Publicacions
Ma emá iques,
Vol
36
(1992),
439-447
.
ON
THE
UNIT-1-STABLE
RANK
OF
RINGS
OF
ANALYTIC
FUNCTIONS
Abs ac
JOAN
JOSEP
CARMONA,
JULIÁ
CUFÍ
ANDPERE
MENAL
In
his
pape
we
p o e
a
gene al
esul
o
he
ing
H(U)
o
he
analy ic
unc ions
on an
open
se
U
in
he
complex
plane
which
implies ha
H(U)
has no
uni -1-s able
ank
and
ha
has
some
o he
in e es ing
consequences
.
We
p o e
also
ha
in
H(U)
he e
is
no
o ally
educible
elemen s
di e en
om
he
ze o
unc ion
.
In oduc ion
Le
A
be a
commu a i e
ing
wi h
uni y
.
A
pai
o
elemen s
(al,
a2)
C-
A
2 is
said
o
be
unimodula
i
he e
exis s (b
l
,
b2)
E
A
2
such
ha
a
l
b,
+
a2b2
=
1
.
We
will
deno e
by U2(A)
he
se
o
all
unimodula
pai s
and
by
Uj(A)
=A
-1
he
se
o
in e sible
elemen s
o
A
.
One
says
ha
he
unimodula
pai
(al,
a2)
is
educible
i i is
possible
o
ind
x E
A
such
ha
al
+
xa2
E
A
-1
.
The
ing
A
is
said
o
ha e
s able
ank
1i
each
unimodula
pai
in
A
is
educible
in
A
.
This
is
a
special
case
o
he
concep
o
s able
ank
n
in oduced
by Bass
[1]
.
This
no ion has
been
use ul
in
ea ing
some
p oblems
in
K- heo y
.
Mo eo e
Vase sh ein
[12]
has
calcula ed
he
s able
ank
o
ings
o
con inuous
unc ions
and
ings
o
di e en iable
unc ions
in
Rn
and
ela ed
i
o
he
opological
dimension
o
he
domain
space
.
Conce ning
o ings
o
holomo phic
unc ions
P
.
Jones,
D
.
Ma shall
and
T
.
Wol
[3]
p o ed
ha
he
disc
algeb a
has
s able
ank
l
.
P e-
iously
L
.
A
.
Rubel
[7]
had
obse ed
ha
he
same
is
ue
o
he
ing
H(U)
o
holomo phic
unc ions'on
he
open
se
U
CC
.
Di e en
p o es
o
hese
esul s
can
be
ound
in
he
pape
o
G
.
Co ach
and
F
.
Suá ez
The
i s
wo
au ho s
dedica e
his
pape
o
he
memo y
o
Pe e
Menal,
who
in o-
duced
hem
o
he
subjec
o
his
pape
du ing
he
las
pe iod
o
his
li e
.
Pa ially
suppo ed
by
DGICYT
PB89-0311
and
DGICYT
PB89-0296
.
44
0

J
.
J
.
CARMONA,
J
.
CUFÍ,
P
.
MENAL
[2]
,
whe e
he
case
o
ings
o
holomo phic
unc ions
o se e al
complex
a iables
is
also
conside ed
.
The
mo e
di icul
p oblem
o
decide
i
he
algeb a
H'(D)
o
bounded
analy ic
unc ions
in
he
uni
disc
D,
has
s able
ank
1
has
ecen ly
been
answe ed
posi i ely
by S
.
T eil
[10]
.
In
[4]
P
.
Menal
and
J
.
Moncasi
in oduced
he
concep
o
uni -1-s able
ank
.
A
unimodula
pai
(al,
a2)
EU2
(A)
is
said
o
be
o ally
educible
i
he e
exis s
an
elemen
uE
A
-1
such ha
al
+ua2
E
A
-1
.
The
ing
A
is
said
o
ha e
uni -1-s able
ank
i
each
unimodula
pai in
A
is
o ally
educible
.
In
[7]
L
.
A
.
Rubel
p o ed
ha
H(C)
has no he
uni -1-s able
ank
p ope y
.
The
ques ion
o
decide
i
he
disc
algeb a
has
uni -l-s able
ank
a ose
and
was
s udied
by
R
.
Mo ini
and
R
.
Rupp
and
by
ou sel es
.
Mo ini
and
Rupp,communica ed
o
us
he
nega i e
answe
o
his
ques ion
and
using
some
ideas
o hei
p oo
we
ob ain
a
mo e
gene al
esul
ha
has
some
o he
in e es ing
consequences
.
This
is
he
con en
o
he
i s
pa
o
he
p esen
pape
.
The
esul
o
Mo ini
and
Rupp
appea s
in
[5]
whe e
he
ques ion
o
cha ac e ize
he
o ally
educible
elemen s
o
he
disc
algeb a
is
also
conside ed
.
They
ind a
su icien
condi ion
o
an
elemen
o be
o ally
educible
in
his
algeb a
.
Gi en
a
ing
A
one
says ha
an
elemen
a
E
A
is
o ally
educible
i
o
each
b
E
A
such
ha
he
pai
(a,
b)
E
U2
(A)
hen
(a,
b)
is
a o ally
educible
pai
.
In
he second
pa
o
his
pape
we
conside
he
o ally
educible
elemen s
o
he
ing
H(U)
.
In
his
case
he
si ua ion
is
comple ely
di e en
om
he
dise
algeb a
because
we
show
ha
he
ze o
unc ion
is
he
only
.
o ally
educible
elemen
o
H(U)
.
We
a e
able
o
ob ain
hese
kind o
algeb aic
p ope ies
o
ings
o
analy ic
unc ions
by
using
deep
heo ems
o
he
unc ion
heo y
o
one
complex
a iable
.
Uni -1-s able
ank
Rom
now
on
we
deal
wi h
he
ing
H(U)
.
I
a
E
H(U)
{0},
hen
we
deno e
by
Z
a
,
he
disc e
closed
se
in
U
o
he
ze os
a
.
Each
ze o
is
conside ed
wi h
he
co esponding
mul iplici y
.
So
when
we
w i e
Z
a
=
Zb
we
mean
ha
a
and
b
ha e
he
same
ze os
wi h
he
same
mul iplici y
.
We
ecall
ha
a
pai
(a,
b)
wi h
a,
b
E
H
(U)
is
unimodula
i
only
i
Z
a
l
Zb
=
0
.
We
can
p o e
he
ollowing
gene al
esul
.
UNIT-1-STABLE
RANK

44
1
Theo em
1
.
Assume
ha
(an),
(bn),
(en),
(d
n
,)
men s
o
H(U)
sa is ying
i)
anbn
+
cndn
=
1 o
all
n
>_
1
and
bn,
dn
a e
in e ible
elemen s
o
H(U)
.
ii)
The
sequences
(a
n
),
(c
n
)
a e
uni o mly
con e gen
on
compac
subse s
o
U
o
a,
c
E
H(U)
{0}
espec i ely
.
Then
ei he
Z
a
n
Z
c
=
0o
Z
a
=
Z,
a e sequences o
ele-
P oo
:
Fi s
we
p o e
ha
(a
nbn
)
is
ano mal
sequence
in
D
Zaa
.
Fix
a
poin
zo
E
D
Zae
.
Then
he e
exis s
a
closed
disc
A
C
U
Zac
wi h
cen e
zo
and a
numbe
S
>
0
such
ha lan(z)cn(z)1
>_ 5,
o
z
E
A,
n
>
,
la ge
enough
.
Conside
he
sequence
gi en
by
(a
nb
n
)
i
n
>_
.
We
will
p o e
ha
his
sequence
is
no mal
in
he
classical
sense
in
A
.
Since
bn
is
in e ible
and an
has
no
ze os
in
A
we
see
ha
anbn
ne e
akes
he
alue
0,
n
>
.
I
an(z)bn(z)
=
1
o
some
z
E
A,
i
ollows
om
i)
ha
cndn(z)
=
0
.
Bu
dn
is
in e ible
and
cn(z)
:y~
0
i
n
>_
and
his
is
a
con adic ion
.
By
Mon el's
Theo em
[8,
p
.
350]
,
(a,,,bn)n>

is
a
no mal
sequence
in
0
and
so
is
(anbn)n>1
.
Since
zo
was an
a bi a y
poin
o
he
open
se
U
Zae,
i
ollows
om
[8,
p
.
51]
ha
(anbn)n>1
is
no mal
in
U
Zac
.
Assume
Z
a
n
Z,
z,~
0and
le
us
ix
a
poin
a E
Z
a
n Z,
Take
a
closed
disc
O1
C
U
wi h
cen e
a
and
such
ha
(O1
{a})
n
Zac
=
0
.
The
esul
will
ollow
i
we
p o e
ha
any
~3
E
Zac
is
a
common
ze o
o
a
and
c
wi h
he
same
mul iplici y
in
a
han
in c
.
Fo
such
a
(~
le
02
be a
closed
disc
wi h
cen e
,P
and
such
ha
(02
{,3})
n
Zac
=
0
.
Le
K
=
01
U~2
.
By
ii)
(cn)
con e ges
uni o mly
o c
in
K
and
(1/c,)
con e ges
uni o mly
on
8K
.
Since
d,
.
=
(1
-
an
b
n
)/c
n
we know
ha
a
pa ial
sequence
o
(d,,)
is
ei he
uni o mly
con e gen
o
uni o mly
di e gen
on
8K
.
In
he
i s
case
his pa ial
sequence
(dn)
is
uni o mly
bounded
in
801
and,
by
he
maximum
modulus
p inciple, also in
O
1
.
Then
a pa ial
sequence
o
(dn(a))
is
con e gen
and
he
co espond-
ing
pa ial
o
((cndn)(a))
ends
o 0
.
Since
b,,
=
(1
-
c,,,dn)/an
and
on
8K
we
ge
ha
same
pa ial
sequence
o (bn)
is
uni o mly
a
a
bounded
on
K
.
The e o e
(an(a)bn(a))
ends
o
0
and
his
oge he
wi h
he
ac
ha
(cn(a)dn(a))
ends
o 0
con adic s
i)
.
The e o e
(dn)
is
uni o mly
di e gen
on
8K
.
Since
d
n
is
in e ible,
by
he
minimum
modulus
p inciple
we
ge
ha
(dn)
is
uni o mly
di e gen
on
K
.
Since
dñ
1
=
anbndn
1
+
c
n
we
ge
ha
(-a
n
bn
dn
1
)
con e ges uni o mly
o
c
in
K
.
Also
(an)
ends
o
a
uni o mly
on
K
and
since
Za,
n
80
2
=
0,
by
Hu wi z's
Theo em
[8,
p
.
158]
he
unc ion
a
has
he
same
ze os
han
c
in
02
.
44
2

J
.
J
.
CARMONA,
J
.
CUFÍ,
P
.
MENAL
The
ollowing
simple
esul
shows
ha
he
hypo hesis
abou
he
in-
e ibili y
o
b,,
and
d
n
canno
be
weakned
.
P oposi ion
1
.
Le
a,
c
E
H(U)
.

Then
he e
exis
sequences
(an),
(b
n
),
(cn),
(d n
)
such
ha (a
n
)
and
(cn)
a e
almos
uni o mly
con e gen
o
a
and
c espec i ely
such
ha
a
n
b
n
+c
nd
n
=
1
and
b
n
is
in e ible
o
all
n
>
1
.
P oo
..
Fo
each
>
0
le
us
conside
he
se
AA=
{z
E
Uja(z)
+
=
0}
n
Z,
.
Since
>,
0Y
implies
AA
n
Ay=
0,
he
se o
A
such
ha
Aa
:,A
0
is
a
mos
coun able
.
The e o e
we
can choose
a
sequence
(
,
n)
o
posi i e
numbe s
ha
con e ges
o 0
such
ha
Aa
n
=
0
.
Pu
an(z)
=
a
(z)+A
n and
c n
=
c,
n
>_
1
.
Clea y
he
sequences
(a
n
)
and
(c
n
)
con e ge
uni o mly
o a
and
c
and
Z
Qn
n
Z,
is
emp y
o
all
n
_>
1
.
Since
H(U)
has
s able
ank
1 his
implies ha
he e
a e b
n
in e ible
and
d

such
ha
anbn
+
c
nd
n
=
1
.
P oposi ion
1
says
in
pa icula
ha
he
opological
s able
ank
o
H(U),
in
he
sense
o
Rei el
[6],
is
2
.
F om
he
Theo em
1
we
can deduce
he
esul
o
Mo ini
and
Rupp
Co olla y
1
.
Le
be a
nonze o elemen
o
H(U)
.
Then
has
some
ze o
in
U
i
and
only
i
he e
is
a posi i e
in ege
n
such
ha
he
un¡-
modula
pai
( ,
1
-
n
2
)
is
no
o ally
educible
in
H(U)
.
P oo
.
Since
(n )
+(1-n
2
)
=
1 i is
clea
ha
( ,
l-n
2
)
is
o ally
educible
when
has
no
ze os
.
Con e sely,
assume
he e
exis
un
E
H(U)
-1
such
ha
n
=
u,
+
1/n
-
2
E
H(U)
-1
o
all
n
>_ 1
.
Then
1
=
u
n
ñ
1
+
(1/n
-
2) ñ
1
.
I
ollows
om
Theo em
1
ha
has
no
ze os
.
Co olla y
2
.
Le
A
be a
sub ing
o
H(U)
.
I
E
A
is
o ally
educible
in
A, hen
has
no
ze os
in
U
o
is
iden ically
0
.
Le
co
:
A
-
B
be
a
ing
homomo phism
.
We
say
ha
cW
has
s able
ank
1
p o ided
ha
o
any
x,
yE
A
wi h
xA+yA=
A
he e
exis s
c
E
S
wi h
cp(x)
+W(y)c
E
B
-1
.
I c
can
be
chosen
o
be
in e ible
in
B,
hen
we
say' ha
cp
has
uni -1-s able
ank
.
Co olla y
3
.
Le
U
be
an
open
se
o
C
and
le
A
be
a
ing
.
I
ep
:
A
-~
H(U)
is
a
ing
homomo phism
wi h
uni -1-s able
ank,
hen
W(A)
C
C
.
UNIT-1-STABLE
RANK

443
P oo
..
Co olla y
1
implies
ha
cp(a)
E
H(U)
-1
when
a
E
A,
a
7¿
0
.
Le
a
:,1
:
0 and
assume
ha
W(a)
is
no
cons an
.
Then
W(a)(U)
con ains
an
algeb aic
numbe and
so
he e
is
a
nonze o
polynomial
P
E
Z[ ]
such
ha
P(W(a))
has
some
ze o
in
U
.
Since
P(W(a))
=
~p(P(a))
we
conclude
ha
P(a)
=
0 and
so
P(cp(a))
=
0
.
This
shows
ha
W(a)
akes
only
ini ely
many
alues
and
so
i
mus
be cons an
which
is
a
con adic ion
.
Co olla y
4
.
Le
EC
C[z]
be
he
se
o
all
polynomials
wi hou
ze os
in
he closed
uni
disc
.
Then
he
ing
R
=
C[z]
has
s able
ank
1
bu
no
uni -1-s able
ank
.
P oo
.-
Clea ly
we
can
iew
R
as
a
sub ing
o
he
disc
algeb a
A(D)
.
I
(a,
b)
is
a
unimodula
pai
in
R
,
he e
exis s
an
elemen
E
A(D)
such
ha
a
+
b
is
in e ible
[2] [3]
.
Since
can be
app oxima ed
uni o mly
by
polinomials
we
can
assume
ha
i sel
is
a
polynomial
.
Then
a
+
b
E
R
and
R
has
s able
ank
1
.
I
ollows
om
Co olla y
3
ha
R
has no
uni -1-s able
ank
.
Ano he
applica ions
o
Theo em
1
a e
some
esul s
ha
gua an ee he
exis en e
o
a
ixed
disc
con ained
in
he
image
o
he
uni
disc
o
each
elemen
o
some
classes o
unc ions
.
In
his
line
we
ecall
he
classical
esul s
o
Bloch
[11,
p
.
262],
Koebe
[9,
p
.
197]
and
also
he
in e es ing
one
e e ed
in
[9,
p
.
502]
.
He e
we
conside
he
class
o
unc ions
g
whe e
is
a
ixed
unc ion
and
g
is
a
holomo phic
unc ion
wi hou
ixed
poin s
in
an open
se
and
also
he
class
o
all
he
uncions
.g
whe e
E
S
and
g
is
as
be o e
.
We
w i e
S
o
he
se
o
all
E
H(D)
such
ha
is
one
o
one
and
(0)
=
0,
'(0)
=
1
.
P oposi ion
2
.

The e
exis s
a
uni e sal
cons an
>
0
such
ha
he
dise
D(0,
)
is
con ained
in
he
image
o e e y unc ion
g,
whe e
E
S
and
g
E
H(D)
has no
ixed
poin s
in
D
.
P oo
..
Assume
ou
conclusion
is
alse
.
Then
o
each
posi i e in ege
n, he e
exis
zn
wi h
1zn1
<
ñ,
n
E
S
and
9n
wi hou
ixed
poin s
such
ha
n9n
-
zn
E
H(D)
-1
.
W i e
9n
(z)
=
z
-
hn(z),
whe e h
n
E
H(D)
-1
.
We
ob ain
1
=
nun
+
(z n
-
z
n) n
I
'

wl h

un,
n
E
HA-1-
Bu
S
is
ano mal
class
[9,
p
.
200]
,
so
he e
exi s
a
pa ial
sequence
o
( n)
uni o mly
con e gen o
some
,
E
S
.
Applying
Theo em
1
we
conclude
ha
Z
=
Z,
,
a
con adic ion
.

444

J
.
J
.
CARMONA,
J
.
CUFÍ,
P
.
MENAL
P oposi ion
3
.
Le
U
be
an
open
se
wi h
0 E
U
and
le
be
a
unc ion
ha
is
nei he
in e ible
no
ze o
.
Then
he e
exiss
a
cons an
=
( )
>
0
such
ha
he
disc
D(0,
)
is
con ained
in
he
image
o
e e y
unc ion
g
,
whe e
gE
H(U)
has
no
ixed
poin s
in
U
.
P oo
.
I
he s a emen
is
no
ue,
hen
o
each
n
he e
would
exis
z,,
1,
wi h
Izn1
<
ñ
and gn
wi hou
ixed
poin s
such
ha
.gn
-
z
n
E
H(U)
-1
.
P oceeding
as
be o e
we
ob ain
1
=
un
+
(z
-
zn) ñ
1
,

wi h

u
n
,
n
E
H(U)
-1
.
Now
we
aply
Theo em
1
and
he
conclusion
ollows
.
The
cons an
ha
appea s
in
P oposi ion
2
is
less
o
iqual
han
1,
by
Koebe's
Theo em
.
Conside ing
he unc ions
(z)
=
z,
g(z)
=
ez
-1
one
can
see
ha
<_
é
.
I
would be
in e es ing
o ind
a
cons uc i e
p oo
o P oposi ion
2
and
he
bes
alue
o
.
To ally
educible
elemen s
We
a e
going
now
o
conside
he
o ally
educible
elemen s
o
he
ing
o
analy ic
unc ions
in
an
open
se
.
Le
A
be a
Banach
algeb a
.
Fo
his
special
case
e e y
elemen
u
E
A
-1
is
o ally
educible
.
In
ac
,
o
each
E
A,
he
pai
(u,
)
is
unimodula
and
u
+
e
E
A
-1
i e
<
11 u
1
11 '
Fo
he
disc
algeb a
A(D)
o med by
he unc ions
which
a e
con inuous
on
D
and
holomo phic
in
D,
Mo ini
and
Rupp
p o ed
ha
each
ou e
unc ion
in
A(D)
is
o ally
educible
[5]
.
The
si ua ion
is
comple ely
di e en
o
he
ing
H(U)
as
he ollowing
heo em
shows
.
Theo em
2
.
Le
U
be
an
open
se
in
C
.
A
unc ion
E
H(U)
is
o ally
educible
in
H(U)
i
and
only
i
is
he
ze o
unc ion
in
U
.
Fo
he
p oo
we
conside
sepa a ely he
wo
di e en
cases
U=
C
and
U
7~
C
.
P oo
o
he
i s
case
:
Le
be
o ally
educible
in
H(C)
.
We
know,
using
Co olla y
2,
ha
=
eh
wi h
h
an
en i e
unc ion
.
I
he
un¡-
modula
pai
(e
h
,
z)
would be
o ally
educible
hen
he e
would
exis
k,
l
E
H(C)
such
ha
ehe
k
+ze
l
=
1
.
ÜNIT-1-STABLE
RANK

44
5
The
unc ion
ze
l
ne e
akes
he
alue
1
and
akes
he
alue
0
only
once
a
he
o igen
.
By
he
g ea
Pica d
Theo em
[8,
p
.
353]
ze
l
mus be
a
polynomial
and
his
con adic s
he
ac
ha
i
ne e
akes
he
alue
1
.
To
p o e
he
Theo em
2
o
U
z,~=
C
i s
o
all
we
ema k
ha
he
exis en e
o
some
E
H(U),
:,A
0 o ally
educible
implies ha
each
unc ion
in
H(U)
-1
is
also
o ally
educible
and
his
exis en e
is
equi a-
len o
an
in e pola ion
p oblem
as
he
ollowing
lemma
shows
.
Lemma
.
Le
U
be
an
open
se o
C
.
Then
he
ollowing
a e
equi alen
.
i)
The e
exis s
a
unc ion
E
H(U)
-1
which
is
o ally
educible
in
H(U)
.
ii)
Fo
each
closed
and
disc e e se
{zn},
coun ing
e e y
z
n
wi h
some
mul iplici y,
he e
exis s
a
unc ion
h E
H(U)
whose
ze os
a e
{z
n
},
wi h
he
co esponding
mul iplici y,
and
such
ha
h
ne e
akes
he
alue
1
in
U
.
iii)
Each
unc ion
g E
H(U)
-1
is
o ally
educible
in
H(U)
.
P oo
o
he
lemma
:
i) =~>
ii)
Le
E
H(U)
-1
be
o ally
educible
.
Gi en
he
se
{zn}
ake
k
E
H(U)
wi h
{zn}
as
i s
ze o
se
.
By
i)
he e
a e
a,
b
E
H(U)
-1
such
ha
a
+
bk
=
1
.
So a
=
11
bk
and h
=
bk
sa is ies
he
equi emen s
o
ii)
.
ii)
==>
iii)
Le
g E
H(U)
-1
and
le
l
E
H(U)
be
a bi a y
.
The
pai
(g,
l)
is
unimodula
.
Le
{z,,}
be
he
ze o
se o
l .
By
ii)
he e
is
some
hE
H(U)
ha
akes
he
alue
1
on
{z
n
}
and
h(z)
:7É
0
.
Now
he unc ions
b
=
11
hh
E
H(U)
-l
and
a
=
s
E
H
(U)
-1
e i y
ag
+
bl
=
1
.
We
need
he
ollowing
esul s
[11,
p
.
215-204]
.
Theo em
(Ahl o s)
.
Le
E
H(D)
and assume
ha
lim
sup

T
( )

=
+oo
.
-1

log
11
1
Then
akes
any
alue
in ini ely
o en
in
D
wi h
one
possible
excep ion
.
He e
T( )
is
he
cha ac e is ic
unc ion
o
Ne alinna
[11,
p
.
196]
.
Theo em
.
Le
E
H(D)
and
le
{zn}
be
i s
ze o
se
.
Assume
ha
-°°
1
(
1-
I
zn
I
)1+P
=
+
oo,
o
all
p
>
0
.
Then
lim
sup,1
io
=
+
oo
.
44
6

J
.
J
.
CARMONA,
J
.
CUFÍ,
P
.
MENAL
P oo
o
he
second
case
:
Assume
i s
ha
U=D
.
Le
(zñ)
be a
sequence such
ha
lim
n
~~
zñ
=
1
and
_,'
1
(1
-
jzñ¡)'+P
=
+
oo
o
all
p
>
0
.
Le
{z

,}
be
he
sequence
o
he
same
poin s
bu
doubling
hei
mul iplici y
.
We
show
ha
ii)
o
he
lemma
is
no
sa is ied
o
{zn}
.
Le
h
E
H(D)
be any
unc ion
ha
anishes
on
{zn}
.
We
ha e
h
=
hó
o
some
ho
E
H(D)
anishing
a
{zñ}
.
By
Ahl o s
Theo em
ho
may
omi
only
one
alue
and
so
i s
squa e
h
akes
any complex
alue
in ini ely
o en
.
Fo
a
gene al
open
se
U
le
A
be
any
disc
A
C
U
such
ha
he e
is
a
poin
a
E
ao
n
aU
.
l
su ices
o
ake
a
sequence
{zn},
z
n
E
A
as
be o e
wi h
lim
e
-,,,
zñ
=
a
.
Fo
his
sequence, condi ion
ii)
o
he
lemma
is
no
sa is ied
in
U,
since
i is
no
sa is ied
on
A
.
Re e ences
1
.
H
.
BASS,
K- heo y
and
s able
algeb a,
Publi
.
Ma h
IHES
22
(1964),
5---60
.
2
.

G
.
CORACH
AND
F
.
DANIEL
SUÁREZ,
S able
ank
in
holomo phic
unc ions
algeb as,
Illinois
J
.
o
Ma h
.
29
(1985),
627-639
.
3
.

P
.
W
.
DONES,
D
.
MARSHALL
AND
T
.
WOLFF,
S able
ank
o
he
disc
algeb a,
P oc
.
Ame
.
Ma h
.
Soc
.
96
(1986),
603-604
.
4
.

P
.
MENAL
AND
J
.
MONCASI,
K1
o
Von
Neumann
egula
ings,
Jou nal
o
Pu e
and
Applied Algeb a
33
(1984),
295-312
.
5
.

R
.
MORTINI
AND
R
.
RUPp,
To ally
educible
elemen s
in
ings
o
analy ic
unc ions,
Comm
.
i
n
Algeb a
( o
appea )
.
6
.

M
.
A
.
RIEFFEL,
Dimension and
s able
ank
in
he
K- heo y
o
C*-algeb as,
P oc
.
London
Ma h
.
Soc
.
46
(1983),
303--333
.
7
.

L
.
A
.
RUSEL,
Linea
composi ions
o
wo
en i e
unc ions,
Ame
.
Ma h
.
Mon hly
85
(1978),
505-506
.
8
.

S
.
SAKS
AND
A
.
ZYGMUND,
"Analy ic
unc ions,"
Else ie
Pub
.
Company,
1971
.
9
.

G
.
SANSONE
AND
J
.
GERRETSEN,
"Lec u es
on
he
heo y
o
unc-
ions o a
complex
a iable,"
Wol e s
Noo dho ,
1969
.
10
.
S
.
TREIL,
The
s able
ank
o
he
algeb a
H°°
equals
1,
P ep in
Lening ad
Uni e si y
(1991)
.
11
.
M
.
Tsu,Ii,
"Po en ial
heo y in
mode e
unc ion
heo y,"
Chelsea
Publishing
Company,
New
Yo k,
1975
.
UNIT-1-STABLE
RANK

44
7
12
.

L
.
N
.
VASERSTEIN,
S able
ank
o
ings
and
dimensionali y
o
opo-
logical
spaces,
Func ional
Anal
.
Appl
.
5
(1971),
102-110
.
Depa amen
de
Ma emá iques
Uni e si a
Au ónoma
de
Ba celona
08193
Bella e a
(Ba celona)
SPAIN
Rebu
el
2de
Ma C
de
1992