Publicacions
Ma emá iques,
Vol
36
(1992),
369-388
.
A
bs ac
THE
ALMOST
ISOMORPHISM
RELATION
FOR
SIMPLE
REGULAR
RINGS
PERE
ARA
AND
K
.
R
.
GOODEARL
Dedica
a la
memó ia
del
nos e
amic,
Pe e
Menal
i
B u al
A
longs anding
open
p oblem
in
he
heo y
o
on
Neumann
eg-
ula
ings
is
he
ques ion
o
whe he
e e y
di ec ly
ini e
simple
egula
ing
mus
be
uni - egula
.
Recen
wo k
on
his
p oblem
has
been done
by
P
.
Menal,
K
.C
.
O'Mea a,
and
he
au ho s
.
To
cla i y
some
aspec s
o
hese
new
de elopmen s,
we
in oduce
and
s udy
he
no ion
o
almos
isomo phism be ween
ini ely
gene a ed
p ojec i e
modules
o e
a
simple
egula
ing
.
0
.
In oduc ion
.
In
he
ew
pas
yea s,
he e
ha e been
some
ad ances
in
he unde -
s anding
o
di ec ly
ini e
simple
egula
ings
.
In
1988,
Menal
and
he
second au ho
[GM,
Theo em
5
.2]
showed
ha
i
R
is
a di ec ly
ini e
egula
algeb a
o e
an
uncoun able
ield,
and
i
R
con ains no
uncoun -
able
di ec
sums
o
nonze o
igh
ideals,
hen
i is
uni - egula
.
As
a
consequence
o
his,
any
s ably
ini e
simple
egula
algeb a
R
o e
an
uncoun able
ield
is
uni - egula
[GM,
Co olla y
5
.4]
.
Mo e
ecen ly,
O'Mea a
p o ed
ha
a
di ec ly
ini e
simple
egula ing
sa is ying
weak
compa abili y
is
uni - egula
[O,
Theo em
1]
.
An
al e na i e
p oo
o
O'Mea a's
Theo em
was
de eloped
by
he
second au ho
in
p i a ely
ci cula ed
no es
[G5]
.
We
ake
he
oppo uni y
o
p esen
his
p oo
he e
.
Ou
s anda d
e e en e
o
he
heo y
o
egula
ings
is
[G1],
and
o
he
heo y
o pa ially
o de ed
abelian
g oups
is
[G4]
.
The
eade
can
e e
o
he e
books
o
any
unde ined
e ms
.
The
esea ch
o
he
i s
au ho
was
pa ially
suppo ed
by
DGICYT
g an
P89-0296,
and
ha
o
he
second au ho by an
NSF
g an
37
0

P
.
ARA,
K
.
R
.
GOODEARL
Le
R
be an
associa i e
ing
wi h
1
.
Deno e
by
P
( esp
.
Po)
he
class o
ini ely
gene a ed
p ojec i e
igh
R-modules
( esp
.
he
class o
nonze o
ini ely
gene a ed
p ojec i e
igh
R-modules)
.
I
R
is
a
egula
ing
hen
L(RR)
will
deno e
he
la ice
o
p incipal
igh ideals o
R
.
Fo
A,
B
E
P,
we
will
w i e
A<B
( esp
.
A
~
B)
i
A
is
isomo phic
o
a
submodule
( esp
.
p ope
submodule)
o
B
.
Fo
a
posi i e
in ege
k and
module
A,
we
le
kA
deno e
he
di ec
sum
o
k
copies
o
A
.
A
ing
R
is
said
o
be
di ec ly
ini e
i
xy
=
1
implies
yx
=
1,
o
x,
y
E
R
.
We
say
ha
R
is
s ably
ini e
i
M

,
(R)
is
di ec ly
ini e
o
all
n >
1
.
I is
no
known
whe he
di ec ly
ini e
egula
ings
a e
s ably
ini e
[G1,
Open
P oblem
l]
.
The
ques ion
is
open
e en
in
he
case
o
simple
egula
ings
.
A
ing
R
is
said o
be
uni - egula
i
o
any
x
E
R
he e
exis s
a
uni
u E
R
such
ha
x
=
xux
.
E e y
uni - egula
ing
is
s ably
ini e,
bu
he e
exis
s ably
ini e
egula
ings
which
a e
no
uni - egula
[G1,
P oposi ion
5
.2
and
Example
5
.10]
.
Howe e ,
he e
a e
some
in e es ing
classes
o
egula
ings o
which
i is
known
ha
di ec
ini eness
implies
uni - egula i y
.
Fo
example,
his
holds
o
egula
ings
sa is ying
gen-
e al
compa abili y
[G1,
Theo em
8
.12],
o
igh
k~o-con inuous
egula
ings
[G2,
Theo em
1 .4],
and
o
k~o-comple e
egula
ings
[Bu,
Co ol-
la y
1
.6]
.
An
ou s anding
ques ion
in
he heo y
is
whe he
a di ec ly
ini e
simple
egula
ing
is
uni - egula
[G1,
Open
P oblem
3]
.
We
say
ha
a
class
o
modules
C
sa is ies
he
cancella ion
p ope y
(wi h
espec
o
he
isomo phism
ela ion)
i
A
®
C=B®C
implies
A=B
o
A,
B,
C
E
C
.
A
egula
ing
R
is
uni - egula
i
and
only
i
P
sa is ies
he
cancella ion
p ope y, see
[G1,
Theo em
4
.5]
.
The
main
esul o
Sec ion
1 s a es
ha
a
di ec ly
ini e
simple
egula ing
is
uni -
egula
i
and
only
i
P
sa is ies
he
cancella ion
p ope y
wi h
espec
o
he
almos
isomo phism
ela ion
(de ined
in
Sec ion
1)
.
We
say
ha
R
is
s ic ly
unpe o a ed
whene e
nA
~
nB
implies
A
~
B
o
A,
B
E
P
and
n
_> 1
.
R
is
unpe o a ed
i
nA
;
:S
nB
implies
A<B o A,BEPandn>1
.
Assume
ha
R
is
a di ec ly
ini e
simple
egula ing
.
I
is
an
open
ques ion
whe he
R
is
(s ic ly)
unpe o a ed
.
S ic ly
unpe o a ed
di-
ec ly
ini e
simple
egula
ings
ha e
a
numbe
o
in e es ing
p ope ies
.
In
pa icula
hey
a e
uni - egula
and,
in
he
non-a inian
case,
hey
a e
close
o
being
ings
o
ma ices
o
any
size
(see
Sec ion
2)
.
Some
echnical
esul s
needed
o
ob ain
he
la e
s a emen
a e
included
in
an
Appendix
.
Le
R
be a
s ably
ini e
simple
egula ing
.
Then
(Ko(R),
[R])
is
a
pa ially
o de ed
abelian
g oup
wi h
o de -uni ,
see
[G1,
P oposi ion
SIMPLE
REGULAR
RINGS

37
1
15
.3]
.
Le
<D
:
Ko(R)

>
A (S(Ko(R),
[R]))
be
he
na u al
map,
see
[G4,
Chap e
7]
.
Fo
any compac
con ex
se S,
deno e
he
s ic o -
de ing
on
A (S)
by
«,
ha
is,
«
g
i
and
only
i
(x)
<
g(x)
o
all
x E
S
.
By
[B3,
Theo em
3
.1
.4]
and
[G4,
Theo em
4
.12],
R
is
s ic ly
unpe o a ed
i
and
only
i
D
([A])
«
<D
([D])
implies
A
~
D
o
A,
D
E
P
.
We
can
conside
he
ollowing
weake
condi ion
:
Fo
any
D
E
L(RR),
he e
exis s
K
>_ 1
such
ha ,
o
A
E
L(RR),
i
K,D([A])
«
4>([D])
hen
A
-<
D
.
We
will
see
in
Sec ion
3
ha
R
sa is ies
his
condi ion
i
and
only
i
R
sa is ies
he
ollowing
p ope y
:
The
doubling
condi ion
(DD)
:
Fo
any
D
E
L(RR),
he e
exis s
K
>_
1
such
ha , o
A
E
L(R
R
),
i
A
<_
D
and
K(D([A])
«
1>([D])
hen
2A~
D
.
Simila ly,
he ollowing
wo
condi ions
a e
equi alen
o
a
di ec ly
ini e
simple
egula ing
R
:
Weak
compa abili y
(O'Mea a)
:
Fo
any
D
E
L(RR)
he e
exis s
n
>_
1
such
ha ,
o
A
E
L(RR),
i
nA
<
R
hen
A
;
:5
D
.
(dd)
:
Fo
any
D
E
L(RR)
he e
exis s
n
_> 1
such
ha ,
o
A
E
L(RR),
i A<DandnA
;
:5
R hen2A<D
.
Obse e,
in pa icula ,
ha
he
doubling
condi ion
implies
weak
com-
pa abili y
in
any
s ably
ini e
simple
egula ing
.
We
close
he
pape
by
s udying
he
e ec
o
imposing
compa abili y
wi h
espec
o
he
(app oxima ely)
almos
isomo phism
ela ion
on a
s ably
ini e
simple
egula
ing
.
1
.
S able
ange
o
simple
egula
ings
.
In
his
Sec ion
we
s udy
he
s able
ange
o simple
egula
ings,
ob-
aining
a
es ic ion
on
he
beha iou
o
he
s able
ange
on
he
amily
o
ini ely
gene a ed
p ojec i e
modules
.
I
is
easy
o
show
by
using
ou
esul s
ha
i
he e
exis s a
simple
egula ing
o
s able
ange
2,
hen
he e
a e
co ne
ings
o
R
wi h
a bi a y
ini e
s able
ange
n>
1
.
So,
he
si ua ion
o
simple
egula
ings
di e s
e y
much
om
he
si ua ion
o
a bi a y
egula
ings,
see
[MM
;
GMM]
.
We
will
apply
he
esul s
on
s able
ange
o
gi e
he
new
p oo
o
O'Mea a's
Theo em
[O,
Theo em
l]
.
Recall
ha
a ing
R
sa is ies
he
n-s able
ange
condi ion
( o
a
gi en
posi i e in ege
n)
i
whene e a
l ,
. . . .
a
n+
l
E
R
wi h
a
l
R+
+an+iR
=
R,
he e
exis
elemen s
bl,
. . . ,
b n
E
R
such
ha
(al
+
an+ibi)R
+
-
.
.
+
(a
n
+
an+ibn)R
=
R
.
37
2

P
.
ARA,
K
.
R
.
GOODEARL
I
n
is
he
leas
posi i e
in ege
such
ha
R
sa is ies
he
n-s able
ange
condi ion,
hen
R
is
said
o
ha e
s able
ange
n,
and
we
w i e
s (R)
=
n
.
I
is
well-known
ha
a
egula
ing
has
s able
ange
one
i
and
only
i
i is
uni - egula
[G1, P oposi ion
4
.12]
.
The
eade
is
e e ed
o
[V]
o
he
basic
p ope ies
o
he
s able
ange
and
o
[W
;MM
;M]
o
he
connec ions
be ween
cancella ion
p ope ies
o
modules and
he
s able
ange
o
he
endomo phism
ings
.
Lemma
1
.1
.
Le
R
be a
non-a inian
simple
egula
ing
.
Then
o
each
P
E
Po
and
o
all
k
>
1,
he e
exis s
Q
E
Po
such
ha
kQ
;j, :
P
.
P oo
.
Clea ly
we
can
assume
ha
P=
eR
o
some
nonze o
idempo-
en
e
E
R
.
Since
R
is
no
a inian,
eR
=
e1R
®
e2R
o
some
nonze o
idempo en s
el,
e2
.
Since
R
is
simple,
e1R
;
:5
n(e2R)
o
some
n
and
so e
1
R=A
l
®
. . .
®
A
n wi h
A
Z:5
e
2
R
by
[G1,
Co olla y
2
.9]
.
Then
eR
=
Al
®
(A2
®
.
.
.
®A
n
®
e2R) and
clea ly
2A1
;Z
eR
.
Now,
he
esul
ollows
by
induc ion
.
The
ollowing
esul
is
pa e ned
a e
an
a gumen
o
Rie el
[R]
.
Theo em
1
.2
.
Le
R
be
a
simple
egula ing
such
ha
s (eRe)
<
k
o
some
k
>
1
and
all
idempo en s
e
E
R
.
Then
R
is
uni - egula
.
P oo
.
I
R
is
a inian,
he
esul
is
well-known
.
So
we
can
es ic
ou sel es
o
he
non-a inian
case
.
Assume
ha Pl
®
nR=
P2
®
nR
o
some
ini ely
gene a ed
p ojec i e
modules
P
l
and P2
.
I
Pl
=
P2
=
0,
we
a e
done,
so
we
can assume
ha
Pl
:7~
0
.
By
Lemma
1 .1,
Pl
-
kQ®
U
o
some
Q
E
Po,
and
clea ly
we
can assume
ha
Q
=eR
o
some
idempo en
e
E
R
.
Now,
we
ha e
kQ®U®nR=P
2
®nR
.
Since
R
is
simple
we
ha e
R®
V
=
sQ
o
some
s
>_ 1
and
since
kQ
U®nR®nV=P2®nRE)nVweha ekQ®U®nsQ=P2®nsQ
.
By
[W, Theo em
1
.2],
kQ
®U=P
2
and
so
P
l
-
P2
.
By
[G1,
Theo em 4
.5],
i
ollows
ha
R
is
uni - egula
.
We
need
he
essen ially-known
ac
ha
he
ini eness o
he
s able
ange
is
Mo i a-in a ian
.
We
include
a
p oo
o his
esul ,
which
is
a
s aigh o wa d
applica ion
o
he
echniques
in
[W]
.
Lemma
1
.3
.
Le
P
and
Q
be
ini ely
gene a ed
p ojec i e
modules
o e
a
ing
R
.
Assume
ha he e
exis s
k
>_ 1
such
ha
Q®U=
kP
and
he e
exis s
i
>
1
such
ha
P®T
--
iQ,
ha
is,
P
and
Q
gene a e
SIMPLE
REGULAR
RINGS

37
3
he
same
ca ego ies
o
modules
.
Then
s (EndR(P))
<
s (EndR(Q))
<
oo
.
P oo
..
We
will
ollow
he
p oo
in
[W,
Theo em
1
.11]
.
Assume
ha
s (EndR(Q))
=
<
oo
.
Se
m
=
(
+
i
-
1)k
.
Le
N=Pl®K=Pi®
. .
.®P
;

®L
.
whe e
Pl
=P
j
'
=P
o
all
j
.
Adding
T
o
his
ela ion
we
ob ain
M
:=N®T=Qim
..
.E)Qj®K=Qi®
.
.
.eQí+T-i®V®L
whe e
Q
p
--
Qe
-
Q
o
all
p,
q
.
Applying
[W,
Theo em
1
.11],
we
ge
a
submodule
B
o
M
such
ha
M=B®K=B®C®L
whe e
C
C
Qi
®
. . .
®
Qi +
-
1
®V
.
Now
we
ha e
N=
(BnN)®K=[(B®C)nN]®L
[(B®C)nK]®L=7 ((BE)
C)nK)®L
.
i
and
only
i
and
also
(B
(D
C) n
N
=
(B
n
N)
e)
[(B
®
C)
n
K]
.
Le
7
be
he p ojec ion
o
N
on o
Pi
®
. . .
m
Pn,
along
L
.
Then
I
ollows
ha
N
=
(B
nN)
®
7 ((B
®
C)
nK)
®
L
.
By
[W,
Theo em
1 .6],
s (EndR(P))
<
m
< oo
.
Rema k
1 .4
.
I
s (EndR(Q))
=
hen
by
he
abo e
p oo
we
ob ain
he
ollowing
bound
:
s (EndR(P))
<
(
+
i
-
1)
k
.
In
pa icula ,
i
e
is
an
idempo en
o
R
and
R
;
:5
n(eR) hen
s (R)
<
s (eRe)
+
n
-
1
.
This
is
exac ly
he
same
bound
ob ained
by
Blackada
o
C*-algeb as,
see
[B1,
Lemma
A6], [B2,
p
.33]
.
Ou
ollowing
esul
is
an
immedia e
consequence
o
Theo em
1 .2
and
Lemma
1
.3
.
Theo em
1
.5
.
Le
R
be
a
simple
egula ing
.
Then
one
o he
ol-
lowing
possibili ies
occu s
:
(1)
R
is
uni - egula
.
(2)
s (EndR(P))
=
oo
o
e e y
P
E
Po
.
(3)
s (EndR(P))
is
ini e
o
e e y
P
E
P
and
he se
{s (EndR(P))
P
E
P}
is
no
bounded
.

37
4

P
.
ARA,
K
.
R
.
GOODEARL
11
Rema k
1
.6
.
Le
R
be
any
simple
ing
which
is
no
s ably
ini e
.
By
simplici y,
we
hen
ha e
2nR
;
:S
nR
o
some
n
>
1
.
By
[W,
Theo em
1
.2],
s (EndR(nR))
=
oo
.
By
Lemma
1
.3,
his
implies
ha
s (EndR(P))
=
oo
o
e e y
nonze o
ini ely
gene a ed
p ojec i e
R-
module
.
Theo em
1
.7
.
Le
R
be
a
simple
egula
ing
.
Assume
ha
whene e
B,
C
l ,
C
2
E
L(R
R
)
wi h
R®B
;
:-~,
R®C
Z
o
i
=
1,
2,
hen
B
;
:S
C
l
®
C2
.
Then
R
is
uni - egula
.
P oo
.
Since
i is
easily
seen ha he hypo hesis
is
inhe i ed
by
he
co ne
ings
o
R,
i
su ices
by
Theo em
1
.2
o
show
ha
s (R)
<
2
.
Le
a,
b,
c
E
R
such
ha
aR
+
bR+
cR
=
R
.
The e
is
an
idempo en
e
E
R
such
ha
cR
=
eR
®
[cR
n
(aR
+
bR)]
.
No e
ha
e
=
c
o
some
E
R,
and
R
=
(aR
+
bR)
®
eR
.
NowRR
=
Cl
®
Dl
=
C2
®
D2
whe e
Cl
=
.annR(a)
and
C2
=
.annR(b)
.
We
obse e
ha
le
mul iplica ion
by
a
induces
an
isomo -
phism
o
Dl
on o
aR,
and
simila ly
D2
-
bR
.
By
using
his
we
see
ha
R®
eR
;Z
R®C
Z
o
i
=
1,
2
.
By
ou
hypo hesis,
we
deduce
ha
eR
;
:S
Cl
®
C2,
so
eR
=E
l
®
E2
wi h
each
E
i
;
:S
C
z
.
De ine
xE
R
such
ha
xDl
=
0
and
xR
=
xCl
=
El
;
no e
ha
x
=
ex
.
Since
aCl
=
0
and
aDl
=
aR,
we
ge
(a
+
x)R
=
aR
+
El
.
Likewise,
he e
exis s
y E
eR
such
ha
(b
+
y)R
=
bR
+
E2
.
Then
we
ha e
(a+c x)R+(b+c y)R
=
(a+x)R+(b+y)R
=
aR+bR+E
l
+E
2
=
aR
+
bR
+
eR
=R
.
This
shows
ha
s (R)
G
2
.
a
We
now
in oduce
a
key
concep
o
his
pape ,
namely
he
almos
isomo phism
ela ion
.
De ini ions
.
Le
R
be
a
egula
ing
and
le
A,
B
E
P
.
We
say
ha
A
is
almos
subisomo phic
o
B,
w i en
A
:5,a
B,
i
o
all
nonze o
C
E
L(RR)
we
ha e
A
<
B®C
.
We
say
ha
A
is
almos
isomo phic
o
B,
w i en
A-
a
B,
i
A
;Za
B
and
B
;
:~a
A
.
We
say
ha
A
is
app oxima ely
almos
subisomo phic
o
B,
w i en
A
;~,aa
B,
i
o
all
nonze o
C
E
L(RR)
he e
exis s
n>
1
such
ha
nA
;~-z
n(B
(D
C)
.
We
say
ha
A
is
app oxima ely
almos
isomo phic
o
B,
w i en
A
-
aa
B,
i
A
;
:Saa
B
and
B
;~Saa
A
.
The
abo e
no ions a e
specially
use ul
whe i
R
is
a
simple
egula
ing
which
is
no
a inian
.
Since
a inian
simple
egula
ings
a e
i ial o
ou
heo y,
we
will
equen ly
assume
ha
ou
simple
egula
ings
a e
no
a inian
.
Lemma
1
.8
.
Le
R
be
a
non-a inian
simple
egula
ing
and
le
SIMPLE
REGULAR
RINGS

37
5
A,
B
E
P
.
Then
:
(a)
A
;i-C
a
B
( esp
.
A
;
:Saa
B)
i
and
only
i
A
;
:S
a
B®C
( esp
.
A
;
:Saa
B
®
C)
o
all
nonze o
C
E
L(RR)
.
(b)
The
ela ion
;!Sa
( esp
.
;
:
-
Z~aa)
is
ansi i e
.
P oo
:
(a)
Assume
ha
A
Z_
S
a
B®C
o
all
nonze o
C
E
L(RR)
.
Fix
a
nonze o
D
E
L(RR)
.
Then
since
R
is
no
a inian
D
=
Dl
®
D
2
o
nonze o
D
I ,
D2
E
L(RR)
.
So
A
;
:s
(B
®
Dl)
®
D
2
=B®
D
.
Thus
A
;5_
5
a
B
.
The
o he
implica ion
is
i ial
.
The
p oo
o
he
ela ion
;!,z
aa
is
analogous
.
(b)
is
p o ed
in
a
simila
way
.
Theo em
1
.9
.
Le
R
be
a
di ec ly
ini e
simple
egula ing
.

Then
he
ollowing condi ions
a e
equi alen
:
(a)
Fo
all
A,B,
C
E P,
A®B
;i5a
A®C
implies
B
;5-a
C
.
(b)
Fo
all
B,
C
E
L(R
R
),
R®B
;
:5
R
®C
implies
B
;!-
:5
a
C
.
(c)
R
is
uni - egula
.
P oo
.
(a)
=>
(b)
:
This
is
clea
.
(b)

(c)
:
Apply
Theo em
1
.7
.
(c)

(a)
:
This
is
immedia e
om
he
cancella ion
p ope y
o
uni -
egula
ings
.
Lemma
1
.10
.
Le
A,
B,
C
be
ini ely
gene a ed
p ojec i e
igh
mod-
ules
o e
a
egula
ing,
such
ha
A®C
=
B®C
.
Le
n
E
N
.
Then
he e
exis
decomposi ions
A
=
A'
®
A"
and
B
=
B'
®B"
and
C
=
C'
®
C"
such
ha
A'
=
B' and
A"
®
C"
=
B"®
C", and
also
n(A"
®
B")
;z5
C
.
P oo
.
By
[G3,
Lemma
2
.2],
he e a e
decomposi ions
A
=
Al,
®
A12
and
B=
B,,
®
B12
and
C=C11
®
C12
such
ha
Al,
=
B,,
and
A12
®
C12
=B
12
®
C12, while
also
A12
=
Cli
.
Applying
his
lemma
epea edly,
we
ob ain
decomposi ions
Ai_1,2
=
A¡,
®A
i
2
and
Bi_1,2
=
Bil
®B
i
and
C
i
_
1
,2
=
Cil
®
Ci2,
o
i
=
2,3,
. .
.,
such
ha
A
¡ , l--
-
B
¡
,
and
A
i
2
®
Ci2
=
Bi
®
Ci2,
while
also
A
i
2
-
Cil
.
Now
se
A1
=
Al,
®A21®
.
.
.
®
A,
and
A2
=
A
n
2,
and
de ine
B,,
B2,
Cl,
C2
simila ly
.
Then
A=
A1®
A2
and
B
=
B
l
®
B2
and
C
=
Cl
®
C2,
wi h
A
1
-B
l
and
A2®C
2
=B2®C
2
.
Since
A2
=A
n
2
<A,-
1,2
<
...
<
A
1
2,
we
also
ha e
nA2
;
:5
A
1
2
®
A22
®
.
.
.
®A
n
2
=
Ci1®
C21®
.
.
.
®
Cnl
=
C
l
.
Finally,
we
Apply
he
abo e
p ocedu e
o
he
isomo phism
B2
®
C2
A2
®
C
2
.
We
ob ain
decomposi ions
B2
=
B3
®
B4
and
A2
=
A3
®
A4
and
C2
=
C3
®
C4
such
ha
B3
=
A3 and
B4
®
C4
=
A4
®
C4,
while
nB
4
;Z
C
3
.
Se
A'
=A
1
®A
3
and
A"
=
A
4
,
and
de ine
B',
B",
C',
C"
37
6

P
.
ARA,
K
.
R
.
GOODEARL
simila ly
.
Then
A=
A'
®
A"
and
B
=
B'
®
B"
and
C=
C®
C",
wi h
A'
-
B' and
A"
®
C"=B"
®
C",
while
also
n(A"
®
B")
<
nA2
®
nB4
,<

Ci®C3=C'
.
E
Now
we
can
gi e
a
di e en
p oo
o
[O,
Theo em
1]
.
Theo em
1
.11
.
(O'Mea a)
Le
R
be a
di ec ly
ini e
simple
egula
ing
sa is ying
weak
compa abili y
.
Then
R
is
uni - egula
.
P oo
. .
We
show
ha
R
sa is ies
condi ion
(b) in
Theo em
1
.9
.
Le
B,
C
E
L(R
R
)
wi h
R
®
B
<R®C
.
Gi en
0
:7~
D
E
L(RR)
he e
exis s,
by
weak
compa abili y,
a
posi i e
in ege
n
such
ha
nT
<R
implies
T<D o anyTEL(RR)
.
By
Lemma
1
.10
he e
exis s
a
decomposi ion
B=
B'
®
B"
such
ha
B'
;
:5
C
and
nB"
;5-
R
.
Consequen ly
B"
<
D
and
B
;
:~
C®D
.
I
ollows
ha
B
~a
C
and
hus
R
is
uni - egula
by
Theo em
1
.9
.
2
.
The
almos
isomo phism
ela ion
.
Le
R
be
a
non-a inian
s ably
ini e
simple
egula
ing
.
By
[B3,
Theo em
3
.1 .4],
he
ela ion
~aa
is
cancella i e,
Le
.
A®B
~aa
A
®
C
implies
B
<aa
C
.
So
he
app oxima ely
almos
isomo phism
classes
o
ini ely
gene a ed
p ojec i e
modules
o m
a
cancella i e
abelian
semi-
g oup
S
(since
i is
easy
o
show
ha
di ec
sum
gi es,
a
well-de ined
ope a ion)
.
Deno e
by
[A]a,
he
class o
A
in
S
.
We
de ine a pa ial
o de
on
S
by
[A]a
<
[B]o
i
and
only
i
A
~aa
B
.
I is
easy o
show
ha
his
ela ion
is
well-de ined
and
ansla ion-in a ian ,
so
ha
S
becomes
a
pa ially
o de ed
abelian
semig oup
.
Also,
he
ela ion
<_ is
cancella i e,
Le
.
x
+
y
<
z
+
y
implies
x
<_
z o
x, y,
z
E
S,
again
by
[B3,
Theo em
3
.1
.4]
.
Le
K
.
'(R)
be
he
abelian
g oup
ob ained
by
adjoining
in e ses
o mally
o
S
.
Because
o
he
cancella i n
p ope y
o
_<,
he
ela ion
x
-
y
<
z
-
i
x
+
<_
z
+
y
o
x, y,
z,
E
S
becomes
a pa ial
o de
in
Kó
(R)
.
I
is
easy o
show
ha his
pa ial
o de
is
ansla ion-in a ian
and
so
Kó
(R)
becomes
a
pa ially
o de ed
abelian
g oup,
in
which
we
ix
he o de -uni
[R]a
.
P oposi ion
2
.1
.
Le
R
be
a
non-a inian
s ably
ini e
simple
egula
ing
.
Le
~P
:
KO(R)
-j
A (S(Ko(R),
[R]))
be he
na u al
map
.
Assúme
ha
oP(Ko(R))
is
endowed
wi h
he pa ial
o de
<
g
i
(x)
<
g(x)
o
all
xES(Ko(R),
[R])
.
Then
4)(Ko(R))
=
Ko
(R)
as
pa ially
o de ed
abelian
g oups
wi h
o de -uni
.
P oo
..
We
ha e
a
su jec i e posi i e
homomo phism
a
:
KO(R)
-~
Ká
(R)
gi en
by
c¿
([A]
-
[B])'=
[A]
a
-
[B]a
.
We
will
show
ha
Ke (a)
_
SIMPLE
REGULAR
RINGS

377
Ke (-D)
.
I
is
clea
ha
-D([A]
-
[B])
=
0 whene e
A
=o,a
B, bu
his
happens
exac ly
when
[A]
Q
=
[B]
o
, .
Con e sely
assume
ha
4>([A]
-
[B])
=
0
.
Le
0
:,~
C
E
L(RR)
.
Then
4)([A
®
C]
-
[B])
»
0
and
so,
by
[G4,
Theo em
4
.12]
he e
exis s
m
>_
1
such
ha
m([A
®
C]
-
[B])
>
0
.
By
using
[B3,
Theo em
3
.1 .4]
we
ha e
ha
he e
exis s
n
_> 1
such
ha
nmB
z5
nm(A
®
C)
.
Consequen ly
B
:Saa
A
.
Analogously
A
<aa
B
and
so
[A]
a
-
[B]a,
=
0
.
I
ollows
ha
we
ha e
a
g oup
isomo phism
-y
:
Ko
(R)
-
<D(Ko(R))
gi en
by
y([A]
a
-
[B]a)
=
<D([A]
-
[B])
.
Since
R
is
no
a inian
y
is
posi i e
.
Con e sely,
i
-b([A]
-
[B])
>_
0
hen
by
he
same
a gumen
as
be o e
we
ob ain ha
B
~S
.
A
and
consequen ly
[A],,
-
[B]o,
>
0
.
Hence o h,
we
will
iden i y
Ko
(R)
wi h
<D(Ko(R))
.
The
p oo
o
he
ollowing
lemma
is
s aigh o wa d
.
Lemma
2
.2
.
Le
R
be a s ably
ini e
simple
egula ing
.
(a)
Le
S
=
lim
M,,,(R)
.
Then
Ko(S)
=
Ko(R)
®Q
.
(b)
Le
S

,
=
lim
M~

k
(R),
o
a
ixed
m
>
1
.

Then
Ko(S
m
)
_
Ko(R)
®
D,

whe e
D

,
=
{a/m
k
1
a
E
Z,
k
>
l}
.
By
[B3,
Theo em
3
.1 .4],
he
ings
S
=
lim
M

(R)
and
all
S

z
=
lim
M

,k
(R)
a e
unpe o a ed
uni - egula
simple
ings
p o ided
R
is
a
s ably
ini e
simple
egula ing
.
I
ollows
ha
Ko(S)
and
all
Ko(S
m
)
a e
simple
dimension
g oups
.
The e o e
we
can use
[G4,
Theo em
14
.14]
and
Lemma
2
.2
o
s udy
he
ques ion o
when
oD(K
o
(R)+)
is
dense
in
A (S(KO(R),
[R]))+,
leading
o
he ollowing
lemma
.
Recall
ha
o
any
pa ially
o de ed
abelian
g oup
G
and any
subg oup
H
o
Q,
he enso
p oduc
G
®
H
is
a
pa ially
o de ed
abelian
g oup
wi h
posi i e
cone
(G
®
H)+
=
{x
®
y
1
xE
G+,
yE
H+}
.
Lemma
2
.3
.
Le
H
=Q
( esp
.
H
=
D

,
.),
endowed
wi h
he
usual
o de
.
Le (G,
u) be a pa ially
o de ed
simple
abelian
g oup
wi h
o de -
uni ,
and
assume
ha
G
®
H
is
a
simple
dimension
g oup
.

Le
d>
G
--->
A (S(G,
u)) be he
na u al
map
.

Then
he
ollowing p ope ies
a e
equi alen
:
(a)
D(G+)
is
dense
in
A (S(G,u))+
.
(b)
Fo
each
0
,-A
x E
G+,
o
each
n
>_ 1
( esp
.
o
each
n
=
mk,
wi h
k
>
0),
and
o
each
e
>
0
he e
exis
y,,
y2
E
G+
such
ha
n<D(y2)
«
-D(x)
K
wD(yl)
nob(y1)
-
d>(x)
<G
e
,
D(x)
-
n-D(y2)
K
6
38
4

P
.
ARA,
K
.
R
.
GOODEARL
1
.

Mo eo e ,
gi en
ini ely
gene a ed
p ojec i e
modules
A,
B,
ei he
nA~
nB
o
some
n>
1,
mB
~
mA,
o
some
m
>
1,
o
A
-aa
B
P oo
:
Since
M,, (R) has
a
unique
ank
unc ion
he
abo e
p oo
applies
e
ini ely
gene a ed
p ojec i e
modules
.
P oposi ion
4
.3
.
Le
R
be a
di ec ly
ini e
simple
egula
ing
sa is-
ying
a-compa abili y
.
Then
R
is
uni - egula
.
P oo
:
T
is
clea
ha
R
sa is ies
2-compa abili y
.
By
[O,
Co olla y
2],
R
is
uni - egula
.
By
a
sligh
modi ica ion
o
he
p oo
in
[G1,
P oposi ion
8
.2],
we
ob ain
he
ollowing
esul
.
P oposi ion
4
.4
.
Le
R
be a
di ec ly
ini e
simple
egula
ing
.
R
sa is ies
he
a-compa abili y
condi ion,
hen so
does
M,,
(R) o
all
n
>
1
.
P oo
:
We
can
assume
ha
R
is
no
a inian
.
We
will
p e e
ha
o
ini ely
gene a ed
p ojec i e
modules
A,
B,
ei-
he
A
<a
B
o
B
<a
A
.
By
induc ion,
assume
he
esul
is
ue
o
p ope
submodules
o (n
-
1)R,
and
le
A,
B
wi h
A,
B
<
nR
.
W i e
A=A
l
®A2,
B=B
l
®B2
wi h
Al,
A2,
B,,
B2
-<
(n-1)R
.
Now
ei he
Al
;
:S
a
Bl
o
Bl
ca
A
l
,
and
ei he
A2
:!Sa
B2
o
B2
ca
A2
.
We
need only
conside
he
case
whe e Al
;~a,
B
l
and
B2
ca
A
2
.
Le
0
:y~
C
E
L(RR)
be
such ha
B
l
®C
~
(n
-
1)R and A2
®
C
~
(n
-
1)R
.
Then
A
l
;
:5
B
i
®C
and
B
2
;
:5
A
2
®
C,
so
B
l
®C
=
B'
®
Bi
,
A
2
m
C
=
AZ
®A2
wi h
B'
-A
l
and
A'
2
-
B2
.
So
ei he
B'
~a
AZ
o
A"
<
aB','
.
Assume
ha
Bi
<a
A2
.
Then
B
l
®
B
2
(D
C=
B'
®
Bi®
A2
;
:Ea
B'
(B
A
2
l'
e
Al
A
l
®
A2
®
C, and
so
Bl
®
B2
Ca
A
l
®
A2,
since
R
is
uni - egula
by
P oposi ion
4
.3
.
P oposi ion
4
.5
.
Le
R
be
a
simple
egula
ing wi h
a
unique
ank
unc ion
N
.
Then
he ollowing a e
equi alen
:
(a)
R
sa is ies
he
a-compa abili y
axiom
.
(b)
R
is
s ic ly
unpe o a ed
.
(c)
Fo
x,
yE
R,
n(xR)
-<
n(yR)
implies
xR
--<
yR
P oo
:
We
can
assume
ha
R
is
no
a inian
.
(a)
==>
(b)
:
Assume
ha
nA
--<
nB
.
I
B
Z-5
a
A
hen
w i e
nB
=
T
l
®T2
wi h
T
l
-
nA
and
T
2
:7¿
0
.
Choose
0
:7~
T
wi h
nT
-<
T2
.
Then
B
;
:5
A®T
so
nB
;
:~
nA®
nT
-<
T
I
®T2
=
nB,
con adic ion
.
So
A
Ga
B
.
Applying

SIMPLE
REGULAR
RINGS

38
5
he
same
a gumen
o
A
®
T,
we
see ha
A®
T
;
:S
o
,
B
.
Consequen ly,
A®T

B®T
and
so,
A
-<
B
because
R
is
uni - egula
.
(b)

(c)
:
Ob ious
.
(c)

(a)
:
Since
R
has
a
unique
ank
unc ion,
we
see
om
P oposi ion
4
.1
ha
ei he
xR
~aa
yR
o
yR
;z
:
a
,
o
,
xR
.
I
xR
;
:Saa
yR,
hen
o
0
:~
C
E
L(RR)
we
ha e
n(xR)
-<
n(yR
®
C),
and
so
xR
-<
yR®
C
p o ided
ha
yR®
C
;i5
R
.
We
can
always
assume
his
excep
in
he
case
whe e
yR
=R
.
Bu
i
yR
=
R
hen
xR
;S
a
,
R=
yR
ob iously
.
Consequen ly,
R
sa is ies
a-compa abili y
.
Appendix
.
We
p o e
he
ollowing
esul
:
P oposi ion
A1
.
Le
R
be a
s ic ly
unpe o a ed
non-a inian
sim-
ple
uni - egula
ing
.
Le
-D
:
Ko(R)
-
j
A (S(K
O
(R),
[R]))
be
he
na -
u al
map
.
Then
D(Ko(R)+)
is
dense
in
A (S(Ko(R),
[R]))+
.
To
p o e
P oposi ion
A1,
i
clea ly
su ices
o
p o e
a
co esponding
esul
o
pa ially
o de ed
abelian
g oups
(Theo em
A3)
.
No e
ha
i
G
is
a
pa ially
o de ed
abelian
g oup
hen
i s
o sion
subg oup
T
is
a
con ex
subg oup,
and
so
G/T
is
a
pa ially
o de ed
abelian
g oup
wi h
espec
o
he
induced
o de ing
.
A
special
case
o
a
esul o
Ellio
[E,
Theo em
4
.5]
says ha
i
G
is
a
s ic ly
unpe o a ed
in e pola ion
g oup,
hen
GIT
is
an
unpe o a ed
in e pola ion
g oup
.
Since
he
p oo
o
his
case
is
much
easie
han
he
p oo
o
[E,
Theo em
4
.5],
we
gi e
he
de ails
.
P oposi ion
A2
.
(Ellio )
Le
G
be
a
di ec ed
s ic ly
unpe o a ed
in e pola ion
g oup,
and
le
T
be
i s
o sion
subg oup
.
Then
G/T
is
a
dimension
g oup
.
P oo
.
I
is
clea
ha
Since
G
is
di ec ed
;
so
is
G/T
.
Fi s
conside
x
E
G
and
n
E
N
such
ha
n(x
+
T)
>_
0
.
I
x
E
T,
hen
x+T
=
0,
and
so
we
may
assume
ha
x
1
T
.
Now
nx+T
=
y+T
o
some
y E
G+,
and y
>
0
because
x q
T
.
Then k(nx
-
y)
=
0
o
some
k
E
N,
whence
knx
=
ky
>
0
.
Since
G
is
s ic ly
unpe o a ed,
x
>
0,
and
hence
x
+T>
0
.
Thus
G/T
is
unpe o a ed
.
Now
conside
xl,
x2,
YI,
y2
E
G
such ha
xi
+T
<
y
j
+T
o
all
i,
j
.
I
x
+T
=
ys
+
T
o
some
, s,
hen
xi
+
T<
x
+
T
<
y
3
-}-
T
o
all
i,
j
.
Hence,
we
may
assume
ha
xi
+
T<
yj
+T
o
all
i,
j
.
Consequen ly,
he e a e
nonze o
elemen s
wij
E
G+
such
ha
xi
+
wij
+
T=
yj
+T
.
The e
is
some
k
E
N
such
ha
k(xi
+
wij
-
yj)
=
0
o
all
i,
j,
whence
386

P
.
ARA,
K
.
R
.
GOODEARL
kxi
<
ky
j
and
so xi
<
y
j
o
all
i,
j
;
by
s ic
unpe o a ion
.
Thus
he e
exis s
z
E
G
such
ha
xi
<
z
_<
yj
o
all
i,
j,
and
hence
xi+T
<
z+T
<_
y
j
+T
o
all
i,
j
.
The e o e
G/T
is
an
in e pola ion
g oup
.
Theo em
A3
.
Le
(G,
u)
be a
s ic ly
unpe o a ed
simple
in e po-
la ion
g oup
wi h
o de -uni ,
such
ha
G+
con ains
no a oms
.
Le
-P
:
G
,
A (S)
be he
na u al
map,
whe e
S=
S(G,
u)
.
Then
1)(G+)
is
dense
in
A (S)+
.
P oo
.
Le
T
be
he
o sion
subg oup
o
G, and
no e
ha
G/T
is
simple
and
ha
he
elemen
u'
:=
u
+T
is
an
o de -uni
in
GIT
.
By
P oposi ion
A2,
G/T
is
a
dimension
g oup
.
Suppose
ha
(GIT)+
con ains
an
a om,
say
x+T
whe ex
E
G+
.
Since
x
canno
be
an
a om
in
G+,
he e
exis s
y
E
G
such
ha
0
<
y
<
x
.
Bu
hen
0
+
T<
y
+
T
<
x
+T
(because
T
n
G+
=
{0}),
con adic ing
oú
assump ion
abou
x+T
.
The e o e
(GIT)+
con ains
no a oms
.
Le
7
:
G
->
G/T
be
he
quo ien
map,
and
se S'
=
S(G/T,
u')
.
The
induced
map
7 *
:
S'
~
S
is
an
a ine
homeomo phism,
and
hence
he
induced
map
7 **
:
A (S) ->
A (S')
is
an
isomo phism
o
o de ed
Banach
spaces
.
The e
is
a
commu a i e
diag am
as
ollows,
whe e
V
is
he
na u al
map
.
Acknowledge nen
.
G
A (S)
G/T
A (S')
Since
(GIT)+
=
7 (G+),
i
su ices
o
p o e
ha
V
((G/T)
+
) is
dense
in
A (S')+
.
Thus
he e
is
no
loss
o
gene ali y
in
assuming
ha
G
is
a
simple
dimension
g oup,
wi h
no
a oms
in
G+
.
Since
G+
has
no
a oms,
G
is
no
cyclic
.
The e o e,
by [G4,
Theo em
14
.14],
4)(G+)
is
dense
in
A (S)+
.
a
I
is
a
pleasu e
o
hank
J
.
Meneas¡
o
his
help ul
commen s
.
No e
added
in
p oo
.
E
.
Pa do
has p o ed
ha
any
non-a inian
s ably
ini e
simple
egula
ing
sa is ies
condi ion (D)
.
SIMPLE
REGULAR
RINGS

38
7
Re e en es
[B1]
B
.
BLACKADAR,A
s able
cancella ion
heo em
o
simple
C*-algeb as,
P oc
.
London
Ma h
.
Soc
.
47
(1983),
303-305
.
[B2]
B
.
BLACKADAR,
Compa ison
heo y
o
simple
C*-algeb as,
in
"Ope a o
Algeb as
and
Applica ion",
D
.E
.
E ans
and
M
.
Takesaki
(eds
.),
LMS
Lec u e
No es
Se ies
135,
Camb idge
Uni
.
P ess,
1988,
pp
.
21-54
.
[B3]
B
.
BLACKADAR,
Ra ional C*-algeb as
and
nons able K- heo y,
Rocky
Moun ain
J
.
Ma h
.
20
(1990),
285-316
.
[Bu]
C
.
BusQuÉ,
Di ec ly
ini e
aleph-nough -comple e
egula
ings
a e
uni - egula ,
in
P oceedings
o
he
Fi s
In e na ional
Mee ing
on Ring
Theo y,
G anada,
Spain
1986,
J
.L
.
Bueso,
P
.
Ja a
and
B
.
To ecillas
(eds
.),
Lec u e
No es
in
Ma h
.
1328,
Sp inge -Ve lag,
Be lin-New
Yo k,
1988,
pp
.
38-49
.
[E]
G
.
A
.
ELLIOTT,
Dimension
g oups
wi h
o sion,
In e na
.
J
.
Ma h
.
1
(1990),
361-380
.
[G1]
K
.
R
.
GOODEARL,
"Von
Neumann
egula
ings,"
Pi man,
Lon-
don,
1979
.
[G2]
K
.
R
.
GOODEARL,
Di ec ly
ini e
aleph-nough -con inuous
egu-
la
ings,
Paci ic
J
.
Ma h
.
100
(1982),
105-122
.
[G3]
K
.
R
.
GOODEARL,
Me ically
comple e
egula
ings,
T ans
.
Ame
.
Ma h
.
Soc
.
272
(1982),
275-310
.
[G4]
K
.
R
.
GOODEARL,
"Pa ially
o de ed
abelian
g oups
wi h
in e -
pola ion,"
Ma h
.
Su eys
and
Monog aphs
20,
Ame
.
Ma h
.
Soc
.,
P o iden e, 1986
.
[G5]
K
.
R
.
GOODEARL,
Unpublished
no es
on
simple
egula
ings,
1990
.
[GM]
K
.
R
.
GOODEARL
ANDP
.
MENAL,
S able
ange
one
o ings
wi h
many
uni s,
J
.
Pu e
Applied Algeb a
54
(1988),
261-287
.
[GMM]
K
.
R
.
GOODEARL,
P
.
MENAL
AND
J
.
MONCASI,
F ee
and
esidually
a inian
egula
ings,
J
.
Algeb a
( o
appea )
.
[M]
J
.
MONCASI,
Rang
es able
en
anells egula s,
Ph
.D
.
Thesis,
Uni-
e si a
Au ónoma
de
Ba celona,
1984
.
[MM]
P
.
MENAL
AND
J
.
MONCASI,
On
egula
ings
wi h
s able
ange
2,
J
.
Pu e
Applied
Algeb a
24
(1982),
25 40
.
[O]
K
.
C
.
O'MEARA,
Simple
egula
ings
sa is ying
weak
compa a-
bili y,
J
.
Algeb a
141
(1991),
162-186
.
388

P
.
ARA,
K
.
R
.
GOODEARL
[R]
M
.
RIEFFEL,
The
cancella ion
Theo em
o
p ojec i e
modules
o e
i a ional
o a ion
C*-algeb as,
P oc
.
London
Ma h
.
Soc
.
47
(1983),285-302
.
[V]
L
.
N
.
VASERSTEIN,
S able
ank
o
ings
and
dimensionali y
o
opological
spaces,
Func
.
Anal
.
Applic
.
5
(1971),
102-110
.
[W]
R
.
B
.
WARFIELD,
JR
.,
Cancella ion
o
modules and
g oups
and
s able
ange
o
endomo phism
ings,
Paci ic
J
.
Ma h
.
91
(1980),
457-485
.
Pe e
A a
:
Depa amen
de
Ma emá iques
Uni e si a
Au ónoma
de
Ba celona
08193
Bella e a
(Ba celona)
SPAIN
Rebu
el
28 de
Gene
de
1992
K
.R
.
Goodea l
:
Depa men
o
Ma hema ics
Uni e si y
o
U ah
Sal
Lake
Ci y
U ah
84112
U
.S
.A
.