing o a fini e AW∗-algeb a, Ann.
Ma h. 65 (1957), 224–240.
4. S. K. Be be ian,“Bae ∗Rings,” Sp inge -Ve lag, Be lin and
New-Yo k, 1972.
5. E. Ch is ensen, Non commu a i e in eg a ion o mono one se-
quen ially closed C∗-algeb as, Ma h. Scand. 31 (1972), 171–190.
6. D. Golds ein, Ricka o de ed ∗-algeb as, Dokl.Uzb. Acad. Nauk
1(1990) (in Russian).
7. D. Golds ein,ℵ0-addable o pa ial isome ies in he Ricka
C∗-algeb as, Dokl. Uzb. Acad. Nauk 10 (1990) (in Russian).
8. K. R. Goodea l, D. E. Handelman and J. W. Law ence,
Affine ep esen a ions o G o endieck g oups and applica ions o
Ricka C∗-algeb as and ℵ0-con inuous egula ings, Memoi s
Ame . Ma h. Soc. 234 (1980).
9. D. Handelman,“Fini e Ricka C∗-algeb as and hei p ope ies,”
Ad . in Ma h. Suppl. S udies 4, Academic p ess, O lando, FL, 1979,
pp. 171–196.
10. D. Handelman, Ricka C∗-algeb as, II, Ad . in Ma h. 48 (1983),
1–15.
11. K. Sai o, On he algeb a o measu able ope a o s o a gene al
AW∗-algeb a, Tohoku Ma h. Jou n. 21 (1969), 249–270.
Ins i u e o Ma hema ics
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