Publicacions
Ma e ná iques,
Vol
36
(1992),
925-954
.
Abs ac
ONE-SIDED
DIVISION
ABSOLUTE
VALUED
ALGEBRAS
ANGEL
RODRÍGUEZ
PALACIOS
We
de elop
a
s uc u e
heo y
o
le
di ision
absolu e
alued
algeb as
which
shows,
among
o he
hings,
ha
he
no m
o such
an
algeb a
comes
om an
inne
p oduc
.
Mo eo e ,
we
p e e
he
exis ence
o
le
di ision
comple e
absolu e
alued
algeb as
wi h
Ie
uni o
a bi a y
in ini e
hilbe ian
dimension
and
wi h
he
addi ional
p ope y
ha
hey na e no nonze o
p ope
closed
le
ideals
.
Ou
cons uc ion
in ol cs
esul s
om
he
ep esen a ion
heo y
o he
so
called
"Canonical
An icommu a ion
Rela ions"
in
Quan um
Mechanics
.
We
also
show
ha
homomo phisms
om
comple e
no med
algeb as
in o
a bi a y
absolu e
alued
algeb as
a e
con ac i e,
hence
au oma ically
con inuous
.
In
memo iam
Es e
a ículo
ha
sido
esc i o
en
homenaje
a
Pe e
Menal
i
B u al
.
Acaso
las
ma emá icas no
sean
el
medio
más
ap opiado pa a
exp esa
el
dolo
po
la
mue e
de
es e
g an
amigo
así
co no
la
ad ni ación
que
su
abajo
me
me ecía
.
Las
ma emá icas
no obs an e
p opicia on
una
bella
amis ad
y
ue on
c edo
y
lenguaje
común
en e
noso os
.
0
.
In oduc ion
A
s ill
unsol ed
old
ques ion
is
ha
o
he
nonassocia i e ex ension
o
he
Gel and-Maza
heo em,
namely
i
any
di ision
no med
(nonassocia-
i e)
algeb a
mus
be
ini o-dimensional
(which
would
imply
dimension
1
in
he
co nplex
case,
and
1, 2,
4,
8
in
he
eal
one,
by a
heo em
o
R
.
Bo
and
J
.
Milno
[6])
.
This
p oblem
was
explici ly
poned
by
F
.
B
.
W igh
[27]
in
1953,
who
in
he
same
pape
ga e
a pa ial
a i ma i e
answe
p o ing
ha
di ision
absolu e
alued
algeb as
a e
ini o-dimensional
.
Ano he
olklo e pa ial
posi i e
esul
abou
his
ques ion
is
ha one-
sided
di ision
comple e
no med
co nplex algeb as
a e
iso no phic
o
he
926

A
.
RODRÍGUEZ
PALACIOS
complex
ield
(sea
[14]),
he
case
o
( wo-sided)
di ision
noncomple e
no med
complex
algeb as
as
well as
ha
o
di ision
(o en
comple e)
no med
eal
algeb as
emaining
open
.
ln
he
con e se
di ec ion, J
.
A
.
Cuenca
[101
has
ecen ly
gi en
examples
o
in ini a-dimensional
one-sided
di ision
absolu e alued algeb as o a
he
ield
o eal
numbe s
.
Since
he
ac
ha
one-sided
di ision
absolu e alued
complex
algeb as
a e
isomo phic
o
he
complex
ield
can
also
be
conside ad
as olklo e
(sea
P oposi ion
2
in his
papa )
;
i
sééms
o
be
easonable
o
look
o
a
s uc u e
heo y
o
a bi a y
one-sided
di ision
absolu e alued
eal
algeb as,
and
in
ac
o
p o ide
such
a
s uc u e
heo y
is
he
main
pu pose
o his
pape
.
To
unde s and
he
pliilosophy
o
ou
wo k,
i is
sui able
o
ake
in o
accoun
ha ,
easily,
le
di ision
absolu e
alued
algeb as
a e
"iso opic"
o
le
di ision
absolu e
alued
algeb as
wi h
a
le
uni
(sea
P oposi ion
4)
.
So
he
.
a en ion
mus
be cen e ed
on
hese
las
algeb as
;
and
hen
he
se
(say
P)
o
all
le
mul iplica ion
ope a o s
on
such
an
algeb a
is
a
subspace
o
bounded
linea
ope a o s
on
he
no mad
space
(say
X)
o
he
algeb a
con aining
he
iden i y
ope a o
and
sa is ying
(*)

11
T(x)
11=11
T
1111
x
11
o
all
T
in
P
and x
in
X
.
lNe
isola e
his
in o ma ion
in
he
i s
sec ion
o
he
pape ,
and
we
p o e
in
Theo em
1 ha ,
i
P
is
a
subspace
o
bounded
linea
ope a o s
on any
eal
no med
space
X
con aining
he
iden i y
ope a o
(say
I)
and
sa is ying
(*),
hen
he
ope a o
no m
on
P
de i es
om
an
inne
p oduc
and
;
o
T
in
P
o hogonal
o
I,
he
equali y
T
2
=-
IIT11
2
I
holds
.
l
ollows
ha
e e y
elemen
in
P
is
an
in e ible
ope a o
on
X
so,
as
a
i s
consequence,
absolu e alued
algeb as
wi h
le
uni
a e
au oma ically
le
di ision
algeb as
.
The
con en
o
he
abo e
e e ed
Theo em
1 is
s a ed
in
he pa-
pe
in
an
equi alen ly- e o mula ed
o m
in ol ing
ce ain
p ehilbe ian
quad a ic
di ision-Jo dan
algeb as
wi h
in olu ion
which,
cu iously,
ha e
been
shown
in
[20]
o
be
he
only
"smoo h
no mad"
nonassocia i e
com-
mu a i e
algeb as
.
F om
his
e sion
o
Theo em
1
we
de i e
in
Sec ion
2
he
main
esul
in
he
pápe
(sea
Theo em
2)
asse ing
ha
"uni al
*-
ep esen a ions"
o
hese
Jo dan
algeb as
on
hei
own
p e-Hilbe
spaces
na u ally
gi e
ise
o
(au oma ically
le
di ision)
absolu e
alued
alge-
b as
wi h
le
uni ,
and
ha
by
his
cons uc i a
me hod
all
absolu e
alued
algeb as
wi h
le
uni
a ise
.
This
las
asse ion
in
Theo em
2
ONE-SIDED
DIVISION
ABSOLUTE
VALUED
ALGEBRAS

927
means
ha
he
no m
o
an
absolu e
alued
algeb a
A
wi h
le
uni
e
comes
om
an
i me
p oduc
(
. 1
.)
sa is ying
(ab
1
c)
=
-(b
1
ac)
and
a(ab)
=
-
11
a
112
b
o
all
a,
b,
c
in
A
wi h
a
o hogonal
o e
.
In
he
hi d
sec ion
we
use
Theo eni
2,
oge he
wi h
he
basic
ac s
abou
he
so-callad
"canonical
an icommu a ion
ela ions"
o
quan um
mechanics
(which
he eade
nay
ind
in
he
i s
pagos
o
[8]),
o
p o e
in
Theo em
3
he
exis en e
o
comple e
absolu e
alued
algeb as
wi h
le
uni
o
a bi a y
in ini e
hilbe ian
dimension,
and
ha
such
al-
.geb as
can be
cliosen
wi h
he
addi ional
p ope y
ha hey
ha e
no
nonze o
p ope
closed
le
ideals
(no e
ha
e e y
le
di ision
algeb a
has
no
non
ze o
p ope
igh
ideals)
.
Since
small
pe u ba ions o
he
p oduc
o
a
le
di ision
comple e
absolu e alued
algeb a
gi e
ise
o
new
le
di ision
algeb as
(P oposi ion
7),
he exis en e
o
a
wide
col-
lec ion o
in ini o-dimensional
le
di ision
non-absolu e- alued
comple e
no mad
algeb as
is
asu ad
.
Le
Lis
emphasize
also
he
exis en e
o
in ini e-
dimensional
comple e
no med
algeb as
on which
he
ope a o s
o
le
and
igh
mul iplica ion
by
any
nonze o
elemen
a e
su jec i e
(a
consequence
o
P oposi ion
8)
.
The
concluding
sec ion
o
he
pape
(Sec ion
4)
is
de o ed
o
p o e
he
au oma ic
con inui y
o
homomo phisms
om
comple e
no mad
algeb as
in o
absolu o
alued
algeb as
.
Mo e
pe ecisely,
we
show
ha such
ho-
momo phisms
a e
con ac i e
.
TI
-
lis is
pe haps
he
i s
non i ial
esul
in ol ing
a bi a y
absolu e
alued
algeb as
.
Tho igh
he
axiom
11
ab
11=11
a
11 11
b
11
seems
o
be
e y
na u al,
he
s udy
o
absolu o
alued
algeb as
has
ecei ed
he
a en ion
o
a
ela i ely
slnall
m
.imbe
o
au ho s
.
Thus
we
a e
only
awa e
o
he
pape s
(ci ad
c onologically)
by
A
.
A
.
Albe
[1]
(1947)
and
[2],
F
.
B
.
W igh
[27],
K
.
U banik
and
F
.
B
.
W igh
[26],
K
.
U banik
[24]
and
[25],
M
.
L
.
El-Mallan
and
A
.
Micali
[19],
and
M
.
L
.
El-Mallah
[15],
[16],
[17]
and
[18]
(1990)
.
We
emphasize
he
esul
in
[26]
(1960)
asse ing
ha
R,
C,
o--0,
and
®
a e he
only absolu o alued
algeb as
wi h
uni
(a
ac
ha ,
as
i
will
be
explained
in
Rema k
4(i),
can
be
easily
de i ad
om
Theo em
2)
.
Also
le
us
men ion
i s
easy
consequence,
p e iously
p o ed
in
[27]
and
al eady
men ioned
a
lle
beginning
o
his
in oduc ion,
ha
di ision
absolu e alued
algeb as
a e
ini o-dimensional
;
as well as
he
ele an
esul in
[19]
showing
ha
absolu o
alued
algeb as
sa is ying
he
iden i y
a(ba)
=
(ab)a
also
mus
be
ini o-dimensional
.
Examples
o
in ini o-dimensional
absolu e
alued
algeb as
we e
known
in
he
classical
li e a u a
(sea
[26],
[24]
;
[3],
and
Rema k
3(i)),
bu pone
o
he
algeb as
in
hese
Examples
a e one-sided
di ision
algeb as
.
1
hopo
ou
papen,
92
8

A
.
RODRÍGUEZ
PALACIOS
as
well
as
he
almos
simul aneous
óne
by
J
.
A
.
Cuenca
[10],
could
con ibu e
as
a
e ulsi e
o
a
subsequen
lou ishing
de elopmen
o
he
heo y
o
absolu e
alued
algeb as
.
1
.
Subspaces
o
linea
ope a o s
whose
nonze o
elemen s
a e
mul iples
o isome ies
In his
sec ion
we
will
deal
wi h
subspaces
P
o
bounded
linea
ope -
a o s
on
a
eal
no med
space
X
con aining
he
iden i y
ope a o
on
X
and
wi h
he
p ope y
ha
11
T(x)
11
=
11
T
1111
x
11
o
all
x
in
X
and
all
T
in
P
.
The
esul s
ob ained
he e
will
become
he
main
ools
o
he
p oo
in
he
nex
sec ion
o
he s uc u e
heo y
o
one-sided
di ision
absolu e alued
algeb as
.
In
ac ,
we
will
p o e
ha
he
ope a o
no m
on
subspaces
o
ope a o s
as
conside ed
abo e
de i es
om
an
inne
p oduc
.
I
will
also
u n
ou
ha
hese
subspaces
a e
ac ually
quad a ic
Jo dan
algeb as
o
ope a o s
whose
nonze o
elemen s
a e
in e ible,
so
ha
hey
co espond
wi h
some
abs ac
ma hema i-
cal
models
p e iously
.
-conside ed
in
he
answe
o
o he
p oblems
and
ha
a e
closely
ela ed
wi h
he
"canonical
an icommu a ion
ela ions"
o
quan um
mechanics
.
In
ac ,
he
possibili y
o
ep esen ing
he canon-
¡cal
an icommu a ion
ela ions
by
means
o
bounded
linea
ope a o s
on
Hilbe
spaces
will
allow
us o
p o e
in
Sec ion
3
he exis ence
o
in ini e-
dimensional
one-sided
di ision
absolu e
alued
algeb as
wi h
addi ional
p ope ies
.
We
begin
ou
a gumen
wi h
he ollowing
(pe haps
well-known)
lemma
.
Fo
a
no med
space
X,
BL(X)
will
deno e
he
associa i e
no med
algeb a
o
all
bounded
linea
ope a o s
on
X
.
Lemma
1
.
Le
X
be
a
Banach
space,
and
T
:
X
->
X
be
a
linea
isome y
which
is
no
on o
.
Then
he
open
uni
ball
i ,
BL(X)
wi h
cen e
T
and
adius
1
con ains
only
ope a o s
wi h
closed
ange
and
which
a e
one- o-one
bu
no
on o
.
P oo
:
Recall
ha
a
linea
ope a o
T
on
X
is
said
o
be
bounded
below
(by
m
>
0)
i
11
T(x)
J¡
>
m
11
x
11
o
all
x
in
X
.
Since,
o
T
in
BL(X),
T
is
one- o-one wi h
closed
ange
i
and
only
i
T
is
bounded
below,
he
lemina
will
ollow
om
he
mo e
gene al
esul ,
we
will
p o e
he e, ha ,
i
T
is
bounded
below by
in
and
is
no
on o,
hen
he
open
ball
in
BL(X)
wi h
cen e
T
and
adius
m
con ains
only
bounded
below
ONE-SIDED
DIVISION
ABSOLUTE
VALUED
ALCE13RAS

92
9
ope a o s
which
a e
no
op o
.
Easily,
o
such
a
T
and
any
S
in
BL(X
)
wi h
II
T
-
S
11
<
m
;
S
is
bounded
below
by
i-
.
II
T
-S
II .
I
a'c ually
II
T
-S
II
<
áand
S
is
su jec i e,
we
ha e
ha
S
is
in e ible
in
BL(X)
wi h
II
S
-I
II<-
1

<
2
,
so
11
T-
S
II
~<
,"z)
<II
S"
-I
II
-I
~
ni-
11
T
-S
~~

i

2
and
so
T
is
in e ible
in
BL(X)
[4,
Theo em
2
.11],
which
is
a
con a-
dic ion
.
In
his
way
we
ha e
p o ed
ha
he
open
ball
in
BL(X)
wi h
cen e
T
and
adius
z
con ains
only
bounded
below
ope a o s
wich
a e
no
op ó
.
l
ollows
ha
he
se
S2
:=
{R
E
BL(X)
:
R,
bounded
below
and
no
o o}
is
open
.
Now
he
open
ball in
BL(X)
wi h
cen e
T
and
adius
m
is
a
connec ed
opological
space
which
is
he
disjoin
union
o
i s
in e sec ions
wi h
SZ
and
wi h
he
se o in e ible
elemen s
o
BL(X)
.
Since hese
in e sec ions
a e
open and
he
i s
one
is
nonemp y,
i
ollows
ha
he
e e ed
ball
is
con ained
in
S2,
as
equi ed
.
A
his
i ase
we
ecall
some
concep s
sui able
o
a
easonably
clea
s a e nen
o
ou
main
esul in
his
sec ion
.
.A
Jo dan
.algeb a
is
a
com-
mu a i e
algeb a
(say
J)
sa is ying
he
"Jo dan
iden i y",
na nely
x2-(y
.2)
=
(x2
.y)
.X
o
all
x,
.y
in
J
.
Tl e
nos
easy
examples
o
.Jo dan
algebas
a e he
so-
called
"Jo dan
subalgeb as
o
associa i e
algebas'',
namely
subspaces
o
an
associa i e
algeb a
which
ase
closed
unde
he
"Jo dan
p oduc "
a
.b
:=
2
(ab
+
ba),
whe e
he
associa i e
p oduc
has
been
deno ed
as
usually
by
jux aposi-
ion
.
In
case
he
associa i e
algeb a
is
he
one
o
all
linea
ope a o s
on a
ec o
space
X,
such
.Jo dan
subalgeb as
a e
called
"Jo dan
algebas
o
ope a o s
on
X"
.
An
algeb a
is
said
o
be
quad a ic
i i
leas
a
uni
1
and,
o
e e y
x
in
he
algeb a
;
he e
exis
A and
l
.c
in
he
base
ield
such ha
x
2
+
x
+
l
.cl
=
0
.
A
no ned
algeb a,
is
a
eal o
co nplex
algeb a
wllose
ec o
space
is
a
'no ned
space
wi h
espec
ó
a
nomas
II
. 11
sa is ying
11
xy x
11
11
y
II
o
al]
x, y
in
he
algeb a
.
A
uni al
no ned
algeb a
is a,
no ned
algeb a
wi h
uni
1 sa is ying
11
1
11=
l
.
Finally
a
s noo h
no ned
.
algeb a
is
a,
uni al
no med
algeb a
whose
uni
is
a
smoo h
poin
o
i s
closed
uni
ball
.
All
asse ions
in
he
ollowing
p oposi ion, ha
a e no
o s aigh o wa d
e i ica ion,
a e
consequences
o
[20,
Theo em
27] (see also
[22,
Sec ion
2])
.

93
0

A
.
RODRÍGUEZ
PALACIOS
P oposi ion
1
.
I
H
is
a
eal
p e-Hilbe
space
and we
conside
on
R1
®
H
he
p oduc
gi en
by
(A1
+
17)
.01
+
~)
:=
(AM
-
(77
I
l))1
+
(a1
+,U
7
7)
o
all
A,
M
in
R
and
1
1,
~
in
H,
hen
R1(D
H
becomes
a
quad a ic
Jo dan
algeb a
o e
R
which,
endowed
wi h
he
p e-Hilbe
no m
11
~ 1
+
1
jj
:=
(~
2
+
11
77
112)1/2
is
a
smoo h
no med
algeb a
.
Mo eo e ,
e e y
eal
smoo h
no med
com-
mu a i e
algeb a
a ises
in
his
way
.
We
will
e e
o
he
no med
algeb as
desc ibed
in
he
abo e
p oposi ion
as
he
"smoo h
no med
Jo dan
algeb as"
.
Rema ks
1
.
i)
An
elemen
x
in
a
Jo dan
algeb a
J
.wi h
uni
is
said
o
be
in e ible
i
he e
exis s
y
in
J
wi h
x
.y
=
1 and x
.y
2
=
y
(see
[13,
De ini ion
1
.11
.51)
.
A
di ision-Jo dan
algeb a
is
a
Jo dan
algeb a
wi h
uni
whose
nonze o
elemen s
a e
in e ible
.
I
is
easy
o
see
ha
all
smoo h
no med
Jo dan
algeb as
a e
di ision-Jo dan
algeb as,
so
hey
a e
eal
no med
di ision-Jo dan
algeb as
unde any
algeb a
no m
(which
mus be
g ea e
han
he
canonical
p e-Hilbe
no m
because
his
las
no m
equals
he
spec al
adius)
.
I
was
p o ed
in
[14]
ha
no
mo e
eal
no med
di ision-Jo dan
algeb as
exis
.
ii)
I
is
well-known
ha
i
A
is
an
associa i e
algeb a
wi h
uni
1
and
J
is
a
Jo dan
subalgeb a
o
A
wi h
1
E
J,
hen
an
elemen
x
in
J
is
in e ible in
J
in
he
abo e
sense
i
and
only
i
x
is
in e ible in
A
in
he
usual sense
and
i s
associa i e
in e se
lies
in
J
[13,
p
.
51]
.
In
he
pa icula
case
o
J
being
(isomo phic
o)
one
o
he
smoo h
no med
Jo dan
algeb as,
his ac
is
almos
ob ious
.
Fo ,
i
o
x
=
A1
+
77
in
J
=
R1
®
H
we
dono e
by
x*
he
elemen
in
J
gi en
by
x*
:=
A1
-
77,
deno ing
by
yux aposi ion
he
associa i e
p oduc
o
A
we
ha e
x*x
=
XX*
=
a
2
1
-
77
2
=
>,
2
1
-
77
.91
=
(A2+
117711
2
)1=11
x
11
2
1,
so
e e y
nonze o
x
in
J
is
in e ible
in
A
in
he
associa i e sense,
wi h
associa i e
in e se
equal
o
11
x
11
-2 x*
(which
o cou se
lies
in
J)
.
iii)
The
no med
space
o a
smoo h
no med
.Jo dan
algeb a
is
ce ainly
a
nonze o
eal
p e-Hilbe
space,
and,
gi en
any
nonze o
eal
p e-Hilbe
space
K,
up
o
isome ic
isomo phisms
he e
is
a
unique
smoo hno med
Jo dan
algeb a
whose
p e-Hilbe
space
is
K
.
This
is
so
because,
chosing
any
no m-one
elemen
u
in
K
and
deno ing
by
H
he
o hogonal
comple-
men
o
Ru
in
K,
H
is
he
only
p e-Hilbe
space
sa is ying
K
=
IRu®
12
H,
ONE-SIDIE,D
DIVISION
ÁBSOLUTE
VÁLUED
ALGEBRAS

93
1
so
he e
is
a
unique
commu a i e
p oduc
en
he
no med
space
o
K
con-
e ing
i
in
a
smoo h
no med
Jo dan
algeb a
whose
uni
is
u
.
Mo eo e ,
he
choice
o
he
no n-one
ele nen
u
in
K
is
i ele an
hanks
o
he
`' o a ion
p ope y"'
o
p e-I-Iilbe
spaces,
namely any
no n-one
elemen
can
be
ca ied
in o
ano he
by
means
o
a
sui able su jec i e
linea
iso n-
e y
.
Indeed,
his
is
clea
o
dimension
1
o
2,
and,
in
he
emaining
case,
i
u
and
'a 'e
no n-one
elemen s
in
K
and
i
we
deno e
by
L
he
linea
hull o
{u,
}
as
well
as
by
cp
a
linea
iso ne y
om
L
,
on o
L
wi h
~o(u)
=
;
hen
he
mappi ig
0
:
l
+
l
1
->
cp(l)
+
1
1
om
K
=
Le)
L'
in o
K
is
a
su jec i e
linea
iso ne y sa is ying
O(u)
_
.
As
a
consequence,
gi en
a
ca dinal
iumbé
k~,
he e
exis s
a
unique
Smóo Ii
comple e'no med
Jo dan'algeb a
wi h
hilbe ian
dimension
equal
e
11
.
Now
we
s a e
and
p o
e
he
main
esul
in
his
sec ion
.
Theo em
1
.
Le ,
X
be
a
nonze o
eal
no med
space,
and
P
be
a
sub-
space
o
bounded
lineal-
ope a o s
on
X
con aining
he
iden i y
ope a o
and
sa is ying
II
T(x)
11=11
T
II
II
.c
II
o
al,l
T
in
P
and
all
x
in
X
.
Thén
P
is
a
Jo dan
algeb a
o
ope a o s
on
X
'which,
endowed
wi h
lié
ope a o
no77n,
is
isome ically
isomo phic
o
some
o
he
smoo h
no7 ned
Jo dan
algeb as
:
P oo
.-
In a
i s
s ep
we
edúce
he
p oo
e
he
pa icula
case
o
X
being
a
Banach
space
.
To
his end,'
conside
he
comple ion
X
o
X
a id,
o
T
-in
BL(X
), le
T
deno e
he
unique
ele nen
in
BL(X
)
which
ex ends
T
.
Ti en
he isome ic
ho nomo phism
T
->
T
om
he
associa i e
no med
algeb a
BL(X)
in o
he
Banach
algeb a
BL(X)
maps
P
en e
a
subspace
(say
P)
o
BL(X)
which
clea ly inhe i s
he
p ope ies
o
P
:
P
is
a
subspace
o
bounded
linea
ope a o s
en
X
con aining
he
iden i y
ope a o
on
X
and
sa is ies
11
S(y)
11=11
S
1111
y
11
o
all
S
in
P
and
all
y
in
X
.
In
his
way,
i
he
heo em
is
ue
in
he
comple e
case
;
he
in o ma ion
i
gi es
abou
1'
is
easily
ans e ed
o
P
.
In
a
second
s ep
we
assume
X
o
be a
Banach
space
and
we
show
ha
hen
e e y
nonze o
ele nen
in
P
is
in e ible
in
BL(X)
.
This
ac
being
clea
i
he
dimension
o
P
is
one,
assume dim(P)
>
2,
so
ha
P {0}
93
2

A
.
RODRÍGUEZ
PALACIOS
wi h
he
opology
o
he
ope a o
no m
is
a
connec ed
opological
space
in
which
he
subse
{T
E
^{0}
:
T
is
in e ible in
BL(X)}
is
ce ainly
open
(see
[4,
Theo em
2
.11])
and
nonemp y
.
Now
his
second
s ep
is
concluded
by
e i ying
ha
he
complemen a y
subse
Q
:=
{T
E
P {0}
:
T
is
no
in e ible in
BL(X)}
is
also
open
in
P {0}
.
Bu ,
i
T
is
in
9,
11
T
11-1
T
is
a
linea
isome y
om
X
in o
X
which
is
no
on o
and
so,
by
Lemma
1,
he
open
ball
in
P {0}
wi h
cen e
T
and
adius
11
T
11
is
con ained
in
9,
hence
9
is
open
in
P {0},
as
equi ed
.
Since
in
wha
ollows
complex me hods
will
be
applied,
i is
sui able
o
summa ize
ou
si ua ion
in
he
ollowing
way
.
P
is
a
eal
subspace
o
a
uni al
complex
(associa i e)
Banach
algeb a
(in
ou
case,
he
no med
complexi ica ion
o
BL(X)
[4,
P oposi ion
13
.3])
ha
will
be
deno ed
by A,
he
uni
o
A
líes
in
P,
and
e e y
nonze o
elemen
T
in
P
is
in e ible
in
A
wi h
11
T
-1
11=11
T
11-1
.
No e
ha
his
las
p ope y
implies
(*)

I
z
1=11
T
11
o
all
T
in
P
and
all
z in
he
spec um
o
T
ela i e o
A
.
The
hi d
s ep
in
ou
p oo
consis s in
showing
ha ,
o
T
in
P R1,
he
eal
linea
hull
o
{l,
T}
wi h
he
es ic ion
o
he
no m
o
A
is
a
copy
o
he
euclidean
space
R2
.
Chose
z in
he
spec um
o
T
ela i e o
A
and
w i e
A
:=
Re(z)
and
S
:=11
T
-Al
11
-1
(T
-Al),
so
ha
S
lies
in
P,
11
S
11
=
1,
and
he
spec um
o
S
con ains
a
numbe
o
he
o m
áE
o
some
E
in
R
.
Then,
o
a bi a y
a,
0
in
R,
a
+
¡E/3
lies in
he
spec um
o
al
+,PS
so,
by
(*),
we
ha e
11
al
+
QS
11
2
=1
a
+
¡Ep
1
2
=
a
2
+
6202
=
=a
2
+
1
aE
12
02
=
a
2
+
11
S
112
X32
=
a
2
+
02,
so ha
(a, /3
)
--->
al +,65
is
a linea
isome y
om
he
euclidean
space
Ii8
2
on o
LinR{1,T}
.
Ou
concluding
s ep
o
he
p oo
o
he
heo em
will
show
as
desi ed
ha
P
is
a
eal
Jo dan
subalgeb a
o
A
isome ically
isomo phic
o
one
o
he
smoo hno med
Jo dan
algeb as
.
The
consequence
ha
he
op-
e a o
no m
on
P
de i es
om an
inne
p oduc
can be
easily
ob ained
om
he
abo e
s ep
applied
o
he
subspaces
o
BL(X)
o
he
o m
{S
-
'T
:
T
E
P}
wi h
S
any
nonze o
ixed
elemen
in
P
;
bu
in
ac
he
ONE-SIDED
DIVISION
ABSOLUTE
VALUED
ALGEBRAS

93
3
p ehilbe ian
na u e
o
P
will
be
eencoun e ed
in
wha
ollows
join ly
wi h
he
emaining
pa
o
he
in o ma ion
.
We
will
use
some
concep s
and
esul s
om
he
heo y
o
nume ical anges
.
Thus,
ecall
ha
an
elemen
a
in
A
is
said o
be
he mi ian
i
(a)
lies
in
R
o
all
in
he
dual
Banach
space
o
A
wi h
11
11=
(1)
=
1
.
Clea ly,
he
se
o
all
he mi ian
elemen s
in
A
is
a
eal
subspace
o
A
so,
i
we
deno e
by
H
he
se
o
hose
S
in
P
such
ha
iS
is
an
he mi ian
elemen
in
A,
H
is
a
subspace
o
P,
and
we
claim
P=
R1
®ez
H
.
Indeed,
ob iously
Vil
l
H
=
0
and,
by
he
hi d
s ep
o
he
p oo ,
e e y
T
in
P
is
o
he
o m Al
+S
o
sui able
A
in
R
and
S
in
P
wi h
11
al
+
OS
II2=
a
2
+
11
S
112
02
o
all
a,
0
in
R,
an
equali y ha implies
~~
A1
+
S
II2=
A2+
11
S
112
and

lim

11
1
+,QS
I1
-
1
=
0
PER {0},P-o
,Q
so iS
is
he mi ian
in
A
[4,
Theo em
10
.10],
and
so
S
lies
in
H,
concluding
he
p oo
o
he
claim
.
A
Theo em
by
B
.
Bollobás
(see
[5,
Theo em
26
.7])
asse s
ha ,
i
a
is
an
in e ible
he mi ian
elemen
o
a
uni al
complex
Banach
algeb a
wi h
(1
a
11=11
a
-I
11=
1,
hen
a
2
=
1
.
This
applies
in
pa icula
o
elemen s
o
he
o m
T,-,',,
S
wi h
S
in
H {0}
o
ob ain
S
2
=
-
11S11
21
o
all
S
in
H,
hence
he
es ic ion
o
he
no m
o
A
o
H
comes
om
an
inne
p oduc
(
. 1
.)
.
In
passing
om
quad a ic
mappings
o
associa ed
symme ic
bilinea
mappings,
we
ind
S
.T
=
-(S
1
T)1
o
all
S,
T
in
H
.
Finally,
o
e e y
Al
+
S
and
pl
+T
in
P
=
R1
®I2
H,
we
ha e
(A1
+
S)
.(M1
+T)
=
(Ap
-
(S
1
T))1
+
AT
+
pS,
so
ce ainly
P
is
a
eal
Jo dan
subalgeb a
o
A
which
is
a
ma e ializa ion
o
a
smoo hno med
Jo dan
algeb a
.
Co olla y
1
.
Le
X
be
a
nonze o
eal
no med
space,
P
be a
subspace
o
BL(X)
sa is ying
11
T
(x)

11
=
11
T

x
~~
o
all
T
in
P
and
all
x
in
X, and assume
ha
some
elemen
in
P
is
in e ible
(wi h
in e so
possibly
ou side
o
P)
.
Then
he
ope a o
no m
940

A
.
RODRIGUEZ
PALACIOS
(iii)
=~>
(i)
This
is
i ial
.
An
almos
di ec
consequence
o
he
implica ion
(i)
=>
(iii)
in
he
abo e
p oposi ion
is
he ollowing
Co olla y
3
.
The
co nple ion
o
a
le
di ision
absolu e
alued
algeb a
is
a
le
di ision
absolu e
alued
algeb a
.
In
iew
o
P oposi ion
4,
o
ha e a
sa is ac o y
heo y
o
le
di ision
absolu e
alued
algeb as
i is
enough
o
s udy
(au oma ically
le
di i-
sion)
absolu e
alued
algeb as
wi h
le
uni
.
To
his
end
i
is
use ul
o
in oduce
some
addi ional
e minology
.
Gi en a
Jo dan
algeb a
J
and
a
ec o
space
X,
a
ep esen a ion
o
J
on
X
will
mean
an
homomo phism
(say
0)
om
J
on o
a
Jo dan
algeb a
o
ope a o s
on
X
.
I
J
has
a
uni
1 and
~b(1)
equals he
iden i y
ope a o
on X,
he
ep esen a ion
0
will
be
called
uni al
.
I
X
is
a
p e-Hilbe
space,
*
is
an
algeb a
in olu ion
on
J,
and
he
ep esen a ion
0
sa is ies
(O(x)( l)
1
~)
=
( 7
1
1
zb(x
*
)(I))
o
all
x
in
J
and
all
l,
~ in
X,
hen
we
will
say
ha
0
is
a
*-
ep esen a ion
.
When
J
and
X
a e
no med,
he
ep esen a ion
0
will
be
called
isome ic
( esp
. :
con ac i e)
i ,
o
all
x
in J,
he
linea
ope a o
O(x)
on
X
is
bounded
wi h
11
O(x)
11=11
x
II
( esp
. : ~~
O(x)
11<11
x
11)
.
F om
now
on
e e y
smoo hno med
Jo dan
algeb a
J
=
81
®
H
will
be
conside ed
as
algeb a
wi h
in olu ion
*
de ined
by
(A1
+
97)*
:=
Al
-
77
.
This
in olu ion
can
be
in insically
cha ac e ized
as
he
only
algeb a
in olu ion
* in
J
such
ha , o
e e y
x
in
J,
x
+
x*
and x
.x*
lie
in
Hl
.
Lemma
2
.
Le
J
be a
smoo h
no med
Jo dan
algeb a,
K
a
nonze o
p e-Hilbe
space,
and
0
be a
uni al
ep esen a ion
o
J on
K
.
Then
he
ollowing
asse ions
a e
equi alen
:
i)

is
a
*- ep esen a ion
.
ii)
~~
V)(x)(k)
11=11
x

k
11
o
all
x
in
J and
k
in
K
.
iii)

is
isome ic
.
i )
z~ is
con ac i e
.
P oo
.
(i)
=
:>
(ii)
Being
J
a
simple
Jo dan
algeb a
and
0
a
uni al
ep esen a ion,
he
ango
o
0
is
a
Jo dan
subalgeb a
o
he
associa i e
algeb a
L(K),
o
all
linea
ope a o s
on
he ec o
space
o
K,
isomo phic

ONE-SIDED
DIVISION
ABSOLUTE
VALUED
ALGEBRAS

941
o
J
and
con aining
he
uni
o
L(K)
(namely,
he
iden i y
ope a o
on
J
;
which
will
be
deno ed
by
I),
hence
by
Rema k
1(ii)
we
ha e
0(x
*
W
(x)
=
II
x
11
,
I
o
all
x
in
J
.
The e o e,
om
he
assu np ion
(i),
we
ob ain
o
a bi a y
kinK
11
VG(x)(k)
11
2
=
(~
b
(x)(k)
10
(x)(k))
=
_
(k
I
O(x*)O(x)(k))
=
(k
111
x
11
2
k)
=11
x
11
2
11
k
11
2
(ii)

(iii)
=>
(i )
These
implica ions
a e clea
.
(i )

(i)
W i ing
J
=
R1®
H,
i is
enough
o
show
ha ,
i
he
uni al
ep esen a ion
0
o
J
on
K
is
con ac i e,
hen
o
any
in
H
and
all
k
I
,
k2
in
K,
we
ha e
(0( 7)(kj)
I
k2)
=
-(k
1,0(,«k2»,
o
equi alen ly
(0( l)(k)
1
k)
=
0
o
all
no m-one
elemen
k
in
K
.
Bu ,
deno ing
by
a
any
posi i o
numbe ,
we
ha e
(0(
,
l)(k)
I
k)
_
((I+c VG(7l))(k)
I
k)
-1
<
11
1
+aO(
,
n)
II
-1
-
a

ce
=
11
0(
1
+
cm)
II
-1
<
11
1
+
0
,
77
II
-1
_
(1
+
n
2
II
7,11
2
)1
/
2
-1
(whe e
o
he
second
inequali y
we
ha e used
he
assump ion
ha
0
is
con ac i e)
.
The e o e
(o(
7
7)(k)
I
k)
C
lim

(1
+
a
2
II
l
1I2)
I/2
-
1
=
0
a
_o+
a
and,
changing
/
by
-
l,
we
ob ain
(0( l)(k)
1
k)
=
0
.
Now
we
s a e
and
conclude
he
p oo
o
he
main
esul
in
his
sec ion
.
Theo em
2
.
I
J
is
a
smoo h
nomned
Jo dan
alyeb a
and
0
is
a
uni al
*- ep esen a ion
o
J
on
he
p o-Hilbe
space
o
J,
hen
he
no med
space
o
J
wi h
p oduc
O
de ined
by
x
O
y
:=
IP(x)(y)
is
an
(au oma ically
le
di ision)
absolu o
alued
alyeb a
wi h
le
uni
.
Mo eo e ,
up
o isome iic
isomo phism,
by
means
o
his
cons uc i o
me hod
all
absolu o
alued
algeb as
wi h,
le
uni
a ise
.
P oo
.
The
e i ica ion
o
he
i s
pa ag aph
in
he
s a emen
is
e y
easy
.
Since
0
is
a
uni al
ep esen a ion,
he
uni 1 o
.J
as
a,
Jo dan
94
2

A
.
RODRÍGUEZ
PALACIOS
algeb a
-becomes
a
le
uni
o
he
p oduc
O,
and,
being
also
0
a
*-
ep esen a ion,
he
ac
ha
J
wi h
p oduc
O
is
absolu e
aluad
is
a
di ec
consequence
o
he
implica ion
(i)
=~>
(ii)
in
Lemma
2
.
Conce ning
he
p oo
o
he
second
pa ag aph
in
i e
heo em,
le
A
be an
a bi a y
absolu e alued
algeb a
wi h
le
uni ,
so
ha
by
Theo em
1
he
se
J
:={L
a
:aEA}
is
a
smoo h
no med
Jo dan
algeb a
(o
bounded
linea
ope a o s
on
he
no med
space
o
A)
.
Since
A
is
an
absolu e alued
algeb a,
he
mapping
u
:
a
->
L
a
is
a
linea
isome y
o n
he
no med
space
o
A
on o
he
one
o
J
and, as
a
consequence
;
he
mapping
0
:
F
->
UFU
-
is
a
uni al
isome ic
(hence
*-,
by
he
i iplica ion
(iii)
==> (i)
in
Lemma
2)
ep esen a ion
o
J on
he
p e-Hilbe
space
o
J
.
The
p oo
will
be
concluded
by showing
ha
A
is
isome ically
isomo phic
o
he
absolu e
alued
algeb a
ob ained
om
he
pai
(
.1,
0) by
he
cons uc i a
me hod
in
he
i s
pa ag aph
.
Bu
he
abo e
conside ad
su jec i e
linea
isome y
u
:
A
->
J
is
also
an isomo phism
om
A
on o
(J,
O),
because
o
a
and
b in
A
we
ha e
u(a)
O
u(b)
=
V)(u(a))(u(b))
=
uL
a
u
-I
(u(b))
=
u(L
a
(b))
=
u(ab)
.
Rema ks
4
.
i)
Mos
o
he
in o ma ion
gi en
by
he
abo e heo em
can be
s a ed
wi hou
in ol ing
Jo dan
algeb as
and
hei
ep esen a-
ions
on
ec o
spaces,
as
ollows
.

The
no m
o
any
absolu e
alued
algeb a
A
wi h
le
uni
e
de i es
o n
an
inne
p oduc
(
.

.)
;
and,
o
a, b,
c in
A
wi h
a
o hogonal
o e,
we
ha e
(ab
1
c)
=
-(b
1
ac)
and

a(ab)
=
-
11
a
112
b
.
ii)
The
Albe -U banik-W igh
heo em on
absolu e alued
algeb as
wi h
uni
can be
easily
de i ed
om
he
abo e
ema k
.
Fo ,
i
A
is
such
an
algeb a
and
1
deno es
i s
uni
elemen ,
aking
b
=
1
in
he
las
equali y
we
ob ain
a
2
= -
11
a
112
1 o
all
a,
in
A
o hogo ial
o
1,
hence
A
is
a
quad a ic
algeb a
.
Mo eo e
he
same
equali y
now
yields
o
La
=
L
a z
o
a,
in
A
o hogonal
o 1,
and
by
symme y
we
ha e
also
R
2
=
Rae
;
hence
A
is
al e na i a
.
Now
A
is
a
di ision
quad a ic
al e na i e
algeb a,
so
i is
isomo phic
o R, C, H,
o
®
by
he
ex ended
F obenius
heo em
(sea
o
example
[12,
Theo em
2
.26])
.
iii)
The
examples
in
Rema k
3(i)
show
obs ensibly
ha
isomo phisms
be ween
absolu e
alued
algeb as
can
ail
o
be
isome ic
o
e en
con in-
uous,
a
pa hology
ha , as
we
will
show
in
Sec ion
4,
only
can
occu
in
absence
o
comple eness
.
Howe e ,
in
he
pa icula
case
o
le -di ision
absolu e
alued
algeb as
i
is
no
di icul
o
de i e
om
P oposi ion
4
and
Theo em
2
ha
isomo phisms
mus
be
isome ic
.
ONE-SIDED
DIVISION
ABSOLUI
E
VALUED
ALCEBRAS

94
3
3
.
Exis en e
o
one-sided
di ision
absolu e
alued
algeb as
By
Co olla y
3
he
comple ion
o a
le
di ision
absolu e alued
al-
geb a
is
a
le
di ision
absolu e
alued
algeb a,
and by
P oposi ion
4
and
Theo em
2
e e y
le
di ision
comple e
absolu e
alued
algeb a
is
a
Hilbe
space
.
Then
one can
ask na u ally
o
hose
ca dinal
numbe s
3`
o
whicli
he e
exis
le
di ision
comple e
absolu e
alued
algeb as
wi h
hilbe ian
dimension
equal
l`
.
Since
in
ini o
dimension
he
answe
is
clea ly
1`2
=
1,
2,
4,
o
8
(see
P oposi ion
3),
we
will
cen e
ou
a en ion
in
he
in ini o-dimensional
case,
and
in aca
we
will
p o e
he exis en e
o
(au oma ically
le
di ision)
comple e
absolu e
alued
algebbas
wi h
le
uni
o
a bi a y
in ini e
hilbe ian
dimension
and
wi h
he
addi ional
p ope y
ha
hey ha e
no
nonze o
p ope
closed
le
ideals
(no e
ha
e e y
le
di ision
algeb a
has
no
nonze o
p ope
igh
ideals)
.
By
in ok-
ing
Theo em
2,
he
e i ica ion o
his
ac
is
equi alen
o
p o e
ha
e e y
in ini o-dimensional
smoo h
comple e
no med
Jo dan
algeb a
has
an
"i educible" uni al
*- ep esen a ion
en
i s
own
Hilbe
space
.
A
his
espec
we
ecall
ha
a
sel -adjoin
se
S
o
bounded
linea
ope a o s
on
a
eal
o
complex
Hilbe space
K
is
said
e
ac
i educibly
on
K
i
he
only
closed
S-in a ian
subspaces
o
K
a e
0and
K
.
While
o
complex
K
his
concep
has
been
widely
s udied,
his
is
no
he
case
o
he
eal
con ex
in
which
we
a e
mainly
in e es ed,
so
we
begin
ou
a gumen
wi h
he ollowing
Lemma
3
.
Le
K
be
a
complex
Hilbe
space,
S
a
sel -adjoin sub-
se
o
BL(K)
ac ing
i educibly
on
K,
and
Q
be
any
nonze o
p ope
S-in a ian
closed
eal
subspace
o
K
.
Then
K
=
Q
®12
áQ
.
As
a
conse-
quence,
S
( ega ded
as
a
sel -adjoin
se
o
bounded
linea
ope a o s
on
he
eal
Hilbe
space
Q)
ac s
i educibly
on
Q
.
P oo
.
Fi s
no e
ha ,
being
Q
n
iQ
a
complex
p ope
closed
S-
in a ian
subspace
o
K
and
ac ing
S
i educibly
on
K,
we
mus
ha e
Q
1
áQ
=
0
.
Deno ing
by n
he
( eal
linea )
o hogonal
p ojec ion
om
K
on o
Q, by
he
S-in a iance
o
Q
and
he
sel -adjoin ness
o S,
i
commu es
wi h
e e y
elemen
in
S
.
Now,
ega ding
complex
numbe s
as
linea
ope a o s
on
K,
7
-
i7 i is
a
bounded
co iplex-linea
ope a o
on
K
commu ing
wi h
he
elemen s
o S,
hence,
by
he
i educibili y
o
S
on
K,
we
ha e
7
-
i7 i
=a
+
i,3
o
sui able
a
and,3
in
R
(see
[7,
P oposi ion
2
.3
.8])
.
Mul iplying
on
he
igh his
equali y
by
7 ,
we
ha e
i
-
i7 i7
=
a7
+
i,(j7
so,
since
Q
7~
0
94
4

A
.
RODRÍCUEZ
PALACIOS
and
Q
n
iQ
=0,
we
ob ain
a
=
1
and
-T i7
=
j7
.
By
aking
adjoin s
in
he
las
equali y
i
ollows
ha
,0
=
0
and
he e o e
-
-
i7 i
=
1
.
Thus,
since
-¡i ¡
is
clea ly
he
o hogonal
p ojec ion
om
K
on o
iQ,
we
ha e
ha
iQ
is
he
o hogonal
complemen
o
Q
in
K,
ha
is
K
Q
® 2
iQ
.
Fo
he
consequence
asse ed
in
he
s a emen ,
no e
ha
any
nonze o
S-in a ian
closed
( eal)
subspace
R,
o
Q
is
a eal
subspace
o
K
which
also
sa is ies
he
assump ions
on
Q,
hence
he
abo e p o ed
ac
abou
Q
applies
o
R
gi ing
clea ly
R=
Q,
and
ce ainly
S
ac s
i educibly
on
Q,
as
desi ed
.
The
exis en e
o
"i educible" uni al
*- ep esen a ions
o
smoo h
com-
ple e
no med
Jo dan
algeb as
on
( eal)
Hilbe
spaces
will
ollow
om
he
abo e lemina
and
he
nex
p oposi ion,
which
con ains
basic
ac s
abou
he
"canonical
an icommu a ion
ela ions",
and
is
aken
almos
li e ally
om
[8,
pp
.
6-11] (see
p ecisely
[8,
P oposi ion
5
.2 .2])
.
P oposi ion
5
.
Gi en
a
complex
Hilbe
space
H,
he e
a e
a
nonze o
complex
Hilbe
space
K
( he
so
called
Fe mi-Fock
space o
H)
and
a
conjuga e
linea
mapping
-
a( )
( he
"annihila ion"
ope a o )
om
H
in o
BL(K)
sa is ying
he
ollowing
he e
p ope ies
:
i)
2a( )
.a(g)*
=
(
ig)I
and
a( )
.a(g)
=
0
o
all
,
g in
H
("canon-
¡cal
an icommu a ion
ela ions"),
whe e
"
."
deno es
Jo dan
p od-
uc
and
I
deno es
he
iden i y
ope a o
on
K
.
ii)
K
is
ini e
dimensional
whene e
H
is
so,
while,
i
H
is
in ini e-
dimensional,
he
hilbe ian
dimension
o
K
equals
he
one
o
H
.
iii)

The
sel -adjoin
se
o
ope a o s
{a(
),
a(g)*

:
,
g
E
H}
ac s
i educibly
on
K
.
Call
a
ep esen a ion
o a
Jo dan
algeb a
on a
nonze o
Hilbe
space
i educible
i
i s
ange
is
a
sel -adjoin
se
o
bounded
linea
ope a o s
ac ing
i educibly
on
he
gi en
Hilbe
space
.
P oposi ion
6
.
E e y
smoo h
.co cple e
no med
Jo dan
algeb a
has
an
i educible
uni al
*- ep esen a ion
on
a
( eal)
Hilbe
space
.
Mo eo e
o
such
an
algeb a
(say
.J)
he
ollowing
asse ions
a e
equi alen
:
i)
J
has
an
i educible
uni al
*- ep esen a ion
on
i s
own
Hilbe
space
.
ii)
J
has
a
uni al
*- ep esen a ion
on
i s
own
Hilbe
space
.
iii)
The
hilbe ian
dimension
o
J
equals
1,
2,
J ,
8,
o
any
in ini e
ca dinal
numbe
.
ONE-SIDED
DIVISION
ABSOLUTE
VALUED
ALCEBRAS

94
5
P oo
.
Fo any
ca dinal
nu be K,
conside
he
complex
Hilbe
space
o
(complex)
hilbe ian
dimension
?~,
le
K
and
a
be
espec i ely
he
complex
Hilbe
space
and
he conjuga e-linea
mapping
om
H
in o
BL(K)
gi en
by
P oposi ion
5,
and
de ine
a
( eal-linea )
mapping
s
om
H
in o
BL(K)
by
s( )
:=
i(a( )
+
a( )*)
.
F om
asse ion
(i)
in
P oposi ion
5
we
ob ain
o
all
and
g
in
H,
so
ha
he
se
s( )
.s(g)
=
-
Re(
1
g)I
J
:={AI+s( )
:AER,
EH}
is
a
eal
Jo dan
algeb a
o
ope a o s
on
K
ha ,
algeb aically
conside ad,
is
acopy
o
he
smoo h
comple e
no med
Jo dan
algeb a
o
( eal)
hilbe -
ian
dimension
21`
+
1
(ac ually
one
can
sea
ha ,
when
endowed
wi h
he
ope a o
no m,
his
copy
is
o en
an
iso ne ic
copy,
bu
his ac
is
i ele an
o
ou
a gu nen )
.
Mo eo e ,
lince clea ly
S( )*
=
-s( )
o
all
in
H,
J
appea s
uni ally
*- ep esen ad
on
he
eal
Hilbe
space
KR
unde lying
K
.
In
a
i s
ins an e,
aking
in o
accoun
ha
he
smoo h
comple e
no med
Jo dan
algeb a
o
hilbe ian
di nension
2?~
can
be
uni ally
*-embedded
in
he
one
o
hilbe ian
dimension
2?~
+'1
and
applying
asse ion
(ii)
in
P oposi ion
5,
his
a gu nen
shows
ha
any
ini o-dimensional
smoo h no mad
Jo dan
algeb a
can
be
uni ally
*- ep esen ad
on
a
.
nonze o
ini a-dimensional
( eal)
Hilbe
space
and,
o
ob ain
i educible
uni al
*- ep esen a ions,
i is
enough
o
pass
o
he
es ic ion o
he
ope a o s
in
he
ango
o
he
exis ing
ep esen a ion
o a
subspace
which
is
inimal
among
he
nonze o
subspaces
ha
a e
in a ian
unde
i e
ango
o
he
gi en
ep esen a ion
(such
a
n inimal
subspace
always
exis s
because
o
he
ini e
dimensionali y)
.
Re aking
he
ini ial
a gu nen
in
i e
in ini a-dimensional
case,
J
is
he
smoo h
comple e
no mad
Jo dan
algeb a
o
a bi a y
in ini a
hilbe ian
dimen-
sion
1`
(=
21~
+
1
in
his
case),
and
he
iden i y
ope a o on
J
is
a
uhi al
*- ep esen a ion
o
J
on
he
Hilbe
space
K
R
which,
in
iew
o asse -
ion
(ii)
in
P oposi ion
5,
has
also
hilbe ian
dimension
equal
o
?~
.
I
his
ep esen a ion
is
no
i educible,
by
he
de ini ion o
J
he e
us
exis
a
nonze o
p ope
elosed
eal
subspace
Q
o
K
in a ian
unde
i e
sel -adjoin
se o
bounded
complex-linea
ope a o s
S
:=
{s( )
:
E
H}
.

946

A
.
RODRÍGUEZ
PALACIOS
Bu ,
i
ollows
om
he
de ini ion
o
he
mapping
s
and
he
conjuga e-
linea i y o
he
mapping
a
ha
a( )-_
s(Z )-is( )
anda( )*=-
s(i )+is( )
2

2
o
all
in
H
.
These
equali ies,
oge he
wi h
asse ion
(iii)
in
P opo-
si ion
5,
show
ha
S
ac s
i educibly
on
K
(in
he
complex
sense)
.
By
Lemma
3,
S
ac s
i educibly
on
Q
(in
he
eal
sense)
ha
is,
he
mapping
T
->
T/Q
om J
in o
L(Q)
is
an
i educible
uni al
*- ep esen a ion
o
J
on
he
eal
Hilbe
space
Q
which,
in
iew
o
he
equali y
K
=
Q®iQ
(sea
again
Lemma
3)
and
he
in ini e-dimensionali y
o
K,
is
also o
hilbe ian
dimension
equal
o
k~
.
Thus
;
in
any
case,
he
in ini e-dimensional
smoo h
comple e
no med
Jo dan
algeb a
J
o
hilbe ian
dimension
?~
has
an
i e-
ducible
uni al
*- ep esen a ion
on a
Hilbe
space
o
hilbe ian
dimension
IZ
.
Since
he
ini a-dimensional case
has
been
conside ad
p e iously,
his
concludes he
p oo
o
he
i s
pa ag aph
in
he
p oposi ion,
and
e en
p o es
he
implica ion
(iii)
=>
(i)
in
he
in ini e-dimensional
con ex
.
To
inish
he
p oo
o
his
implica ion
no e
ha ,
i
o
i
=
1,
2, 4,
8
we
deno e
by
A
i
he
absolu a
alued
algeb a
R,
C,H,
®
espec i ely,
he
smoo h
no med
Jo dan
algeb a
JZ
o
dimension
i
can be
ecognized
as
he
Jo dan
algeb a
o
ope a o s
on
A
i
gi en
by
{L
a
:
al
E
A
Z
},
and
hen
he
iden i y
mapping
on
Ji
is
an
i educible
uni al
*- ep esen a ion
on
he
Hilbe
space
o
A
i
which
o
cou se
has
dimension
i
.
Since
he
implica ion
(i)
=~> (ii) is
clea ,
le
us
conclude
he
p oo
o
he
p opo-
si ion
showing
ha
(ii)
=>
(iii),
namely,
i
a
smoo h
comple e
no med
Jo dan
algeb a
J
has
a
uni al
*- ep esen a ion
on
i s
Hilbe
space
;
any
ini e
dimension
di e en
om
i
=
1,
2, 4,
8
mus
be
excluded
o
J
.
Bu
his
ollows
om
he
i s
pa ag aph
in
Theo em
2
oge he
wi h
he
implica ion
(iii)
=~>
(ii)
in
P oposi ion
3
.
91
Wi h
P oposi ion
4
and
Theo em
2,
he
abo e
p oposi ion
leads
di-
ec ly o
he
ollowing
Theo em
3
.
Le
di ision
comple e
absolu a
aluad
algeb as
o
hilbe -
ian
dimension
?~
exis
i
and
only
i
H
equals
1,
2, 4, 8,
o
any
in ini e
ca dinal
numbe
.
Mo eo e ,
o
such
a
ca dinal
l~
he e
exis
in
ac (au-
oma ically
le
di ision)
comple e
absolu a
alued
algeb as
wi h
le
zeni
o
hilbe ian
dimension
k~
wi h
he
p ope y
ha
hey
ha e
no
nonze7o
p ope
closed
le
ideals
.
Rema ks
5
.
i)
Since
he
comple ion
o
a
smoo h no med
Jo dan
algeb a
is
a
smoo h
no med
Jo dan
algeb a,
i
ollows
om
he
i s
pa ag aph
in
P oposi ion
6
ha
e e y
smoo h no mad
Jo dan
algeb a
ONE-SIDISD
DIVISION
ABSOLUTEVALUED
ALGEBRAS

94
7
has
a
uni al
*- ep esen a ion
on
a
Hilbe
space
.
Wi h
he
i nplica ion
(i)
=>
(ii)
in
Le nma
2,
his
shows
ha
e e y
smoo h
no med
Jo dan
algeb a
can
be
iewed
as
a
subspace
P
o
bounded
linea
ope a o s
on
a
sui able
no med
space
(which
ac ually
can
be
chosen
o
be
a
Hilbe
space)
sa is ing
he
assump ions
in
Theo em
1
.
Now
ce ainly
we
a e
su e ha
Theo em
1
canno
say
mo e
.
ii)
E e y
in ini e-dimensional
smoo h
comple e
no med
Jo dan
algeb a
J
has
noni educible
uni al
*- ep esen a ions
on
i s
Hilbe
space
.
Fo
;
i ~b is
any
uni al
*- ep esen a ion
o
J on
i s
Hilbe
space
;
he
napping
V)
®
0
om
J
in oL(J
®l2
J),
gi en
by
'O
®
~(
X)
(y,
z)
=
(0
(X)
(Y),
0(x)(z))
o
al]
x, y,
z
in
J,
is
a
noni educible
uni al
*- ep esen a ion
o
J on
he
Hilbe
space
J
(D
¿2
.I
which
has
he
sa ne
hilbe ian
dimension
ha o
J
.
Via
Theo em
2,
his
ac e iec s
on
he
exis en e
o
comple e
absolu e
alued
algeb as
wi h
le
uni
(o
a bi a y
in ini e
hilbe ian
dimension)
ha ing
nonze o
p ope
closed
le
ideals
.
iii)
I
A
is
an
in ini e-dimensional
le -di ision
absolu e
alued
algeb a,
hen
all
ope a o s
o igh
mul iplica ion
on
A
a e
nonin e ible
.
This
ollows
om
i nplica ion
(i)
==>
(iii)
in
P oposi ion
3
.
The
es o
his
sec ion
will
be
de o ed
o
ob ain
so e
in e es ing
consequences
o
he
exis en e
o
in ini e-dimensional
one-sided
di ision
absolu e alued
algeb as
.
Tl e
i s
esul
we
will
p o e
in
his
di ec ion
is
ha
small
pe uba ions
o
he
p oduc
o
a
.
le
di ision
comple e
abso-
lu o
alued
algeb a
gi e
ise
o
new
le
di ision algeb as,
hus
p o iding
in
iew
o
Theo em
3 a
e y
wide
collec ion o
in ini e-dimensional
com-
ple e
no med
le
di ision
algeb as
.
The
a bi a i y
o
he
pe u ba ion,
oge he
wi h
he s uc u e
heo y
o
le
di ision
absolu o alued
alge-
b as
(P oposi ion
4
and
Theo em
3),
shows
ha
he
algeb as
ob ained
by
his
p ocedu e canno
be
in
gene al
absolu o
alued
algeb as
.
As
a
ma e
o
ac ,
we
will
ealize
ha
all
hese
algeb as
ail
o
be
di ision
algeb as
.
P oposi ion
7
.
Le
A
be
a
le
di ision
comple e
absolu o
alued
al-
geb a,
le
El be
any
con inuous
bilinea
p oduc
on
he
Banach
space
o
A,
and
le
d deno e
he
dis an e
om
l]
o
he
p oduc
o
A
.
Thenwe
ha e
:
i)
I
d
<
l
.,
he
ec o
space
o
A
wi h
he
p oduc
111
is
a
le
di ision
algeb a
.
ii)
I
A
is
in ini e-dimensional
and
d
<
1,
hen
all
ope a o s
o
igh
mul iplica ion,
on
A
7-ela i e
o
he
p oduc
0
a e
nonin e?
,
ible
.
94
8

A
.
RODRÍGUEZ
PALACIOS
P oo
..
Fo
a in
A,
le
us
deno e
as
usual
by
L
a
and
R
a
espec i ely
he
ope a o s
o
le
and
igh
mul iplica ion
by a
ela i e o
he
ini ial
p oduc ,
and
by
Lo
and
R
.°
he
ones
ela i e
o
he
p oduc
El
.
I
d
<
1,
since
A
is
a
le
di ision
absolu e
alued
algeb a,
o
e e y
a
in
A {0}
we
ha e
ha
L
a
is
in e ible
and
~~
La
-
L°

lic
d
11
a
jj<jj
a
11=11
La
1

11
-1
so
by
comple eness
o
AL°
is
in e ible,
and
(i) is
p o ed
.
Le
us
assume
A
in ini e-dimensional
and d
<_
1
.
Then,
o
a
in
A
wi h
11
a
11=
1,
R
a
is
a linea
isome y
o m
A
in o
A
which
is
no
on o
(see
Rema k
5
(iii))
and
~~Ra
-
Ro
li~d11ajj<
1
,
so
Lemma
1,
oge he
wi h
he
ac
ha
he
se
o
nonin e ible
elemen s
o
a
Banach
algeb a
is
closed,
gi es
ha
Rao
is
no
in e ible,
and
(ii)
ollows
.
Gi en an
algeb a
A
and
an
elemen
A
in
he
base
ield,
he
A-mu a ion
o A,
deno ed
by
A(
A
),
is
de ined
as
he
algeb a
wi h
he
same
ec o
space
ha
o
A
and
p oduc
gi en
by
(a,
b)
->
Aab
+
(1
-
A)ba
.
Co olla y
4
.
Le
A
be
a
le
di ision
comple e
absolu e
alued
algeb a,
and
A
be
a
eal
numbe
.
Then
we
ha e
:
i)
I
A
>
2,
hen A1`1
is
a
le
di ision
algeb a and,
i
in
addi ion
A
is
assumed
o
be
in ini e-dimensional,
hen
all
ope a o s
o
igh
mul iplica ion
on
A(A)
a e
nonín e ible
.
ii)
I
A
=
2
a ad
A
is
in ini e-dimensional,
hen
all
ope a o s
o
le
(= igh )
mul iplica ion
on
A(
-
)
a e
nonin e ible
.
iii)
I
A
<
2,
hen
Ahl
is
a igh
di ision
algeb a and,
i
in
addi ion
A
is
assumed
o
be
in ini e-dimensional,
hen
all
ope a o s
o
le
mul iplica ion
on
A(A)
a e
nonin e ible
.
P oo
..
(i)
and
(ii)
ollow
di ec ly
om
P oposi ion
7
by
aking
as
he
p oduc
0
he
one
gi en
by
allb
:= ab
+
1

ba,
while
(iii)
ollows
om
(i)
applied
o
1
-
A
aking
in o
accoun
ha
he
opposi e
algeb a
o
A('
--
)
is
A(
A
)
.
ONE-SIDED
DIVISION
ABSOLUTE
VALUED
ALGEBRAS

94
9
Rema k
6
.
A
consequence
o
he
abo e
co olla y
is
he
well-known
ac
ha
all
A-mu a ions
wi h
A
7~

o a
ini e-dimensional
absolu e
á
alued
algeb a
a e
di ision
algeb as
.
Ou
concluding
esul
in
his
sec ion,
oge he
wi h
Theo em
3,
will
show
he exis en e
o
in ini e-dimensional
comple e
no med
algeb as
such
ha
he
ope a o s
o
le
and
igh
mul iplica ion
by any
no lze o
elemen
a e
su jec i e
.
Un o una ely,
in
ou
examples
all
hese ope a o s
will
ail
o
be
one- o-one
.
Ou
a gumen
begins
wi h
he
easy obse a ion
in
he
ollowing
le nma
.
By
in olu ion
on a
eal
( esp
. :
complex)
ec o
space
we mean
a
linea ( esp
. :
conjuga e-linea )
ope a o
on
he
space
wi h
squa e
he
iden i y
ope a o
.
Lemma
4
.
Le
A
be
a
eal
o
complex
comple e
absolu e
alued algeb a
whose
Banach
space
is
a
Iiilbe
space,
le
-
be
an
in olu ion
on
he
ec o
space o A,
and
de ine
a
new
(bilinea )
p od-ac
0
on
A
by
a0b
:=
Rb
(a)
(whe e,
as
usual,
R
.b
deno es
he
ope a o
o
igla
mul iplica ion
by
b
ela i e
o
he
inicial,
p oduc )
.
Then,
o e e y b
in
A {0},
che
ope a o
R°
o
igh
mul iplica ion
by
b
ela i e
o
he p oduc
E
is
su jec i e
.
P oo
.
No e
ha
R°
=
Rb,
so
i
is
eno lgh
o
p o e
ha
R*
is
a
su jec i e
ope a o
o
any no m-one
elemen
b
in
A
.
Bu ,
being
A
an
absolu e
alued
algeb a,
o
such
a
b,
R
.h
is
a linea
isome y
so,
since
he
Banach
space
o
A
is
a
Hilbe
space,
we
ha e
R*R
=
I
( he
iden i y
ope a o
on
A),
hence
ce ainly
R*
is
su jec i e
.
Ou
ollowing
obse a ion,
which
is
also
o easy
e i ica ion,
in ol es
s anda d
e minology
o
H*-algeb as
.
1{ollowing
[23]
and
[111,
a
le ,
se?ni-H*-algeb a
will
be
a
eal
o
complex
Hilbe
space
A
oge he
wi h
a
con inuous
bilinea
p oduc
on
A
(deno ed
usually
by
jux aposi ion)
and
an
in olu ion
- on
he ec o
space
o
A
sa is ying
o
all
a,
b,
c
in
A
.
(ab
1
e)=(b1ác)
Lemma
5
.
Le
A
be
a
eal
o7-
complex
le
semi-H*-algeb a
whose
in olu ion
-
is
isome ic,

_and
de ine
a
new
p oduc
0
on
A
by
a0b
-
R*
(a)
.

Then
-
is
an
algeb a
in olu ion
on
(A,[
:])
( ha
is,
a0b
=
bOa
o
all
a,
b
in
A)
.