Publicacions
Nla e ná iques,
Vol
36
(1992),
793-760
.
HOW
TO
SOLVE
AN
OPERATOR
EQUATION
A
bs ac
MAI2TIN
MATHILU
This
a icle
summa izes
a
se ies
o
lec u es
cleli e ed
a
he
Ma h-
e na
. ics
Depa nen
o
he
Uni e si y
o Leipzig,
Ce inany
in
Ap il 1991,
which
we e
e
o e iew
ecl niques
o
sol ing
ope a o
eclua ions
en
C*-algeb as
connec ed
wi h
me hocis
de eloped
in
a
Spanish-Ge nan
esea ch
p ojec
en
"S uc u e
ancí
Applica ions
o
C*-Algeb as
oí'
Quo ien s"
(SACA)
.
One
o
he
esea che s
in
his
p ojec
was
P o esso
Pe e
Menal
un il
his
unexpec ed
dea h
his
Ap il
.
To
his
me no y
his
pape
shall
be
dedica ed
.
1
.
In oduc ion
Sol ing
equa ions belongs
o
l e
undamen al
asks
o ina hema ics
.
Many
p oblems
in
he
sciences
load
e
equa ions
in ol ing
numbe s,
map-
pings
;
nd
o he
guan i ies
.
In
aca,
i
equendy
occu s ha
e en ually
a
dues ion
can
be
ph ased
as
an
"equa ion",
al hough,
a
i s ,
i
appea ed
ú
be
o a
a he
di e en
na u e
.
To
ind
a
solu ion
o
an
equa ion
gene -
ally
implies
bo h
he
exis ence
as well as
he
unidueness
p oblem
.
The e
is
no
uni e sal
p ocedu e
o
sol ing
;
bu
he
de ices
in en ed
seem
,
o
be as
mani old
as
he
possible
ques ions,
asid
only
allow
a
a he
ough
classi ica ion
such
as nu ne ical,
app oxima i e,
algeb aic
me hods
e c
.
Flowe e ,
i is
always
an
i npo an
s op
o
de e mine
he
co mon
ea-
u es
in
sol ing
a
ce ain
class
o exa ples
o
he
ai e
o
de eloping
a
machine y
which
enables
o
handle
a speci ied
collec ion
o
equa ions
a
one
iene
.
In
he
p esen
pape
;
we
will
be
conce ned
wi h
equa ions
wi hin
a
non-commu a i e
in ini e
dimensional
se ing
.
To
be
mo e
speci ic,
hey
will
be
o
he
o m
'Chis
pape
is
pa
o
a
esea ch
p ojec
suppo ed by
lic
DAAD
.
744

M
.
MATHIEU
whe e,
o
each
`pa ame e '
a,
T«,~1,

,
, en
is
a
linea
ope a o
on
a C*-
algeb a
A
(wi h
ce ain
addi ional
p ope ies)
and
we
a e
looking
o
elemen s
xj
E
A
sol ing
he
equa ion
(1) (o
be e , his
sys em
o
equa ions)
.
We
will
i s ly
collec
some
examples
o
ques ions
which
can
be
ph ased
in
an
equa ion
such
as
(1),
hen
desc ibe
a
gene al
ool
o
ackle
hem,
and
inally
indica e
solu ions
which
yield
answe s
o
he
ques ions
lis ed
.
As
a
common
ea u e,
he
ques ions
in
Sec ion
2
lead
o
equa ions
in
a
C*-algeb a
;
ha
is,
we
a e
looking
o
ce ain
elemen s
in
a
C*-algeb a
sol ing
he
equa ion,
while
he condi ions
ypically
a e
o mula ed
in
e ms
o
ope a o s
de ined
on
he
C*-algeb a
.
Needless
o
say
ha
he e
a e
many
mo e
ins an es
which
can be
se led
by
he
p oposed
me hods
.
2
.
Examples
We
ha e
selec ed
ou
examples
om
he
ollowing
ou classes o
op-
e a o s
on C*-algebas
:
de i a ions,
comple ely
posi i e
ope a o s,
cen-
alizing
mappings,
and
gene a o s o
dynamical
semig oups
.
2 .1
.
De i a ions
.
Le
A
be a
C*-algeb a
and
S
a
de i a ion
on A,
Le
.
a
linea
mapping
om
A
in o
i sel
sa is ying
Leibniz'
ule
S(xy)
=
x6(y)
+
b(x)y
o
all
x,
y E
A
.
Each
de i a ion
S
is
au oma ically
bounded
whence
i is
meaning ul
and
wo hwhile
o
know
unde
which
ci cums ances
8
is
a
compac
ope a o ,
wi h
espec
o
he
no m
o
a
weake
opology
.
He e,
we
ask
when
S
is
weakly compac , ha
is,
when
does
S
map
he
uni
ball
o
A
in o
a
subse
whose
closu e
is
compac
wi h
espec
o
he
weak
opology
on
A
.
(This
is
mo e
closely
ela ed
o
he
poin
o
iew
aken
in
his
pape
han
he
no m
compac
case,
which,
howe e ,
can
be
ea ed
simila ly
.)
Specialize
o
he
case
A=
B(H),
he
algeb a
o
all
bounded
linea
ope a o s
on
some
Hilbe
space
H
.
Since
B(H)
is
he second
dual
o
K(H),
he
closed
ideal o
all
compac
ope a o s
on
H,
and
S
is
con inuous
wi h
espec
o
he
Q(B(H),K(H)*)- opology,
S
coincides
wi h
(S
j
)**,
he
second
adjoin
o
he
es ic ion
81
o S o
K(H)
.
I
is
well
known
ha
6
1
is
weakly
compac
i
and
only
i
(81)**
maps
K(H)**
in o
K(H)
[15,
VI
.4
.2]
.
Mo eo e ,
by
Gan mache 's
heo em
[15,
VI
.4
.8],
61
is
weakly
compac
i
and
only
i
(ó1)**
is
weakly
compac
.
Pu ing
all
his
oge he
yields
ha
8
=
(S1)**
is
weakly
compac
i
and
only
i
SB(H)
C
K(H)
.
In
he
gene al
case
we
ha e
o
eplace
K(H)
by
he
ideal
K(A)
o
all
compac
elemen s
in
A
;
and,
using
app op ia e
ep esen a ions,
we
ob ain
he
ollowing,
c
.
[23,
Theo em
2
.7]
.
HOW
TO
SOLVE
AN
OPERATOR
EQUATION

74
5
P oposi ion
1
.
A
de i a ion
S
on
a
C*-algeb a
A
is
2ueakly
compac
i
and
only
i
b**A**
C
K(A)
.
Again,
K(A)
is
b-in a ian
and
hus
b
induces
a
de i a ion
b
en
he
gene alized
Calkin
algeb a
A/K(A)
.
Co olla y
2
.
I b
is
weakly
compac , hen b
=
0
.
Suppose
b
we e
inne ,
Le
.
b
=
b
.,
whe e
6,(x)
=
xa
-
ax,
and
he
elemen
a
belonged
o
K(A)
.
Then, b
is
weakly
compac by
[41,
The-
o em
3
.1]
.
On
he
o he
hand, b**
is
always
inne
by
Sakai's
heo em
.
The e o e, he
o iginal
ques ion
o
weak
compac ness
o b
leads
o
he
ollowing
ope a o
equa ion
.
(1
.1)

Can
&
=
0 be
sol ed
in
K(A)?
Going
one
s op
u he
we can
ask
a
simila
ques ion
o
he
p oduc
6162
o
wo
de i a ions
51,
62
on
A
(which
;
in
gene al,
is
no
longe
a
de i a ion)
:
when
is
5162
(weakly)
compac ?
This
ques ion should be
e-
la ed
o
11e
Dun o d-Pe is
p ope y
o a
commu a i e
C*-algeb a
which
implies
ha
T1T2
is
a
compac
ope a o
whene e
T1,
T
2
a e
weakly
com-
pac on
A
.
By
simila
a gumen s
as abo e,
i
can
be
o mula ed
in e cos
o
ope a o
equa ions
as
ollows
.
(1
.2)

Can
&
i
b,
12
=
0
be
sol ed
in
K(A)?
Ques ions
o his
kind
a e
s udied
in
[25]
and
[27]
.
2
.2
.
Comple ely
posi i e
ope a o s
.
Recall
ha
a
linea
mapping
T
on
a
C*-algeb a
A
is
said
o be
com-
ple ely
bounded
i
he no nls
JIT,,11
o
he canonical
ex ensions
T,,
o
T
o
he
ma ix
algeb as
Al,,(A)
o e
A
a e
all
bounded
by
some
eal
numbe ,
and
T
is
comple ely
posi i o
i all T,,
a e
posi i o
ope a o s
on
M

(A)
.
The
p o o ypes
o
comple ely
bounded
ope a o s
a e
11e
eleme a a y
op-
e a o s
gi en
conc e ely
as
mappings
o
he
o m
5
:x~-+ xb
j
wi h
x
E
A,
a1,
. . . ,
a,
l
,
b1,
.
.
. ;
b,
L
E
1Vl(A),
whe e
A11(A)
deno es
11e
mul iplie
algeb a,
o
A
.
This
is
jus i ied
by
he
ep esen a ion
heo em
o
comple ely
bounded
ope a o s
and
he
ac
ha ce ain
comple ely
bounded
ope a o s
can
be
app oxima ed
746
.
M
.
MAT1-HEU
by
elemen a y
ope a o s,
c
.
[12]
.
A
na u al
ques ion
in
his
con ex
is
:
wha
does
a
comple ely
posi i e
elemen a y
ope a o
S
look
like?
Al hough
his
is
in ol ing
ineguali ies,
we
immedia ely
a e
led
o
an
ope a o
equa ion
.
Deno e by
M
a
,b
he
( ino-sided)
mul iplica ion
x
>->
axb
.
I
S
=
1
1Vl
a
~,b~
is
posi i e,
i is
he mi ian-p ese ing
om
which
111
l
b -'
a
AI
a

b~
j-1
ollows-
As
a esul
we
a e o
conside
he
ollowing
ope a o equa ion
.
(1
.3)

Which
elemen s
.x
j
,
y
j
EM(A)
sol e
Z
:"
1
A
j
,
yj
=
0?
This
ques ion
has
eme ged
o
be
no
only an
example,
bu
o
unda-
men al
signi ican e
o
ou
app oach,
c
.
[28]
.
2
.3
.
Cen alizing
mappings
.
Le
R
be a
ing
.
An
addi i e
mapping
F
:
R
->
R
is
cen alizing
i ,
o
e e y
.x
E R, we ha e
[x,
F(x)]
=
xF(x)
-
F(x)x
E
Z(R),
he cen e
o
R
.
In
many
cases
;
he
exis en e
o
ce ain
cen alizing
mappings
yie1ds
commu a i i y
c i e ia
o
R
.
Fo
example,
i
R
is
a
p ime
ing
;
hen
R,
is
commu a i e
i
he e
is
a
non-ze o
cen alizing
de i a ion
on
R
[38,
Theo em
2],
see
also
[30],
o
i
he e
is
a
non-iden ical
cen alizing
au omo phism
on
R
[31,
Theo em]
.
In
he
con ex
o
ope a o
algebas,
he e
a e
analogues
o
he e
esul s as
ollows
.
P oposi ion
3
.
The e
is
no
non-ze o
cen 7-alizing
de i a ion
on a
C*-algeb a
.
This
seems
o
be a
olklo e
ex ension
o
Singe 's
classical
esul
ha
he e
a e
no
non-ze o
de i a ions
on
commu a i e
C*-algebas
.
In
ac ,
i
b
is
a
cen alizing
de i a ion
on
a
C*-algeb a
A,
i
easily
ollows
ha
bA
C_
Z(A)
.
Hence,
he
es ic ion
6
1
o
S o
Z(A)
anishes
so ha
62
=
0
.
The
iden i y
2
b(x)y5(x)
=
6
2
(Zyx)
-
xb
2
(yx)
-
b
2
(xy)x
+
xb
2
(y)x

(x,
y
EA)
he e o e
yields
Aló(x),a(x)
=
0
o
all
x E
A,
whence
b
=
0
.
The
case
o
au omo phisms
equi es
some mo e wo k
and
was
i s
s udied
by
Mie s
.
o
equi alen ly,
o
equi alen ly,
HOW
TO
SOLVE
AN
OPERATOR
EQUATION

74
7
P oposi ion
4
.
[32,
Theo em
5]
Le
ce
be
a
cen alizi ~g
*-
au omo phism
on
a
on
Neumann
algeb a
A
.
The e
is
a
cen al p ojec-
ion
e
E
A
such
ha
a(e)
=
e,
aJA,
=
idA,
and
A(1-e)
is
commu a i e
.
Whe he
his esul
emains
ue
o
a bi a y
(no
necessa ily
*-
p ese ing)
au omo phisms
was
answe ed
only
ecen ly
by
B esa ,
who
also
ob ained
a
gene al
s uc u e
heo em
o
cen alizing
mappings
en
on
Neumann
algeb as
as
ollows
.
P oposi ion
5
.
[8,
Theo em
2
.1]
Le
F
be a cen alizing
addi i e
mapping
on
a
on
Neumann
algeb a
A
.
Then
he e
exis
an
elemen
c
E
Z(A)
and an
addi i e
mapping
(
:
A
--->
Z(A)
such
ha
F=
L,
+
(
.
He e
and
in
he
sequel,
we
will
deno e
by L,
he
le
mul iplica ion
x
-->
ax
and
by
R
.
a
he
igh
mul iplica ion
x
H
xa
.
We
will
now
e o mula e
bo h
he
assump ion
as
well
as
he
conclusion
in
e ms
o
ope a o
equa ions
.
This
will
enable us
o
ob ain an
ex ension
o
B esa 's
esul o
a bi a y
C*-algeb as
in
Sec ion
4
.
Obse e
a
i s
ha
e e y
cen alizing
addi i e
mapping
F
on a
C*-
algeb a
A
is
in ac
commu ing,
Le
.
[x,
F(x)]
=
0
o
all
x E
A
[9,
P oposi ion
3
.1]
.
Replacing
xby x
+
y
he e o e gi es
[x,
F(y)]
+
[y,
F(x)]
=
0

(x,
y
E A)
(2)

6F(y)
-
6
y
F
=
0

o
all
y
E
A
.
Secondly,
i
F=
L,
+
~
whe e
A
is
a
C*-subalgeb a
o a
C*-algeb a
B
wi h
cen aliza
C,3
(A), c
E
CB(A)
and ~
:
A
.
->
CB(A),
hen
[x,
F(y)]
_
[x,
cy]
+
[x,
«y)]
=
[x, cy]
o
all
x,
yE
A
.
Hence
[x,
F(y)
-
cy]
=
0

(x,
y E A)
5F(y)-cy
=
0

o
all
yE
A
.
Con e sely,
i
c
E
CB(A)
sa is ies
(3),
hen
~
=
F-L,
de ines
an
addi i e
mapping
om
A
in o
CB(A)
.
As
a esul
we
a i e a
he ollowing
ques ion
.
(1
.4)
Suppose
ha
F
sa is ies (2) o
all
yE
A
.
Is
he e
an
elemen
c
E
CB(A)
o
a
`sui able'
C*-algeb a
B
con aining
A
sa is ying
(3) o
all
y
E
A?
No e
ha
(3)
p ecisely
is
a
sys em
o
ope a o
equa ions
o
he
o o
(1)
pa ame ized
by
all
elemen s
in
A
.

74
8

M
.
MATHIEU
2
.4
.
Gene a o s
o
dynamical
semig oups
.
Le
A
be a
uni al
C*-algeb a
.
A
bounded
he mi ian-p ese ing
linea
ope a o
L
:
A
->
A
wi h L(1)
=
0
is
called
comple ely
dissipa i e
i ,
o
allnEN,
These
ope a o s
a e he
gene a o s
o
no m-con inuous
one-pa ame e
semig oups
(T ) ER
+
o
uni al
comple ely
posi i e
ope a o s
T,,
en
A
;
which
desc ibe
he
i e e sible
dynamics
o
open
quan um
sys ems,
o ,
equi alen ly,
se e
as
ansi ion
ope a o s
o
non-commu a i e
Ma ko
p ocesses
.
In
many
conc e e
si ua ions,
hey
a e
buil
om wo
p o o-
ypes
:
he
comple ely
posi i e
ope a o s
and
he
he mi ian-p ese ing
gene alized
inne
de i a ions
Sk
,
k
.
=
R
.k
+
Lk
.
.
The
con e se
ques ion,
when
a
gi en
comple ely
dissipa i e
ope a o
L
can
be
decomposed
in o
L,(x*x)
?
x
*
L,,(x)
+L
,
,(x
* )
x

(x
E
M,,
(A»
.
a e
wo
decomposi ions
.
Then,
pu ing
a
=
k1
-
k
2
,
we
ha e
(
5
)

S
.,a-+01-02=0
.,
Thus,
we
may
ask
(1
.5)

Unde
which
condi ions
does
(5)
imply
ha

0?
A
mo e
gene al
ques ion
would
be
which
a
in
A"
sol e
he
equa ion
wi h
0
comple ely
posi i e
om
A
in o
some
possibly
la ge
C*-algeb a
B
and
k
E
B
was
i s
s udied
by
Co ini,
Kossakowski
and Suda shan
[18]
and
Lindblad
[21]
;
and
ela ed
o
cohomological
p ope ies
o
A
in
[22]
and
[11]
.
I
A
C_
B(H),
hen
a
decomposi ion
(4)
o
L
always
exis s
wi h
OA
CA"
and
k E
A"
.
In
gene al
;
his
decomposi ion
will
no
be
unique
.
The
uniqueness
p oblem
can
be
e o mula ed
in
e ms
o
an
ope a o
equa ion
as
ollows
.
Suppose
ha
L
=
01
+
4,,k,
=
02
+
8kz,kz
3
.
De ices
All he
abo e
equa ions
(1
.1)
h ough
(1
.5)
can
be
subsumed
unde
he
gene al
o m
(1)
.
To
mo i a e
ou
ools o
sol ing
hem
; .
le
us
How
o
SOLVE
AN
OP-ERATOR
.
EQUATION

74
9
u he mo e
conside
a
special
case
o
(1
.3)
.
Le
A = B(H)
and
b
E
A
be
gi e
.
(1
.3')

Which
a
E
A
solee
AI,,b
=
0?
In
ou
pa icula
si ua ion,
he
answe
is
quickly
eaclied
.
I L1I a, b
=
0,
hen
axb~
=
0
o
all
x E
A
asid
1
E
H
.
I
b
=
0,
ob iously
all
a E
A
a e
solu ions
.
I
b
7~
0
;
pick
~
E
II
wi h
b~ :7~ 0 and
no e
ha
b~
is
cyclic
o
A,
i
.e
.
Ab~
=
H,
and
hus
a
=
0
.
Clea ly, his
me hod
only
wo ks
in
he
p esence
o
a
Hilbe
space
on
which
A
ac s' ansi i ely
eno gh', e
.g
.
i
A
is
i educible
.
The
algeb aic
me hod
p esen ed
now
wo ks
wi hou
unde lying space
.
I
is
con enien
o
eph ase
(1
.3)
using
he ollowing
concep
.
Fo
e e y
C*-algeb a
A
we
le
ú(A)
be
he
algeb a
o
all
ele nen a y
ope a o s
on
A
.
We
de ine
a
su jec i e
algeb a
llo o no phism
(6)

0
:
M(A)
®
A11(A)"1>
--,
ú(A),

B(a
(9 b)
=
111,,b
whe e
A11(A)
®
AJ(A)"
deno es
he algeb aic enso
p oduc
o
M(A)
wi h
i s
opposi e
algeb a
.
The
p oble n
now
is
o
de e mine
hc ke nel
o
0
.
The
ollowing
was
p e ed
in [24,
Pa
1
;
Co olla y
4
.4]
.
(7)

0
is
injec i e
i
and
onlg
i
A
is
p ime
.
Since
p imi i e
C*-algeb a5
a e p ime,
i
is
emp ing
e
use ep esen-
a ion
heo y
in
o de
o
app oach
hc
gene al case
o n
he
special
one
.
Howe e ,
as
i
eme ged,
he e
may
be
p oblems
in
pu ing
he
'local'
in o ma ion
oge he
o
ob ain
a,
'global'
pic u e
.
I ,
seems
ad a, a,geous
o
iew
he
p ime
C*-algebas
as
he
building
blocks,
which
esul s in
ega ding
a
C*-algeb a
as a
se nip i ne
algeb a
a he
i an
a
se-misim,ple
one
.
In
ac ,
simila
ecliniques
a d
esul s as
hose
desc ibed
below
a e
a ailable
in
he
se ing
o
se nip i ne
ings
.
The
ideal
s uc u e
o a
p ime
algeb a
is
dis inguislied
by
he
ac
ha
e e
y
non-ze o
ideal
is
essen ial,
Le
.
in e sec s
cae]
- )
o he
non-
ze o
ideal
non- i ially
.
This
allows
o
"nie e
a
. ound
o n
one
place
o
ano he '''
wi hin
he
C*-algeb a wi hou
loss
o
in o ma ion
.
Fo
an
a bi a y
C*-algeb a
.
A
we
he e o e
deno e
by
1,
and
1,
he
collec ions
o
all
essen ial
and
all
closed
essen ial ideals o
A,
espec i ely
.
No e
ha
he e
a e
di ec ed
dow wa ds
by
inclusion,
Le
.
1
7
12
E
1,
i nplies
ha
1
1
nI
2
E1,
.
Fo
e e y
se nip i ne
ing
R,,
he
nul iplie
ing
AI(R)
is
de ined
by
i s
uni e sal
p ope y
ha
Id
is
an
essen ial
ideal
in
M(R)
and
he e
is
75
0

M
.
MATHIEU
a
unique
ex ension
p
o
he
inclusion
p
:
R
-4
M(R)
which
makes
he
ollowing
diag am
commu a i e
;
whene e
R
is
an
ideal in
ano he
ing
S,
R

--'->

M
(R)
in
o he
wo ds,
M(R)
is
he
(abs ac )
idealize
o
R
.
Usually,
AII(R)
is
cons uc ed
ia
double
cen alize s
o
R
.
Mo eo e ,
p
is
injec i e
i
and
only
i
R
is
essen ial in
S
.
Now,
i
I,
J
E
1,
and
J
C_
I,
hen
J
will
be
an
essen ial ideal in
M(I)
whence,
om
he
abo e,
he e
is
a
unique
injec i e
*-homomo phism
p»
:
M(I)
->
M(J)
making
he ollowing
diag am
commu a i e
J
pes,
M(J)
We
may
desc ibe
pjj
as " es ic ing
he
double
cen alize s"
.
By
means
o
his,
we
ob ain
a
di ec ed
sys em
{M(I)
;
p j,
J
C
I} o
C*-algebas
and
inclusions,
and
i s
algeb aic
di ec
limi
alglim
M(I)
along
1
.,
will
be
deno ed
by
Qb(A)
and
called
he
bounded
symme ic
algeb a
o
quo-
ien s
o
A
.
This
is
a
p e-C*-algeb a
wi h
comple ion
Qb(A)
-
=
lim
M(I)
deno ed
hence o h
by
MI,,(A)
and
called
he
local
mul iplie
algeb a
o
A
.
Fo
each
I E
1
e
le
P(I)
deno e
he
Pede sen
ideal o
I
[37,
5
.6]
.
Using
he
ac
ha
P(I)
is
*-in a ian ,
belongs
o
1,
and
ha
P(I)P(J)
=
P(I)
1
P(J)
o
all
I,
.I
E
1,
we
de ine
Q,
(A)
=
alglim
M(P(I))
along
1
e
and
obse e
ha
his de ini ion
leads
o
he
symme ic
algeb a
o quo-
ien s
o
A
as
de ined
(sligh ly
di e en ly)
in
ing
heo y
.
l
ollows
ha
Qb(A)
embeds
as
a
*-subalgeb a
in o
Q
.,(A)
and
is
in
ac
he
bounded
pa
o
Q,(A)
[2,
Theo em
1 .3]
.
A
s onge
ela ion
be ween
Qb(A)
and
Q,
(A)
p o ed
in
[3,
Theo em
2] is
ha
Q,
(A)
is
he
cen al
localiza ion
o
Qb(A)
.
Rema ks
.
The
cons uc ion
o
M
1o
,(A)
was
i s
pe o med
by
Ped-
e sen
[361
and
Ellio
[16]
unde
he
name
o
essen ial
mul iplie s
.
They
used
i
o
s udy
ope a o
equa ions
o
he
o m
5
=
ba,

aE
Mlo,
:
(A),
and
How
To
SOLVE
AN
OPERATOR
EQUATION

75
1
a
=
AJ
.,
i
.,

u,
E
Ah
o
,(A)
uni a y,
ha
is,
o ob ain
inne ness
o
de i a ions
8
and
*-au o no phis as
a
in
A4h
o
,(A)
.
In
pa icula ,
Pede sen
p o ed
ha
(8)
always has
a
solu ion
i
A
is
sepa able
[36,
P oposi ion
2]
.
A
abou
he
sa ne
i ne,
Kha chenko
in oduced
he
sy mne ic
ing
o
quo ien s
o
semip ime
ings
and
used
i
in
pa icula
in
Galois
he-
o y
[19]
;
[20]
.
This
heme was
u he
pu sued
by
Pass nan
[34], [35],
Mon goine y
[33],
and
o he s
.
I
is
o
be
seen
in a
long
adi ion
go-
ing
back
o
l e
30's in
in es iga ing
gene al
ings
o
quo ien s,
c
.
[40]
.
The
basic
idea
-
o
enla ge
a
gi en
'domain'
by
addi ional
'nunnbe s'
(=' ac ions', 'quo ien s') in
o de
o
be able o
sol e
mo e
equa ions
-
also
se es
as he
mo i a
. ion
o
ou
app oa,ch
o
ope a o
equa ions
.
In
he
la e
80's,
M,
o
,(A)
was
edisco e ed
independen ly
by
A a
[2],
[3]
a ld
he
au ho
[26],
[29]
which
hen
launched
a
join
esea ch p ojec
on
he s uc u e
and
applica ions
o local
mul iplie s
[4],
[5],
[6]
;
a
co n-
p ehe isi e
accoun
o
his
is
o
be
gi en
in
[7]
.
We
will
now
compile
come
o
he
basic
p ope ies
o
Ah
o
,(A)
.
P oposi ion
6
.
Le ,
A
be
a
C*-algeb a
uwi h
local
nul iplie
algeb a
Ai1,,
:
(A)
.
(i)
A
is
commu a i e
i
and
only
i
A1h,,(A)
i
.s
com nu a i e
.
(ii)
A
is
p ime
i
and
only
i
M1,(A)
has
i ial,
cen e
.
(iii)
Fo
each
I
E
Z,
and
each
uni iza ion
B
o
A
we
ha e
A z
.,
;([)
=
A4" ,,
.(A)
=
NI,~
.(B)
.
(i )
Le ,
l
be
he
p imi i e
spec um
o
A
.

I
A
is
disc e e,
hen
H,,,,(A)
=
A4-(A)
.
( )
I
A
is
an
AW*-al
g
geb a,,
hen
Ah
o
,(A)
=A
.
om
(7)
and
(ii)
in
he
abo e
p oposi ion
we
see
ha
he ke nel
o 0
is
closely
ela ed
o
he
cen e
Z =
Z(A1h,,(A))
o
1VI
lo
,(A)
.
I
is
he e o e
impo an
o
analyse
i s
s uc u e
.
The
ollowing
was
p o ed
in
[5,
Theo em
1
and
Co olla y
1]
and
can
be
iewed
as
a
local
e sion
o
he
well-known
Dauns-Ho inann
heo em
iden i ying
he cen e
Z(111(A))
o
NI
(A)
wi h
he
algeb a,
C(3Á)
o
all
con inuous
complex- alued
unc ions
on
he
S one-Cech
compac i ica ion
3 l
o
l
.
P oposi ion
7
.
Fo
e e y C*-algeb a A,
he
cen e
Z
o
A1h,
(A)
is
an
AW*-algeb a
and
can
be
i,den i ied
wi h C(
li
E
n1
0Í),
whe e
he
in e se
758

,
,
.
;M
.-,MATHIEU
5
.
Conclusion
We
hope
ha
he- esul s
desc ibed
abó e
may
gi e
some
e iden e
ha
he
local
mul iplie
álgeb a
can
se e
as
a
`uni e sé',
in
which
ope a o
equa ions
on
C*-algeb as,
a
leas
hose
o
he
o o
(1),``can~be
sol ed'
by a
uni ied
me hod
.
Re e en es
l
.
._
C
. .
A
.,AKEMANN
.ANDS
.
WRIGHT,
Compác
and
weakly-compac
de i a ions
o

C*-algeb as,
'Paci ic
J
.
Ma h
.
85
(1979),
253-259
.
2
.

P
.
ARA,
The
ex ended
cen oid
o
'C*-algeb as
;
A ch
.
.=Ma h
.
1
54
(1990)
1
358-364
.
3
.

P
.
ARA
;
On
he'
sy nmé ' ic
álgeb a
'b
qúo ien s
o
a
C*-algeb a,
Clasgow,
Ma h
.
J
.
32
.
(1990),
3777379-
4
.

P
.
ARA
AND
.
M
.
MATIAIEU,
On
ul ap ime
Banach
algeb as wi h
non-zé d
soéle'P oc
.
R
. .
.I
:-
Acad~
91Á
(1991),'
7
89-98
.

`
5
.

P
.
ARA
AND
=M
. .
MAT
HIEU,
A
local
e sion
o
he
Dauns~Ho mann
heo em,
Ma h
.
Z
.
208
(1991),
349-353
.
6
: .
P
- .
ARA
AND
M
.
'MATHI£U,
An
applica ion
o
local
mul iplie s
-
o
cen aliziiig
. imappings-on
.C*-algeb as
;
Qua
.
.J
.
.Ma h
.,
Ox o d,
o
appea
.
7
.

P
.
ARA
AND
M
.
MATHIEU,
Local
mul iplie s
o C*-algeb as,
in
p epa a ion
:
', .

1

',
8
.

M
.
BRE§AR,
-Cen alizing
,
mappings
ón,
on,
Neumann
algeb as,
P oc
.
Amen
Ma h
.
Soc
.
111
:(1991),,501-510
.

-

,
.
9
.

M
.
BRESAR,
Cen alizing
mappings
and
de i a ions
in
p ime
ings,
J
.
Algeb a
;
o
appea
. .
7
:
,
'
10
.
A
.
CHATTERJEE
.
AND
R
.
R
.
SMITH,
The
cen al
Haage up
enso
p ód'ú~
and
náps
be wéén
' ón'
1Veümánn
algeb as,'
J
.
Func
.
Anal
.
.,
,

o
-appeá
..

.

.

,
.

.

,

,

. ;

.,
11
.
E
.
CHRISTENSEN
ÁND
D
.'
E
- .
EVAÑs,
Coho nology
o
ope á o
alge-
b as
and
quan um
dynamical
semig oups,
J
.
London
Ma h
.
Soc
.
20
12
.

E
.
CHRISTENSEN
AND
A
.
M
.
SINCLAIR,
A
su ey
`o
.
comple ely
bounded
ope a o s,
Ball
.
L
.ondon
Ma h
.
Soc
.
21
(1980),'417-448'
.
13
.
-E
.
B
.
-
DA i
;ES,
.-Uniqueness
o
; he
s anda d
.
o m
o£
he,
gene a o
o
a
quan um-dynamical
.semig oup,
Rep
.
Ma h
:
Yhys
.-
17
(1980)
;
249-255
.

,
,
.
- .-

HO«
o
SOLVE
AN
OPER,ATOR
.
EQUATION

759
14
.
D
.
DECKART
AND
C
.
PEARCY,
On
ma ices
o e
he
ing
o
con-
inuous
complex
alued
unc ions
en a
S onian
space,
P oc
.
Ame
.
Ma h
.
Soc
.
1
4
(1963),
322-328
.
15
.
N
.
DUNFORD
AND
J
.
T
.
SCI-IWARTz,
"Linea
ope a o s,"
Pa
1,
In e science,
New
Yo k,
1958
.
16
.
G
.
A
.
ELLIOTT,
Au omo phisms
de e mined
by
mul iplie s
on
ide-
als
o
a
C*-algeb a
;
J
.
Func
.
Anal
.
23
(1976),
1-10
.
17
.
T
.
TI2
.EIBERCER
.
AN
D
M
.
MATHIEU,
Uniqueness
o
a
Lindblad
de-
composi ion,
in
P oc
.
In
.
Wo kshop,
on
"Elemen a y
Ope a o s
and
Applica ions",
Blaubeu en,
June
1991,
Wo ld
Scien i ic,
Singapo e,
1992,
pp
.
179-187
.
18
.
V
.
GORINI,
A
.
KOSSAKOWSKI
AND
E
.
C
.
G
.
SUDARSHAN,
COm-
ple ely
posi i e
dynamical
semig oups
ci
N-le el
sys ems,
J
.
Ma h
.
Phys
.
17
(
.1976),
821-825
.
19
.
V
.
K
.
KHARCHENKO,
Gene alized
iden i ies
wi h
au omo phisms,
Algeb a
i
Logika
14
(1975),
215-237,
English
ansl
.
(1976),
132-148
.
20
.
V
.
K
.
KHARCHENKO,
Galois
heo y
o
semip ime
ings,
Algeb a
i
Loyika
16
(1977)
;
313-363,
English
ansl
.
(1978),
208-258
.
21
.
G
.
LINDBLAD,On
he
gene a o s
o
quan um
dynamical
se ni-
g oups,
Com mcn
.
Ma h
.
Phys
.
48
(1976)
;
119-130
.
22
.
G
.
LIND13LAD,
Dissipa i e
ope a o s
and
cohomology
o
ope a o
algeb as,
Le
.
Ma h
.
Phys
.
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