Publicacions
Ma emá iques,
Vol
36
(1992),
485-488
.
Abs ac
ON
ONE-SIDED
DIVISION
INFINITE-DIMENSIONAL
NORMED
REALALGEBRAS
JOSÉ
ANTONIO
CUENCA
MIRA
Dedica ed
o
he
memo y
o
Pe e
Menal
In
his
no e
we
in oduce
he
concep
o
Cayley
homomo phism
which
is
closely ela ed
wi h
hose
o
composi ion
algeb a
and
no malized
o hogonal
mul iplica ion
.
The
key
esul
shows
he
exis en e
o
ce ain
ypes
o
Cayley
homomo phisms
o in ini e
dimension
.
As an
applica ion
we
p o e
he
exis en e
o
le
di ision
in ini e-dimensional
comple e
no med
eal
algeb as
wi h
le
uni y
.
Le
K
be a
ield
o
cha ac e is ic
di e en
om
2,
V
and
W
ec o
K-spaces
(no
necessa ily
wi h
ini e
dimension)
e e yone
endowed
wi h
a
nondegene a e
symme ic
bilinea
o m
.
These
o ms
will
be
deno ed
by
(
1
)
.
We
shall
say
ha
(V,
-,
e)
is
a
Cayley
iad
(o e
V)
i
-
is
an
in olu i e
isome y
o
V
and
e
an
elemen
in
V
such
ha
é
=
e
.
We
deno e
by
G(W)
he ec o
subspace
o
he
linea
maps
T
in
EndK
(W
)
which
ha e
an
adjoin
map
T*
wi h
espec
o
(
1
)
.
The
linea
mapS
V
-j
G(W)
ca ying
e e y
x o
S
x
is
said o
be
a
Cayley
homomo phism
(b ie ly,
homomo phism)
om
he
Cayley
iad
o
W
i
he
ollowing
condi ions
a e
sa is ied
:
1)
S
y
o5
x
=
(xix)
Id
o
any
x in
V
(whe e
Id
deno es
he
iden i y
ope a o
on
W)
.
2)
Sx
=
S,
3)
S,
=
Id
.
I
is
easy
o
show
ha
2)
is
equi alen
o
he
ollowing
condi ion
:
2')
Fo
any
x, y,
z in
V
we
ha e
(Sx(y)1
Sx(z))
_
(x1
x)
(y1
z)
.
Linea izing
equali y
1)
we
ob ain
S
x
oS
y
+
S
g
oS
x
=
2(xl y)
Id
486

J
.
A
.
CUENCA
MIRA
o
any
x,
y
in
V
.
On
he
o he
hand
1),
2)
and
3) yield o
(eje)
=
1
.
Assume
ha
V=
W
is
a
composi ion
K-algeb a
whose
symme ic
bilinea
o m
is
(1)
and
wi h
Cayley
an iau omo phis n
-
.
I
e
is
he
uni y
o
V
hen
he
map
sending
e e y
elemen
x
in
V
o
he
ig h
mul iplica ion
ope a o
R
.,
is
a
Cayley
homomo phism
om
he
Cayley
iad
(V,-,
e)
o
V
.
Con e sely,
le
V
be
a
ini o-dimensional
ec o
space
endowed
wi h
a
nondegene a e
symme ic
bilinea
o m,
assume
(V,
-,
e)
a
Cayley
iad
o e
V
and
S
:
V
->
G(V)
=
EndK(V)
a
Cayley
homomo phism
om
he
iad
o
V
.
As
in
[5,
p oo
o
P oposi ion
1
in
p
.
25],
we
can
de ine
a
p oduc
on
V
o
which
V
becomes
a
composi ion
algeb a
.
Cayley
homomo phisms
a e
also closely
ela ed
wi h
no malized
o hogonal
mul iplica ions
[1,
p
.
140-158]
.
Theo em
1
.
Le
K
be
an
o de ed
ield
whe e
e e y
posi i e
elemen
has
a squa e
oo ,
H
a
ec o
K-space
wi h
coun able
in ini e
dimension
which
is
endowed
wi h
a
posi i e
de ini e
symme ic
bilinea
o m
.
Then
he e
a e
a
Cayley
iad
(H,
-,
e)
o e
H
and
an
homomo phism
om
his
iad
o
H
.
Mo eo e
-
only
ixes
he
ec o
line
spanned
by e
.
P oo
.
I
is
well
known
ha
H
has
an
o hono mal
basis
eo,
el)
.
.
.
,
en
,. . .
(see o
ins an e
[2,
Theo em
29])
.
Fo
any
in ege
n>
0
le
V
n
( esp
.
W,)
be
he
subspace
o
H
spanned by
he
n
+
1
( esp
.
2n)
i s
elemen s
o his basis
.
Le
e
=
eo
.
Deno e by
-
he
linea
ope a o
on
H
ixing
eand
sending
he
emainde s
elemen s
o
his
basis
in o hei
opposi es
.
Fo
each
n
he
map
-
can
be
es ic ed
o
an
in olu i e
au omo phism
o
V
nwhich
we
shall also
deno e
in
he
same
way
.
E e y
(V
n
,
-,
e) is
a
Cayley
iad
.
We
shall
p o e
by
induc-
ion
ha
o
e e y
n
he e
exis s a
Cayley
homomo phism
S(n)
om
his
iad
o
W

,
which
sa is ies
he
ollowing
p ope y
:
o
all
m
<
n
and
any
x,
E
V,
yE
W,
n
we
ha e S(
n1
(y)
=
SX
)(y)
.
Ob iously he
s a e-
men
is
ue
o
n
=
0
.
Assume
now
ha
he e
exis s
an
homomo phism
S(n)
om
(V
n
,
-,
e)
o
W
n
sa is ying
he
equi ed
p ope y
.

We
shall
de ine
S(n4- )
.
Fo
his,
i s ly
we
obse e
ha
he e
exis s
an
isome y
b
n
om
W

,
on o
he
o hogonal
subspace
o
Wn
ela i e
o
W,
+
,
.
So
e e y
elemen
in
Wn+
can
be
w i en
in
a
unique
way
as
x
+
bn
(y)
wi h
x,
y
E
W
n
.
An
a bi a y
elemen
in
Vn+1
is
a
sum
u+Ae
n+
wi h
u
E
V
n
and A
E
K
.
We
deno e
by
S(+áe)
+1
he
linea
map
om
Vn+
o
Wn+
de ined
in
he
ollowing
way
Su+áe
.+1
(x
+
bn(y))
=
S(n)
(x)
-
Ay
+
bn(S(n)
(y)
+
ñx)
.
The
map
S(n+ )
:
Vn+
->
G(Wn+1)
=
End
K(W
.
+1
)
gi en
by
u
+
Aen+
-->
Su+Aen
+1
is
an
homomo phism
om
he
Cayley
iad
ONE-SIDED
DIVISION
NORMED
REALALGEBRAS

487
(V,+1,
-,
e)
o
Wn+i
which
sa is ies
he
equi ed
p ope y
.
Fo any
ele-
men s
x, y
E
H
he e
exis
a
V
n
such ha
x,
yE
V
n
.
The
elemen
S~
n
l
(y)
does
no
depend
o
he chosen
ec o
space
V
n and
will
be
deno ed
by
Sx(y)
.
E e y
S
.,
is
a
linea
ope a o
on
H
which
has
adjoin
ope a o
equal
o S,
and
he
map
x
-,
S
x
om
H
o
£(H)
is
an
homomo phism
om
he
Cayley
iad
(H,
-,
e)
o
H
.
a
Theo em
2
.
Le
H
be
an
in ini e-dimensional sepa able
eal
Hilbe
space
.
Then
he e
exis
a
Cayley
iad
(H, -,
e)
and
an
homomo phism
om
his
iad
o
H
.
P oo
.
The e
exis s
in
H
a
comple e
o hono mal
sys em
o
coun able
in ini e
ca dinal
.
Le
H'
be
he
ec o
subspace
spanned by
his
se
.
By
heo em
1,
he e
a e
a
Cayley
iad
(H',
-,
e)
and
an
homomo phism
S'
om
his
iad
o
H'
.
We
also
deno e
by
-
he
con inuous
ex ension
o
-
o
H
.
I
x,
y E
H
and
{xn},
{yn}
a e
sequences
o
elemen s
in
H'
such
ha
x
=

lim
xn
,
y
=

lim
y,
hen
we
ha e
ha
{S,',
:,
(Y
.)}
n-ce n-oo
is
a
Cauchy
sequence
in
H
.
The
limi
does
no
depend
o
he
chosen
Cauchy
sequences
and
is
deno ed
by
Sx(y)
.
Fo
any
x,
yE
H
we
ha e
115.(y)jj
=
IIxil IIy1I
.
Mo eo e
all
S
x
is
con inuous
wi h
adjoin
map
S~
ú
.
So
he
linea
mapS
:
H
-~
£(H)
gi en
by
S
:
x
-->
S~,
is
an
homomo phism
om
he Cayley
iad
(H,
-,
e)
o
H
.
a
We
ecall
ha
a
(nonassocia i e)
algeb a
V
is
a
le
di ision
algeb a
i
o
any
nonze o elemen
x
in
V
he
le
mul iplica ion
ope a o
L
',
is
in e sible
.
In
a
simila
way
igh
di ision algeb as
can
be
de ined
.
The
eal
algeb a
V
is
said
o
be an
absolu e
alued
algeb a
i i is
no med
and
i
sa is ies
11xyjj
=
lixil jjyjj
o
any
x,
yE
V
.
I
H
is
a
eal
Hilbe
space,
(H,
-,
e)
a
Cayley
iad
o e
H
and
S
a
Cayley
homomo phism
om
he
iad
o
H,
hen
H
wi h
he
p oduc
de ined
by xy
=
S,,(y)
becomes
a
le
di ision
algeb a
wi h
le
uni y
e
which
is
absolu e
alued
.
So
we
ha e
Theo em
2'
.
In
e e y
sepa able
eal
Hilbe
space
wi h
in ini e
di-
mension
can
be
de ined
a
(nonassocia i e)
p oduc
wi h
which
H
becomes
a
le
di ision
absolu e
alued
algeb a
wi h
le
uni y
.
Rema k
1
.
Theo em
T
shows
he
exis en e
o
eal
le
di ision
com-
ple e
no med
algeb as
wi h
in ini e
dimension
.
I
was
conjec u ed
by
F
.
B
.
W ig h
[4]
ha
e e y
no med
di ision
algeb a
o e
he
eals
is
ini e-dimensional
.
Rema k
2
.
Le
M
be a
se , L
an
ul a il e
on
M
and
{Hy}7Em
a
amily
o
eal
Hilbe
spaces
which
a e
le
di ision
absolu e
alued
48
8

J
.
A
.
CUENCA
MIRA
algeb as
wi h
le
uni y
.
Le
H
be
he
l°°-sum
o his
amily
and
N
he
ideal o
he
elemen s
(x
j
.,
F
m
such
ha
lim
11x711
=
0
.
Then
N
is
closed
and
HIN
is
a
no med
algeb a
which
will
be
deno ed
by
(H i)u
and
called
he
no med
ul ap oduc o
{H
.
y
}
wi h
espec
o
7d
(see
[6])
.
Mo eo e
o
any
elemen
[(x7)]
in
he
ul ap oduc
we
ha e
11[(x7)]11
=
lim
~Ix711-
u
So
his
ul ap oduc
is
a
Hilbe
space
.
I is
easy o
show
ha
(H7)u
is
a
le
di ision
absolu e
alued
algeb a
wi h
le
uni y
.
We
can
ob ain
om
his
ha
he e
exis eal
Hilbe
spaces
wi h
in ini e
Hilbe
dimen-
sion
big
enough
o e
which
we
can
de ine
a
p oduc
endowing
i
wi h
a
s uc u e
o
le
di ision
absolu e
alued
algeb a
wi h
le
uni y
.
Finally
we
obse e
ha
A
.
Rod íguez
[3]
has
gi en
he
s uc u e
heo y o
le
di ision
absolu e alued
algeb as
wi h
le
uni y
.
Re e en es
1
.

D
.
HUSEMOLLER,
Vib e
bundles,"
Sp inge -Ve lag,
New
Yo k,
sec-
ond
edi ion,
1975
.
2
.

I
.
KAPLANSKY,
"Linea
algeb a
and
geome y,"
Chelsea,
New
Yo k,
1974
.
3
.

A
.
RODRÍGUEZ,
One-sided
di ision
absolu e alued
algeb as,
o
ap-
pea
.
4
.

F
.
B
.
WRIGHT,
Absolu e
alued
algeb as,
P oc
.
N
.A
.S
.
39
(1953),
330-332
.
5
.

K
.
A
.
ZHEVLAKOV,
A
.
M
.
SLIN'KO,
I
.
P
.
SHESTAKOV,
A
.
I
.
SHIR-
SHOV,
"Rings
ha
a e nea ly
associa i e,"
Academic
P ess,
New
Yo k,
1982
.
6
.
H
.
EINRICH,
Ul ap oduc s
in
Banach
space
heo y,
T
.
Reine
Angew
.
Ma h
.
313
(1980),
72-104
.
Depa amen o
de
Álgeb a,
Geome ía
y
Topología
Uni e sidad
de
Málaga
Apa ado
59
29080
Málaga
SPAIN
Rebu
el
13
de
Gene
de
1992