Pub
.
?1a
.
UAB
Vol
.
29
Ns
2
.3
No
.
1984
PARTIALLYORDERED
GROTHENDIECK
GROUPS
K
.
R
.
Goodea l
1
Mo i a ion
o he
s udy
o
pa ially
o de ed
abelian
g oups
has
come om many
di e en
pa s
o ma hema ics,
o
ma hema ical
sys ems
wi h
compa ibleo de
and
addi i e
(o
linea )
s lk'c u es
a e
qui e
common
.
This
is
pa icula ly
e iden
in
unc ional
analysis,
whe e
spaces
o
a ious
kinds
o
eal- alued
unc ions
p o ide
impe us
o
in es iga ing
pa ially
o de ed
eal
ec o
spaces
.
In
he
pas
decade,
he
obse a ion
ha
a
G o hendieck
g oup
(such as
K
D
o
a
ing
o
algeb a)
o en
possesses
a
na u al
pa ially
o de edabelian
g oup
s uc u e
has led
o
new
di ec ions
o in es iga ion,
whose
goals
ha e been
o
de elop
s uc u e
heo ies
o
ce ain
ypes
o
pa ially
o de ed
abelian
g oups o
he
poin whe e
e ec i e
applica ion
o
a ious
G o hendieck
g oups
is
possible
.
Such
ecen
de elopmen s
in
he
a ea
o
pa ially
o de edabelian
g oups
a e he
subjec
o
his no e
.
We
p esen
a
ske ch
o
he
cons uc ion
o
G o hendieck
g oupsas
abelian
g oups
equipped
wi h
p e-o de ings
ha
a e
o en
pa ial
o de ings, oge he
wi h
b ie
ske ches
o
se e al
si ua ions
in
which
he
heo y
o
pa ially
o de edabelian
g oups
can
be applied,
ia
he
G o hendieck
g oups
KO
,
o
he
s udy
o
ce ain
ings
and
C*-algeb as
.
This
discussion
is
somewha cu so y,in
he
in e es o
a oiding
echnicali ies,
and o
easons
o
space
.
Fo
1)
This
exposi o ydiscussion
con ains
an
expanded
e sion
o
se e al
lec u es
gi en
a
he
Cen e
de
Rece ca
Ma emá ica,
Ins i u
d'Es udis
Ca alans,
Ba celona,
du ing
No embe
and
Decembe
1984,
and
he
au ho
wishes
o
hank
he CRM o
a anging
his
e y
pleasan
isi
.
An
ea lie
e sion
o his ma e ialappea ed
in
[13] in
connec ion
wi h
a
lec u e
gi en
a
he
In e na ional
Symposium
on
Algeb a
and I s
Applica ions
in
Delhi
in
Decembe
1981
.
he
sane
easons,
we do
no
discuss
he
ecen
applica ions
o
pa ially
o de ed
abelian
g oups o
opological
Ma ko
chains
[27,
3,
20, 21]
o o
posi i epolynomials
and
compac
g oup
ác ions
[22, 23,
25, 26]
.
A
G o hendieck
g oup
is
an
in a ian
a ached
o
a
collec ion
o
objec s, such as
he
ec o
bundles
on
a
gi en
opological
space, he
ini ely
gene a ed
p ojec i e
modules
o e
a
gi en
ing,
o
he
p ojec ion
ope a o s
associa ed
wi h
a
gi enC*-algeb a
.
The
cons uc ion
only equi es
a
collec ion
o
objec s
equipped
wi h
a
no ion o
isomo phism
and
wi h
some
means
o
combiningany
wo
objec s
om
he
collec ion
in o
a
hi d
objec
.
While
i is
adi ional
o
de elop
G o hendieck
g oups
using
sho exac
sequences
as
he
means
o
combining
objec s,
in
many
impo an
applica ions
his
educes
o
di ec sums
(o
di ec
p oduc s)
.
Since
he cons uc ion
p ocess
is
simple
using
di ec
sums,
we
shall
es ic
ou
discussion
o
ha
case
.
Thus
o
ou
basic
da a we ake
a se P
o
objec s
in
some
ca ego y,
such ha
P
has
a
ze o
objec and
e e y
ini e
se
o
objec s
in P
has
a
di ec sum
(cop oduc )
in P
.
The
easies
algeb aic
sys em
o
build
using
his
da a
is an
abelian
semig oup,
whose
elemen s
a e he
isomo phism
classes
o
objec s
in
P,
and
whose
ope a ion
is
addi ion
induced
om
he di ec sumope a ion
in
P
.
Howe e ,
he
semig oup
ob ained
in
his
manne
need
no
ha e
cancella ion,
and
so
canno
always
be
embedded
in an
abelian
g oup
.
This
p oblem
can
be
ci cum en ed
ei he by
educing
he
semig oup
modulo
a
sui able-cong uence
ela ion,
o ,
mo e
con enien ly,
by
using
an
equi alence
ela ion
on
P
sligh ly
coa se
han isomo phism,
as
ollows
.
7
8
Objec s
A,B
E P
a e
said o be
s ably
isomo phic
(in
P)
i 'and
only i
he e
is
an objec

C
E P
such ha
MC
=
BOC
.

S able
isomo phism
is an
equi alence
ela ion
on
P,
and
we
w i e
[A]
o
he
s able
isomo phismclass
o an
objec
A E
P
.
Le
G o (P)
deno e
he
collec ion
o
all
s able
isomo phism
classes
in
P
.
The
di ec
sum
ope a ion
in P
induces
an
addi ion
ope a ion
in
G o (P)
,
whe e
[A]+[B]
=
[A®B]

o
all
A,B
E P,
and
using
his
ope a ion,
G o (P)
+

becomes
an
abelian
semig oup
wi h
cancella ion
.

By
o mally
adjoining
addi i e
in e ses
o
G o (P)
,
we
ob ain
an
abelian
g oup
G o (P),
he
G o hendieck
g oup
o
P
.

(Reminde
:
i
sho exac
sequences
a e
a ailable
in
P,
he
G o hendieck
g oup
cons uc ed
om
P
using
sho exac
sequences
may
well
be
di e en
om
he
g oupcons uc ed
he e
.)
All
elemen s
o
G o (P)
ha e
he
o m
[A]-[B]
o
A,B
E
P,
and
elemen s
[A]-[B]
and
[C]-[D]
in
G o (P)
a e
equal
i
and
only
i
A®D
is
s ablyisomo phic o
B®C
.
Fo
example,
i P is
he
collec ion
o
all
eal
(complex)
ec o
bundles
on some
compac
Hausdo
space
X,
hen
G o (P)
is
he
eal
(complex)
K-g oup
K
0
(X)
.
Fo
ano he
example,
le
R
be
a
ing wi h
1,

and
le
P
be
he
collec ion
o
all
ini ely
gene a ed
p ojec i e
igh
R-modules
(i .e
.,
all
di ec summandso
ee
igh
R-modules
o ini e
ank)
.
In his
case,
G o (P)
is
he
algeb aic
K-g oup

K
0
(R)
.

Al e na i ely,

K
0
(R)

may
be
cons uc ed
by
aking
P
o
be
he
ca ego y
o
all
ec angula
ma ices
o e
R
.
Then
he
objec s
in
P
a e all
idempo en
squa e
ma ices
o e
R,
and he
di ec
sum
o
idempo en
ma ices
e
and
is
he
block
ma ix
0
)
.
In
case
R
is a
C*-algeb a,
he
se
o
idempo en ma ices
o e
R
may
be
educed
o
he se
o
sel -adjoin
idempo en
ma ices
(see [11,
Chap e
19])
.
In
o de
no
o
lose
ack
o
he
o iginal
semig oup
G o (P)
+
inside G o (P),
we
make
i
he
"posi i e
cone"
o
.
a
n
o de
.
ela ion
.
Namely,
o x,y
E
G o (P), we de ine
x
<_-
y i
and
only
i
y-x
lies
in
G o (P)
.
This ela ion
is a
p e-o de
(i .e
.,
a
e lexi e,
ansi i e
ela ion)
which
is
in a ian
unde ansla ion
( ha
is,
x
5-
y
implies
x+z
=
y+z)
.
The
combined
s uc u e
(G o (P),+,5)
is
called
a
p e-
o de ed
abelian
g oup
.
In
case
he
ela ion
<
is
a
pa ial
o de
(i
.e
.,
an
an i-symme ic
p e-o de ),
G o (P)
is
a
p
a ially
o de ed
abelian
ou
.
Va ious
ela i ely
mild
assump ions
on
P
will
o ce
G o (P)
o
A
common
assump ion
is
ha
all he
objec s
in
P a e
"di ec ly
ini e",
i
.e .,
objec s
A,B
EP
can
sa is y
A®B
-
A
only
i B is a
ze o
objec
.
Fo
example,i
P is
he
collec ion
o
all
ini ely
gene a ed
p ojec i e
igh
modules
o e
a
uni al ing
R,
he
objec s
in P
a e
di ec ly ini e
i
and
only
i
all
squa e
ma ices
o e
R
ha
sa is y
xy
=
1
also
sa is y
yx
=
1
.
pa icula ,
his
holds
i R is
commu a i e,
o i
R
is a
di ec ed
union
o
ini e-dimensional
algeb as,
o
i
R
is
noe he ian
on
I also
holds
i
R
is a
uni -
egula
ing,
meaning
be
pa ially
o de ed
.
ei he side
.
ha
o
any
x
E
R
he e
exis s
a
uni
(in e ible
elemen )
u E
R
such
ha
xux
= x,
o
in
ha case
he
objec s
in
P
may
be
cancelled
om
di ec
sums
[18,
Theo em
2
;
9,
Theo em
4
.5]
.
In
he
case ha
Pis
he
collec ion
o
all
ini ely
gene a ed
p ojec i e
igh
modules
o e
a
uni al ing R,
he module
R
plays
a
special
ole
in
P,
o
e e y
objec
in
P
is
isomo phic
o
a
di ec
summand
o
a
ini e
di ec
sum
o
copies
o
R
.
As
a
consequence,
[R]
plays
a
special
ole
in
K
0
(R)
:
gi en
any
x
EKG
(R)
he e
exis s
a
posi i e
in ege
n
such ha
x 1
n[R]
.
8
0
In
By i ue o his
p ope y,

[R] is
called
an
o de
-
uni
o
K
0
(R)
.
We
may
make
K
0
in o
a
unc o
om
he
ca ego y
o
ings
wi h
uni
(and
uni al ing
homomo phisms)
in o
a
ca ego y
whose
objec s
a e
all
pai s
(G,u)
whe e
G
is a
p e-o de ed
abelian
g oup
and
u is
an
o de -uni
in
G
.
The
app op ia e
mo phisms
in
his
la e
ca ego y
a e
no malized
posi i ehomomo phisms
:
(G,u)
-+
(H, ),
ha
is,
g oup
homomo phisms
:
G

H
such
ha
(G
+
)
c
H+
and
(u)
=
.

We call his
ca ego y
he
ca ego yo
p e-
o de ed
abelian
g oups wi h
o de
-
uni
.
Any
uni al ing
homomo phism

:
R
->
S
induces
a
unc o
(-)ORS
om
he
ca ego y
o ini ely
gene a ed
p ojec i e
igh
R-modules
o
he
ca ego y
o ini ely
gene a ed
p ojec i e igh
S-modules
.
Since
his
unc o
p ese es
di ec
sums,
i
in
u n
induces
a
posi i e
homomo phism

K
0
(~)

:
K
0
(R)
->
K
0
(S),

such ha
K0(~)([A]-[B])
=
[A®RS]-[B®RS]
o all
ini ely
gene a ed
p ojec i e
igh
R-modules
A
and
B
.
As
K0
(¢)([R])

= [S],

we
see
ha

K
0
(¢)

is a
no malized
posi i e
homomo phism
om
(K0(R),[R])
o
(K0(S),[S])
.
Since
he
app op ia e unc o ialp ope ies
a e
clea , we ob ain
a
unc o
om
he
ca ego y
o
ings
wi h
uni
o
he
ca ego y
o
p e-o de ed
abelian
g oups
wi h
o de -uni ,
gi en
by
he
assignmen s
R¡->
(K0(R),[R])

and
K0
W
The cons uc ion
o

K
0
(R)

o
a
ing

R

wi hou

1

is
no
based
ob ained
om
he
"uni i ica ion"
o R,
on
he
abelian
g oup
7LxR,
(m, )(n,s)
=
(mn,ms+n + s)
.
wi h
he se {0}xR,
on some
class
o£
non-uni al
p ojec i e
modules
bu
ins ead
is
namely
he
ing
R
1
based
wi h
mul iplica ion
gi en
by
he
ule
The
o iginal
ing
R
may
be
iden i ied
which
is
an
ideal
o
R
1

1
,

and
hen
R /R
-
7l
.

Then

K
0
(R)

is
de ined
o
be
he
ke nelo

K0
o
he
na u al
map
R
1
1

7L,

ha
is,
he
subg oupo

K0
(R
1
)

consis ing
o
hose
elemen s

[A]-[B]

in

K0 (R
1 )

o
which

A/AR

and

B/BR

a e
(s ably)
isomo phic
abelian
g oups,
and

K
0
(R)

is
equipped
wi h
he
p e-o de ed
abelian
g oup
s uc u einhe i ed
om

K
0
(R
1)
.

In
gene al,

K
0
(R)

need
no
ha e
an
o de -uni
.

To ake
he
place
o an
o de -uni ,
we
may use he subse

D(R)

o

K
0
(R)

consis ing
o
hose
elemen s

x E
K
0
(R)

o
which

0
<--
x
<-_
[R
1
]
.

In
he
con ex
o
he
ca ego yo
no -necessa ily-uni al
ings,
K0
may
be iewed
ei he
as a
unc o
in o
he
ca ego y
o
p e-o de ed
abelian
g
. oups
and
posi i e
homomo phisms,
o as
a
unc o
in o
a
ca ego ywhose
objec s
a e
all
pai s
(G,D)
whe e
G
is a
p e-o de ed
abelian
g oup
and
D is
a
sui able
subse
o
G
+
.
G o hendieck
g oups
ha ing
been cons uc ed,
wo
basic
me a-
ques ions
a ise
:
Wha
so o
in o ma ion
abou
he
da a
Pis
s o edin G o (P),
and
how
may
his
in o ma ion
be
e ie ed?
Fo
ins ance,
since
G o (P)
is
an
in a ian
o
P,
he e
can
be
si ua ions
in
which
i
can
be
p o ed
ha da a
P
and
P'
a e no
equi alen
by
showing
ha
G o (P)
and
G o (P')
a e no
isomo phic
.
Also,
by
i s
cons uc ion
G o (P)
e lec s
he
a angemen
o di ec
sum
decomposi ions
in
P,
and
we
may ask
how
much
o
he
di ec
sum
decomposi ion
s uc u e
o
P,
and
wha
o he s uc u alin o ma ion
abou
P,
may
be
eco e ed
om
G o (P)
.
By
way
o
illus a ion,
we
shall
discuss
a
numbe
o
si ua ions
in
which
hese
ques ions
ha e
been
success ully
answe ed
.
Due
o
he
au ho 's
bias, hese
examples
a e
KO
's
o
ce ain
ings
and
C*-algeb as,
bu
he
pa e ns
o
hese
examples
a e
o be
expec ed
in
G o hendieckg oups
o
o he
ma hema icalsys ems
.
8
2
a
IRRATIONALROTATIONALGEBRAS
.
Le
Tbe
he
uni
ci cle in
he plane,
and
le
C(T)
deno e
he
algeb a
o
all
con inuous
complex- alued
unc ions
on
T
.
We
may
iew
he
unc ions
in C(T)
as
bounded
linea
ope a o s
on
he
Hílbe
space

L
2 (T)

(whe e
a
unc ion om

C(T)

ac s
on
unc ions
om

L
2
(T)

by
mul iplica ion)
.
Gi en
a
posi i e eal
numbe

a,

ie

p
a
:
T
-+
T

be
coun e clockwise
o a ion
h ough
he
angle
i a
.
The
ule
poc*( )
= pa
hen
de ines
a
bounded
linea
ope a o
pc *
on
L
2
(T),
and
we
le
Aa
deno e
he
no m-closed
sel -adjoin
subalgeb a
o
bounded
linea
ope a o s
on
L
2
(T)
gene a ed
by
C(T)
and
p
a*
.
This
algeb a
is
known
as
" he
ans o ma ion
g oupC*-algeb a
o
he
o a ion
pa
"
.
I is
ai lyeasy o
dis inguishamong
he
algeb as

Aa

o
a ional
a,
and
i is
also
easy
o
dis inguish
he
a ional
cases
om
he
i a ional
cases
.
Howe e ,
dis inguishing
among
he
i a ional
cases
is a
sub le
p oblem,
which
was
only
sol ed
when
Pimsne ,
Voiculescu,
and
Rie el
calcula ed
K0
o
hese
algeb as
.
Namely,
o
any
i a ional
numbe
a E
(0,1),
he e
is
an
isomo phism
(in
he
ca ego y
o
p e-o de ed
abelian
g oups wi h
o de -uni )
om
(K
0
(A
CC),[A
a
])

on o

(
71
.
+a
7L
,1)
,

whe e
he
subg oup

71+aF

o

IR
is
gi en
he
usualo de ing
[28,
Co olla y
2
.6
;
29,
Co olla y
1
;
30,
Theo em
1]
.
Thus
i
A
a
-
A6
o
some
i a ional
numbe s
a,s
E
(0,1),

he e
mus
be an
isomo phism


o

(
7L+aZ
,
1)

on o

(
lZ+OF-
,
1)
.

(Since
he
opologies
in

71+a1Z

and

7+gJ-
may
be
de ined
in
e ms
o
he
o de ing,
mus
also be
a
homeomo phism
.)
A e
app oxima ing
elemen s
o

71+a1Z

and

lZ+BZ
by a ionalnumbe s,
clea ing
denomina o s,
and
using
he
ela ion
(1) =
1,
i
is
easily
seen
ha

7L+a/Z
=

1+1
1
.

F om his
a
quick
compu a ion
leads
o
he
conclusion
ha
ei he
S=
a
o
S=
1-a,
whence ei he
p
s
=
pa
-1
o ps
=
pa
F-algeb a
ha
is a
union
o
a
coun ableascending
sequence
o
ma icial
subalgeb as
(equi alen ly,
any
F-algeb a
ha
is
isomo phic
o
a
di ec
limi
o
a
coun able
sequence
o
ma icial
F-algeb as
and
F-algeb a
homomo phisms)
.
I is
easily
checked
ha
ma icial
algeb as
a e
uni - egula
.
Hence,
any uni alul ama icialF-algeb a
R

is
uni - egula ,
and
so

K0
(R)

is
a
pa ially
o de ed
abelian
g oup
.
Using
F-algeb auni i ica ions,
i
ollows
ha
K
0
o
any
ul ama icialF-algeb a
is
pa ially
o de ed
.
These
pa ially
o de ed
abelian
g oupsmay
be
used
o
classi y
ul ama icial
F-algeb as, ollowing
a
me hod
o
Ellio
[7]
.
I
R
and
S
a e
uni al
ul ama icial

F-algeb as,
hen
R
c
S

i
and
only i
he e
exis s an
isomo phism
o
(K0(R),[R])

on o

(K0(S),[S])
in
he
ca ego yo
p e-o de ed
abelian
g oups
wi h
o de -uni
[7,
Theo em
4
.3
;
9,
Theo em
15
.26]
.
I
R
and
S
a e
non-uni al
ul ama icial
F-algeb as,
hen
R
=
S

i
and
only
i
he e
exis s
an
o de ed
g oup
isomo phism
o

K
0
(R)

on o

K0
(S)

mapping

D(R)
on o
D(S)
[7,
Theo em
4
.3]
.
The
pa ially
o de edabelian
g oups
which
can appea
as
KU o
ul ama icial
F-algeb as
a e
jus
hosewhich
a e
isomo phic
(as
o de ed
g oups)
o
di ec limi s
o
coun ablesequences
o ini e
p oduc s
o
copieso
71

[7',
Theo ems
5
.1,
5
.5]
.
Howe e ,
i is
8
4
O
ULTRAMATRICIAL
ALGEBRAS
.
Fix
a
ield
F
.
A
ma icial
F-algeb a
is
any
F-algeb a
ha
is
isomo phic
o
a
ini e di ec
p oduc
o
ull
ma ix
algeb as
o e
F
.

(In
case
F is
algeb aically
closed,
he
ma icial
F-algeb as
a e
exac ly
he
ini e-dimensional
semisimple
F-algeb as
.)
An ul
ama icial
F-algeb a
is
any
usually
impossible
o
checkdi ec ly
whe he
a
gi en
pa ially
o de ed
abelian
g oup
is isomo phic
o
such
a
di ec
limi
.
Some
ob ious
p ope ies
o
he e
di ec
limi s a e
ha hey
a e
coun able,
hey
a e
di ec ed
(upwa d
and
downwa d),
and
hey
a e
unpe o a ed
(any
x
sa is y
-
_ng
nx
?0
o
some
nE
(N
also
sa is ies
x ?
0)
.
A
mo e
undamen alp ope y,
also
easily
checked,
is
he
Riesz
in e pola ion
p ope
"" :
gi en
any
xl'x2'yl'y2
such ha
xi
---
y
j
o all
i,j,
he e
e
;
:is s z
such ha
x
i
<-_
z
<_-
y
j
o all
i,j
.
The di ec
limi so (sequences
o )
ini e
p oduc s
o
copies
o
1
we e
cha ac e ized
by
E os,
Handelman,
and
Shen
as
exac ly hose
(coun able)
pa ially
o de edabelian
g oups
which
a e
di ec ed
and
unpe o a ed
and
which
sa is y
he
Riesz
in e pola ion
p ope y
[5,
Theo em
2
.2
;
11,
Co olla y
21
.8]
.
Pa ially
o de ed
abelian
g oups
wi h
he
la e
h eep ope ies
a e now
called
dimensiong oups
.
Thus, gi en
a
pa ially
o de ed
G
and
an
o de -uni
u E G,
he e
exis s
an
(G,u)
on o
(K0(R),[R])
o
some uni al
F-algeb a
R
i
and
only
i
G
is a
coun able
Fo he
non-uni al
case,
eplace- he
o de -uni
u
such ha
e e y
elemen
o
G
abelian
g oup
isomo phism
o
ul ama icial
dimension
g oup
.
by
an upwa d
di ec ed
subse
D
c
G
and
such
ha
any
elemen
o
G+
which
lies
below
an
elemen
o
D
mus
lie
in
D
.
Then
he e
exis s an
o de ed
g oup isomo phism
o
G
on o
K
0
o some
ul ama icial
F-algeb a
R,
wi h
D
mapping
on o
D(R),
i
and
only
i G is a
coun able
dimension
g oup
.

(These
esul s
a e
ob ained
by
combining
Ellio 's
esul s
[7,
Theo ems
5
.1, 5
.5]
wi h
hose
o
E os,
Handelman,
and
Shen
[5,
Theo em
2
.2]
.
Fo
example,
he
subg oup

{a/2
n
1
a
E
71
is
a
sum
o
elemen s
om
D,
and
n
E
M
}

o
Fo
example,
he
au ho
and
Handelman
showed ha
any
nonze o
pa ially
o de ed
abelian
g oup
wi h
an
o de -uni
has
a
leas
one
s a e
[14,
Co olla y
3
.3
;
9,
Co olla y
18
.2]
.
Since
K
0
o
any
nonze ouni - egula
ing
is
nonze o
andpa ially
o de ed,
i
ollows
ha
any
nonze o
uni - egula
ing
has
a
leas
one
pseudo- ank
unc ion
[14,
Co olla y
3
.5
;
9,
Co olla y
18
.5]
.
The
au ho
and
Handelman
also
de eloped
a ious
c i e ia
o
a
nonze o
pa ially
o de edabelian
g oup
G
wi h
an
o de -uni
u
o ha e
a
unique
s a e
.
Fo
ins ance,
his
happens
i
and
only
i
he eexis
in ege s
s
.>
>
0
such ha
gi en
any
x,y
E G
+
wi h
x+y
=
u,
he e
is some
n E
[N
o
which
ei he n x
1
nsy
o
n y
<
nsx
.
As
a
consequence,
a
nonze o
uni - egula
ing
R
possesses
a
unique
pseudo- ank unc ion
i
and
only
i
he e
exis
in ege s
s >
> 0
such ha
gi en
any
o hogonal
idempo en s
e,
E
R
wi h
e+
=
1,
he e
is
some
n
E
11`1
o
which
ei he
he
di ec sum
o n copies o eR embeds
in
he
di ec sum
o
ns
copieso
R
o
he
di ec
sum
o n copies o
R
embeds
in
he di ec
sum
o ns copies o eR
[14,
Theo em
4
.6
;
9,
Theo em
18
.6]
.
o
PSEUDO
-
RANK
FUNCTION
SPACES
AND
TRACE
SPACES
.
The
collec ion
P(R)
o
all
pseudo- ank
unc ions
on
a
egula
ing
R
(wi h
1)
can
be
iewed
as
a
subse o
he
eal
ec o
space

IR
R

o
all
eal
alued
unc ions
on

R
.

I

1R
R

is
gi en
he
p oduc
opology,
i is
a
locally
con ex
Hausdo
linea
opological
space,
and

JP(R)
is a
compac
con ex
subse
o
RR
[9,
P oposi ion
16 .17]
.
In
ac ,
IP(R)

is a
a he
special
kind
o
compac
con ex
se
known
as
a
"Choque
simplex"
[9,
Theo em
17
.5]
.
(Choque
simplices
a e
in ini e-
dimensional
analogs
o
classical
ini e-dimensional
simplices,
and may
be
cha ac e ized
as
exac ly
hose
compac
con ex
subse s
o
locally
92

con ex
Hausdo linea
opological
spaces
which
a ise
as
in e se
limi s
o
ini e-dimensional
simplices
.)
In
a
simila
ashion,
he
collec ion
S(G,u)

o
all
s a es
on
a
p e-o de ed
abelian
g oup
(G,u)
wi h
o de -uni ,
called
he
s a e
space
o

(G,u),

is
a
compac
con ex
.
subse o
he
p oduc
space

[ñ
G
[9,
P oposi ion
17
.11]
.
I
G is a
dimension
g oup
(mo e
gene ally,
i
G
sa is ies
he
Riesz in e pola ion
p ope y), hen
S(G,u)
is
a
Choque simplex
[17,
Theo em
1
.2 .5
;
5,
P oposi ion
1
.7]
.
Fo he
case
o
S
o
he
egula
ing R,
he
canonical
bijec ion
be ween
he
á a e space
S(K0(R),[R])

and he
pseudo- ank
unc ion
space

JP(R)
is
an
a ine
homeomo phism
[9,
P oposi ion
17 .12], i
.e
.,
an
isomo phism
in
he
ca ego y
o
compac
con exse s
.
Hence,
o
ealize
a
gi en
Choque simplex
K
as

FP(R)
o
some
egula
ing
R,
i
su ices
o
ealize
K
as
S(K0(R),[R])
.
In
he
me izable
case,
his
was
done
by
he
au ho
[8,
Theo em
5
.1
;
9,
Theo ems
17
.19,
17
.23]
.
We
ske ch
an
easie
p oo
o
his
esul ,
aking
ad an age
o
ou
abili y
o
ealize
any
coun able
dimension
g oup
as
K
0
o an
ul ama icial
algeb a
.
Thus
le
K
be
an
a bi a y
me izable
Choque
simplex,
and
le
A (K)
be
he
pa ially
o de ed
eal
Banach
space
o
all
a ine
(i
.e
.,
con ex-combina ion-p ese ing)con inuous
eal- alued unc ions
on
K
.
(The
o de ing
in
A (K)

is
he
poin wise
o de ing
o
unc ions,
and
he
no m
is
he
sup emum
no m
.)
F om
he
me izabili y
o
K, i
ollows
ha
A (K)
is
sepa able
.
Also, since
K
is a
Choque
simplex,
A (K)

sa is ies
he
Riesz in e pola ion
p ope y
[4,
Théo éme
;
32,
Theo em
5]
.
Consequen ly,
we
may
cons uc
a
coun able
dense addi i e
subg oup
G
o
A (K)

such
ha
G
con ains
he
cons an unc ion
1
and
G
has he
Riesz
in e pola ion
p ope y
.
Then
G is a
dimension
g oup
and
1
is
an
o de -uni
in
G
.
Since
G
is
dense
in
A (K),
he
es ic ion
map
S(A (K),1)
-
S(G,1)
is
an
a ine
homeomo phism
.
On
he
o he
hand,
a
s anda d olklo e
esul
is
ha
he
e alua ion
map
K
->
S(A (K),1)

is
an
a inehomeomo phism,
and
hus
K
is
a inely
homeomo phic
o
S(G,1)
.
As
he e
exis s
a
uni al
ul ama icial
algeb a
R
o
which
(K0(R),[R])
-
(G,1),
we
conclude ha

FF>(R)
is
a inely
homeomo phic
o
K
.
By
using
he
s ic
o de ing
on
A (K)

(unde
which
<
g
only i
(x) < g(x)
o
all
x E
K),
we
can
ensu e ha
he
dimension
g oup
G
is
simple,
whence
Ris a
simple
algeb a
.
simplices
as
ace
spaceso uni al
C*-algeb as
.
Gi en
a
uni al
complex
C*-algeb a
A,
he se
Asa
o
sel -adjoin
elemen s
o
A
becomes
a
pa ially
o de ed
eal ec o
space
wi h
posi i e
cone
and
o de -uni
1
[11,
P oposi ion
6
.1]
.
A
s a e
on
Ais
any linea
unc ional

A
y
C

which
es ic s
o
a
s a e
on

(A
sa'
l)
.

Since
all
s a eson

(A
sa,1)

ex end
uniquely
o s a es
on

A,

he
collec ion
o
all
s a eson

A

may
be
iden i ied
wi h
he
s a e
space
o

(A
sa
,1)
.

A
acial
s a e
on
A
is
any
s a e
such ha
(xx*)
=
(x*x)
o
all
x E A,
and
a
no malized
ini e
ace
on
A
is
he
es ic ion
o
A

o
any
acial
s a e
.
The
ace
space
o
A
is
he
collec ion
sa
T(A)
o
all
no malized
ini e
aces
on
A
.
We
iden i y
T(A)
wi h
he
collec ion
o
all
acial
s a es
on A,
which
is a
compac
con ex
subse o

S(A
sa
,1)
.

In
ac ,

T(A)

is
a
Choque simplex
[31,
Theo em
3
.1
.18]
Fo
a
ma ix
algeb a
Mn
(C),
he
ace space
T(M
n
(E))
is a
single on,
as is
he
s a e
space
o

(K
0
(Mn((E)),[Mn(
C)])
.

Using
he
94
Pa allel
p ocedu es
can
be used
o
ealize
me izable
Choque
AS
a
.=
{x
E
A
sa

1

spec um(x)
c

R
+}
obse a ion
ha
he
unc o s
T(-)
and S(K
O
con e
ini e
p oduc s
o ini e
cop oduc s
and
con e
di ec
limi s
o
in e se
limi s,
i
ollows
ha
he
ace
space
o
any
uni al
complex
AF
C*-algeb a
A
is
a inely
homeomo phic
o
S(K0(A),[A})
[1,
Co olla y
3
.2}
.
Gi en
any
me izable
Choque
simplex
K,
he e
is a
coun able
simple
dimension
g oup
(G,u)
wi h
o de -uni
such ha
S(G,u)
is
a inely
homeomo phic
o K, as
indica ed
abo e
.
Hence,
by
choosing
a
simple
uni al
complex
AF
C*-algeb a
A
o
which
(K0(A),[A})
-
(G,u),
we
ob ain
Blackada 's
esul
ha
any
me izable
Choque
simplex
K
is
a inely
homeomo phic
o
T(A)
o
some
simple
uni al
complex
AF
C*-algeb a
A
[1,
Theo em
3
.9}
.
o
METRICALLY
COMPLETE
REGULAR
RINGS
.
A
no m-like
unc ion
N*
may
be
de ined
on
any
egula
ing
R
(wi h
1)
by
se ing
N*(x)
equal
o
he
sup emum
o
he
alues
N(x)

o
N
E
{P(R)
.
I is
easily
checked
ha
he
ule
d(x,y)
=
N*(x-y)
hen
de ines
a
pseudo-me ic
d
on
R [12,
Lemma
1
.2]
.
In case
d is a
me ic
and
R
is
comple e
wi h
espec
o d,
we
say
ha
R is
N*-
comple e
.
Fo
ins ance,
i
he e
exis s
a
posi i e
in ege
n
such ha
all
nilpo en
elemen s
x
E R
sa is y
x
n
=0,
hen
N*(y)
k
1/n o all
nonze o
elemen s
y
E R,
and
so
R is
N*-comple e
[12,
Theo em
1
.3}
.
This
occu s,
o
ins ance,
i
R
can
be embedded
in a
di ec
p oduc
o
nxn ma ix
ings
o e
di ision
ings
.
The
unc ion
N*
on
R
co esponds
o
a
no m-like
unc ion
on
K0
(R),
de ined
using
s a es
in
place
o
pseudo- ank
unc ions,as
ollows
.
Gi en
any
p e-o de ed
abelian
g oup

(G,u)
`wi h
o de -uni ,`wemay
de ine
Ilxll
=
sup{Is(x)I
:
s E
S(G,u)}
o all
x E
G
.
Al e na i ely,

Ilxll
may
be
compu ed
as
lixll
=
in {k/n
l
k,n
E
IN and
-ku
<_
nx
<_
ku}
[17,
Lemma
1
.6
.1]
.
Then
II-ll
is a
nónnega i e
eal- alued
unc ion
on G
such ha
IImxII
=
ImI
.llxll
and
llx+yll
<
llxll+lly!I
o
all
m E
71
and
all
x,y
E
G
[ibid]
.
I
he
pseudo-me ic
d'
on
G
de ined
by
d'(x,y)
=
lix-yll
is
ac ually
a
me ic,
and
i G is
comple ewi h
espec
o
d',
we
say
ha
(G,u)
is
no m
-comp
le e
.
Hecause
o
he
canonical
bijec ion
be ween

FP(R)
and
S(K0(R),[R]),
i
ollows
ha
i¿[xR]II
= N
*(x)
o all
x
c'
R,
and
so
N*-comple eness
o
Ris
ela ed
o
no m-comple eness
o
.
B
:
0
(R)
.
Speci ically,
in case
he
egula
ing
R
is
N*-comple e,
he
au ho
p o ed
ha (K0(R),[R])
is
an
a chimedeanno m-comple e
dimension
g oup
[12,
Theo em
2
.11]
.

(Fo
a
pa ially
o de ed
abelian
g oup
G
o
be
a chimedean
means
ha
whene e
x,y
E
G
wi h nx
S y
o
all
posi i e
in ege s
n,
hen
x
<_
0
.)
Consequen ly,
some
s uc u e
heo y
o
N*-comple e
egula
ings
may
be
ob ained
om
co esponding
s uc u e
heo y
o
a chimedeanno m-comple e
dimension
g oups
.
Fo
example,
he e
exis s
a
comple e
ep esen a ion
o
any
a chimedeanno m-comple e
dimension
g oup
(G,u)
in
e ms
o
a ine
con inuous eal- alued
unc ions
on
i s
s a e space
.
The
only
es ic ions
placed
on
he
unc ionsappea ing
in
his
ep esen a ion
a e
he
alues
allowed
a
disc e e
ex emal
s a es
.

(A
s a e
s
on
(G,u)
is
disc e e
i
s(G)

is a
disc e esubg oup
o IR
.
A
poin
x
in a
con ex
se
K
is
ex emali
x
does
no
lie
in
he
in e io o
any
line
segmen
wi hin
K
.)
Se ing
A
= {q E
A (S(G,u))
l
q(s)
E
s(G)

o all
disc e e
ex emals a es
s},
he
au ho
and
Handelman
p o ed ha
he
e alua ion
map
G
-
A (S(G,u))
gi es
an
isomo phism
o
(G,u)
on o
(A,1)

(as
o de ed
g oups wi h
96
o de -uni )
[15,
Theo em
5
.1]
.
This a ine con inuous unc ion
ep esen a ion
o
a chimedean
no m-comple e
dimension
g oups in u n
p o ides
an a ine
con inuous
unc ion
ep esen a ion
o
K
0
o
he
N*-comple e
egula
ing
R
.
Unde
he
canonical
a ine
homeomo phism
be ween
S(K0(R),[R])
and
?(R),a
disc e eex emal
s a e
s
on
(K0(R),[R])
co esponds
o
an
ex emalpseudo- ank unc ion
P
wi h
a
disc e e
ange
o alues
.
Speci ically,
i
s(K
0
(R))
=
(1/m)j

o
some
m
E
H,
hen
P(R)

=
{0,1/m,2/m,
.
..
.
l},

and
his occu s
i
and
only
i

R/ke (P)

is
isomo phic
o
an
mxm ma ix
ing o e
a
di ision
ing
.
Se
BP
=
(1/m)3
:
in
his
case,
and o
all
o he
ex emal
pseudo- ank
unc ions
P
se
BP
=
F
R
.
Then
he e
is
a
na u al
isomo phism
o
(K0(R),[R])
on o
(B,1),
whe e
B
=

{q
E
A (IP
(R))

q
(P)

E
Bp
 o
all
ex emalpseudo- ank
unc ions
P}
[12,
Theo em
4 .11]
.
To
gi e
an
easy
applica ion
o
his
a ine con inuous
unc ion
ep esen a ion
o
K
0 (R),
assume,
o
some
ixedposi i e
in ege
,
ha
all simple
a inian
ac o
ings
o
R
(i
he e
a e
any)
a e x
ma ix
ings
(o e
some
o he
ings,
no
necessa ily
o e di ision
ings)
.
Then
i
P
is an
ex emal
pseudo- ank
unc ion
and
R/ke (P)
is
isomo phic
o an
mxm ma ix
ing
o e
a
di ision
ing,
mus
di ide

m,

whence

l/ E BP
.

As
a
esul ,
he
cons an
unc ion

l/
belongs
o
he
g oup
B
gi enabo e
.
F om
he
isomo phism
o
(K0(R),[R])
on o
(B,1),
i
ollows
ha
[R]
=
[C]
o
some
[C] E K
0
(R)

.

Since

R

is
uni - egula
[12,
Theo em
2 .3],
he
module
R
is
isomo phic
o
a
di ec sum
o
copies
o
C,
whence
he
ing
R
is
isomo phic
o a
x
ma ix
ing
(o e
he
endomo phism
ing o

C)

[12,
Co olla y
4 .14]
.
In
pa icula ,
i R
has
no
simple
a inian
ac o
ings,
hen
R
is a
x
ma ix
ing
o all
posi i e
in ege s
.
he
collec ion
L(R
R
)
o
p incipal
igh
ideals
o ms
a
la ice,
wi h
ini e
in e sec ions
o
ini e in ima
and
ini e sums
o ini e
sup ema
[9,
Theo ems
1
.1,
2
.3]
.
The
ing
R is
said
o
be
-
con inuous
i£
he
la ice

L(R
R
)

is

'Ng
-con inuous
in
he
sense
ha
(a)
e e y
coun ablesubse
o
L(R
R
)
has
an
in imum
and
a
sup emum
in
L(R
R
);

(b)
whene e
A
E
L(R
R
)
and
B
1
<-
B2
~-l
.
.
.
in
L(R
R),
hen

A n
(V
Bi
)

=

(An
B
i
)
;

(c)

whene e

A
E
L(R
R
)

and
B
1
?
B2
.
1

. . .

in

L(R
R
)
,

hen

AV
(A
Bi
)

=

MA
V Bi
)
.

(Since

L(R
R
)
is
an i-isomo phic
o
he
la ice
o
p incipal
le idealso
R
[9,
Theo em
2 .5],
his de ini ionis
le - igh
symme ic
.)
Equi alen ly,
R
is ieo
-con inuous
i
and
only
i
gi en
any coun ablygene a ed
igh
(le )
ideal
I
o
R,
he e
exis s
a
p incipal
igh
(le )
ideal
J
=) I
such ha
e e y
nonze o
igh
(le )
ideal
con ained
in J
has
nonze oin e sec ion
wi h
I
[9,
Co olla y
14
.4]
.
Handelman
p o ed ha
e e y
!?
C
-con inuous
egula
ing
is
uni -
egula
[19,
Theo em
3
.2
;
9,
Theo em
14
.24],
and he au ho p o ed
ha
e e y
iz
o
-con inuous
egula
ing
is
N*-comple e
[12,
Theo em
1
.8]
.
Hence,
he
s uc u e
heo ies
o
a chimedean
no m-comple e
dimension
g oups
and
N*-comple e
egula
ingsyield
a
s uc u e
heo y
o
X
o
-con inuous
egula
ings
.
Howe e ,
a
s uc u e
heo y
o
-con inuous
egula ings
was
i s
de i ed
om
a
s uc u e
heo y
o
mono one
a-comple e
dimension
g oups,
as
ollows
.

(A
pa ially
o de ed
se
P
is
mono one a-comple ep o ided
ha
e e y
ascending
9
8
o
ALEPH-NOUGHT-CONTINUOUS
REGULAR
RINGS
.
In
any
egula
ing
R,
(descending)
sequence
x
1 :1
x
2
5
. .
.
(x
1
?
x
2
?
.
.
.)
in
P
which
is
bounded
abo e
(below) in P
has
a
sup emum
(in imum)
in P
.)
Handelman,
Higgs, and
Law ence
p o ed
ha
K
0 o
any
-con inuous
egula
ing
Ris a
mono one 6-comple e
dimension
g oup
[24,
P oposi ion
2
.1],
and
ha such g oups
a e
a chimedean
[24,
Theo em
1
.3]
.

Since

K0
(R)

is
a chimedean, hey ob ained
){ke (P)

I
PE

[P
(R)
}
=
{0},
om
which
i
ollows
ha
he
in e sec ion
o
he
maximal
wo-sided
ideals
o R
is
ze o
[24,
Theo em
2
.3]
.
I
M
is any
maximal
wo-
sided
ideal
o
R,
hen
he
exis ence
o
a
s a e
on
(K0
(R/M),[R/M])
implies
he
exis ence
o
a
pseudo- ank
unc ion
P on
R/M,
and
ke (P)
= {0}
because
R/M
is a
simple
ing
.
As
a
consequence,
R/M
con ains
no
uncoun able
di ec
sums
o
nonze o
p incipal
igh
o
le
ideals,
and
using
his
coun abili y
condi ion,
Handelman
p o ed
ha
R/M
is a
igh
and
le
sel -injec i e
ing
[19,
Co olla y
3
.2]
.
Thus,
since
he
in e sec ion
o
he
maximal
wo-sided
ideals
o
R is
ze o,
R is a
subdi ec
p oduc
o simple
igh
and
le
sel -injec i e
ings
.
A
s uc u e
heo y
o
mono one C-comple e
dimension
g oups
was
de eloped
by
he
au ho ,
Handelman,
and
Law ence
[17]
and
applied
o
K0
(R)
.
Fo
example,
he
a ine
con inuous
unc ion
ep esen a ion o
such g oups
led
o
a
comple e
ep esen a ion
o

K
0
(R)

in
e ms
o
a ine
con inuous
unc ions
on

[P(R)

[17,
Theo em
11
.15
.1]
.
As
a
consequence,
i
all
simple
ac o
ings
o
R
a e
x
ma ix
ings
( o some
ixed
posi i e
in ege
),
hen
R is a
x
ma ix ing
[17,
Theo em
II .15
.3]
.
o
FINITE
RICKART
C*-
ALGEBRAS
.

A
Ricka
C*-
algeb a
is a
C*-algeb a
A in
which
he
igh
annihila o
o
any
elemen
x
( ha
is,
he
igh
ideal
{a E
A
1
xa
=
0})
equals
he
p incipal
igh
ideal
gene a ed
by
some
p ojec ion

p

( ha
is,

p =
p*
= p
2
) .

This
is
a
gene aliza ion
o
he
concep
o
an
AW*-
algeb a
,
which
is a
C*-algeb a
in
which
he
igh annihila o
o
any
subse
is
a
p incipal
igh
ideal
gene a ed
by
a
p ojec ion
.
In
pa icula ,
all on
Neumann
algeb as
(W*-algeb as)
a e
Ricka
C*-algeb as
.
A
ini e
C*-algeb a
is
a
uni al
C*-algeb a
A
such ha
all
elemen s
x E
A
sa is ying
xx*
=
1
also
sa is y
x*x
= 1
.
The
K- heo y
o
a
ini e
Ricka
C*-algeb a
A
can
be
in es iga ed
wi h
he
aid
o an
auxilia y

?~
O
-con inuous
egula
ing

R

which
is
alsó
*-
egula
,
i .e
.,
he e
is an
in olu ion
*
on
R
such ha
e e y
p incipal
igh ideal
o
R
is
gene a ed
by
a
p ojec ion
.
Handelman
p o ed
ha

A

is a
*-sub ing
o an

?~
O
-con inuous
*- egula
ing

R
such
ha
he
only
p ojec ions
in
R
a e
hose
in
A
[19,
Theo em
2
.1]
.
The
ing
R
is
essen ially
unique
(up
o
a
*- ing
isomo phism
which
is
he
iden i yon
A),
and
is
called he
egula
ing o
A
.
Handelman
also
p o ed
ha
he
inclusion
map
A ~
R

induces
an
isomo phism
o

(K0(A),[A])

on o

(K0(R),[R])
.

(A
p oo
o he
case
ha
A
has
no
one-dimensional
ep esen a ions
is
gi en
in
[13,
Theo em
5
.2]
.)
In
pa icula ,

K0
(A)

is a
mono one

Q-comple e
dimension
g oup,
and
he
s uc u e
heo y
o
such g oups
yields
a
co esponding
s uc u e
heo y
o
A,
in
exac ly
he
same
manne
as
o
?ZO
-con inuous
egula
ings
.

(Howe e ,
his
s uc u e heo y
was
i s
de i ed
om
he
s uc u e
heo y
o

c¿
0
-con inuous
egula
ings,
ia
he
egula
ing
o
A
.)
Fo
example,
A
is
a
subdi ec
p oduc
o
simple
AW*-algeb as,
and
so
A
can
be
embedded
in
a
ini e
AW*-algeb a
[24,
Theo em
3
.1]
.
Fo
ano he
example,
i
he
dimension
o
e e y
ini e-dimensional
i educible
ep esen a ion
.o
A (i
he e
a e
any)
is
di isible
by
a
ixed
posi i e
in ege
,
hen
A
is a
x
ma ix
10
0
ing o e
some
o he
ini e
Ricka
C*-algeb a
[17,
Theo em
111
.16
.8]
.
REFERENCES
1
.
B
.
E
.
Blackada ,
"T aces
on
simpleAF C*-algeb as"
J
.
Func
.
Anal
.
38 (1980)
156-168
.
2
.
O
.
B a eli,
"Induc i elimi so
ini e-dimensional
C*-algeb as"
T ans
.
Ame
.
Ma h
.
Soc
.
17
1
(1972)
195-234
.
3
.
J
.
Cun z
and
W
.
K iege ,
"Topological
Ma ko
chains
wi h
dicyclic
dimension
g oups"
J
.
eine
angew
.
Ma h
.
320
(1980)
44-51
.
4
.
D
.
A
.
Edwa ds,
"Sépa a ion
des
onc ions
éelles
dé inies
su
un
simplexe
de
Choque "
C
.
R
.
Acad
.
Sci
.
Pa i
s261 (1965)
2798-2800
.
5
.
E
.
G
.
E os,
D
.
E
.
Handelman,
and
C .-L
.
Shen,
"Dimensiong oups
and
hei
a ine
ep esen a ions"
Ame
.
J
.
Ma h
.
102 (1980)
385-407
.
6
.
E
.
G
.
E osand
C
.-L
.
Shen,
"App oxima ely
ini e
C*-algeb as
and
con inued
ac ions"
Indiana
Uni
.
Ma h
.
J
.
29 (1980)
191-204
.
7
.
G
.
A
.
Ellio ,
"On he
classi ica ion
o
induc i e
limi s
o
sequences
o semisimple
ini e-dimensional
algeb as"
J
.
Algeb a
38 (1976)
29-44
.
8
.
K
.
R
.
Goodea l,
"Algeb aic
ep esen a ions
o
Choque
simplexes"
J
.
Pu e
Applied
Algeb a
11
(1977)
111-130
.
9
.

,
_Von
Neumann
Regula
Rings
London
(1979)
Pi man
.
10
.

,
"A inian
and
noe he ian
modules
o e
egula
ings"
Communic
.
i
n
Algeb a
8 (1980)
477-504
.
11
.

,
No es
_on
Real
_and
Complex
_C*-
Algeb as
Nan wich
(Cheshi e)
(1982)
Shi a
.
12
.

,
"Me icallycomple e
egula
ings"
T ans
.
Ame
.
Ma h
.
Soc
.
27
2
(1982)
275-310
.
13
.

,
"Pa ially
o de ed
G o hendieck
g oups"
in
Algeb a
_and _I s
Applica ions
(H
.
L
.
Manocha
and
J
.
B
.
S i as a a,
Eds
.),

pp
.

71-90
New
Yo k
(1984)
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