Internat. J. Math. & Math. Sci.
Vol. I0, No.3 (1987) 513-517
513
ON M-IDEALS IN
B(.lpZ
CHONG-MAN CHO
Department of Mathematics
College of Natural Science
Hangyang University
Seoul 133, Korea
(Received May 8, 1986)
n
ABSTRACT.
< p, r < =, X
For
I
i=l
i
p r
), {n i} bounded, the space K(X) of all com-
pact operators on X is the only nontrlvlal M-ideal in the space B(X) of all bounded lin-
ear operators on X.
Compact operators, hermltlan element, M-ideal.
KEY WORDS AND PHRASES.
1980 AMS SUBJECT CLASSIFICATION CODE.
Primary 46A32, 47B05, secondary 47B05.
INTRODUCTION.
1.
Since Alfsen and Effros [I
introduced the notion of an M-ideal, many authors have
studied M-ideals in operator algebras.
It is known that K(X), the space of all compact
operators on X, is an M-ideal in B(X), the space of all bounded linear operators on X,
if X is a Hilbert space or
,
C
P
Smith and Ward [2] proved that M-ideals in a
(I < p < ).
-algebra are exactly the closed two sided ideals.
< p <
proved that, for
, K(P
Smith and Ward
is the only nontrlvlal M-ideal in
[3], and Fllnn [4]
B(p).
n
of this paper is to generalize this result to B(X), where X
r<
and {n
i}
a bounded sequence of positive integers.
I
i=l
@9
i
p r
The purpose
), for
< p,
In this proof, the ideas and
results of [4], [2], [5] and [3] are heavily used.
NOTATIONS AND PRELIMINARIES.
2.
If X is a Banach space, B(X) (resp. K(X)) will denote the space of all bounded llnear operators (resp. compact linear operators) on X.
A closed subspace J of a Banach space X is an L-summand (resp.
is a closed subspace
II
x
+ y
II
llxll
+
5
of X such that X is the algebraic direct sum of J and
IlYll
(resp.
llxll
max
{llxll, IIYlI})
P: X
X is an L-projectlon (resp. M-projectlon) if
llxll
llPxll,
II (
M-summand) if there
p)xll
for very x
x.
llxll
for
x e J, y e
llPx II
+
II( I
3,
and
3.
A projection
P)x
II
(resp.
C.M. CHO
514
A close4 subspace J of a Banach space X is an M-ideal in X If
J+/-
{x
e X
x
’Ij
--0}
is an L-summand in X
(Xi)i=
If
llxll
_< P <_
,
pXi
llx illp) I/p <
7.
is the space of
i--I
(x i)i=l,
all sequences x
and
sequence of Banach spaces for
is a
su.p[llx
ill}
1
<
e
xi
with the norm
Xi’
.
if p
llxll
_< p <
if
An element h in a complex Banach algebra A with the identity e is hermltlan if
lleiXhll
If
Jl
-
are complementary nontrivial M-summands in A (i.e. A
J2
M-projection of A onto
Ji (i
onto
J2’ e
zJi
A
(e
z)
P(e)
and z
Jl
since I
z)J i-
is hermit+/-an, (e
j
i
P is the
JlJ2),
z
then z is hermit+/-an with z
Jl’
0 [2, 3.2 and 3.4].
1,2) and zJ 2 z
2
z
(e
z)
)Jl(e
z
[6].
for all real
and
2
[2, 3.1],
P is the M-projection of
2) and
i
0.
If M is an M-ideal in a Banach algebra A, then M is a subalgebra of A [2, 3.6].
h
A
is hermit+/-an and h
2
M and M] M
e, then hM
If A is a Banach algebra with the identify
[4, Lemma I].
then A** endowed with Arens mult-
e,
A**
iplication is a Banach algebra and the natural embedding of A into
isomorphism into [6].
hermltlan element z e
If J is an M-ideal in A, then A**
jll
If
J+/-+/- (j+/-l
is an algebra
and the associated
commutes with every other hermit+/-an element of A **
[5 .22].
n
i
< p, r <
and {n i}
where
=I a
P r
bounded sequence of posit+/-rye integers. An operator T e B(X) has a matrix representation with respect to the natural basis of X. From the definition, it is obvious that
any diagonal matrix T e B(X) with real entries is hermitlan.
From now X,
will always denote
E
i=l
Fllnn [4] proved that if M is an M-ideal in B(
matrix, then hM M and Mh M.
and h e B(
P
His proof is valid for X.
a nontrivial M-ideal in B( ), then M a K( ).
P
P
is a diagonal
P
He also proved that if M is
Again his proof with a small modification
is valid for X.
Thus we have observed that if M is a nontrlvial M-ideal in B(X), then M
If M is an M-ideal in a Banach algebra A and h e M is hermltian, then hAh
Indeed, (e
Since
zA** z
hAh A
3.
z)h
M +/-+/-
(e
z)2h
(e
z)
z)h(e
h(e
0
Thus if e e M, then A
MAIN THEOREM.
We may assume that X
m
(
r
h.
M.
m
p
hz
M.
hzAzh e M+/-I. Since h e M,
M +/-I [2: 3.4], zAz & M 1+/- and hence hAh
M.
z) and so zh
K(X).
pr
nk)
n
s) (
p r
d
p
pr
p
nI
r
p
Ank)p
p r
M-IDEALS IN B(X)
Set s
+
m.
+
{I, 2
So
Pjx
_< j _< k,
n
+ nj,
+
o
for every x e X, where i
S.x
n
n
0.
\ Co,
is the indicator function of the set
s.
A
=i__Zlef(i)
i and a fixed j(l < j < k).
(a + n +
O
+
CLAIM:
n_
B
iE__l
.
(B).
Fix (
< s
e
+
<
g(i)
ef (i)
llBll
Since
)i
M
b
For x
then
t
II(x- Pj)x il
IIBxll
<_
ll(I
B)xll !
/
II(I
II
/
Pj)x + Bxll
(I + B)x
aije j e i
such that
n
O
<
# j,
+
+
e M
<
n),
and let g(i)
M.
Hence
for all
e
Mso
II
+
II
IIBxll
<
ilxll-II(I Pj)x II p.
X h
,
Ilxll p
II’xll
p
/
1,2,3...).
that
#(GB) has
(G)
defined by
and
e
+ (i-l)B (i
s
Choose
e B(X)
B,
Sj
k) and s
Suppose B
and AB
_> 8, f(i) e
2, where
I1(
Since Bx has support in
Sj
xll
.
.t
and
have
<_
/
t) 1/p
(I
(3.1)
t) I/p.
(211(I-Pj)xlIP) I/p 21/P(I
I x + Bx II
II
f(i)
To drawaontradiction, we will show that
boe.
IIPjxll p,
t
7
Since
I 0 + II <- II I +BII
d
Let
e B(X) is the operator with
gij aijeje i
where f(i+l)
ef(i),
norm one and attains its norm at A e M.
,
Sj.
B(X).
then M
we may assume that A
B(X)
be the projection on X defined
Pj
By multiplying by diagonal matrices from both sides, and as in Lemma 2 [4],
PROOF.
II*II
Let
where n runs over
If M is an M-ideal in B(X) and contains A
LEMMA I.
(aii)i,l
o
(n + 8N),
(el)i=
with respect to
(aij)
Let N be the set of all natural numbers,
Sj
be the unit vector basis for X.
(ei)i__l
matrix
+
n < s
<
+ n.
n. +
and 8
s
e} and, for
+ n o + n j-1
by
m
515
<
I Pjx II
+
II
(I
Pj)x
+ Bx
Hence
II-< tl/p + 21/P(l-t) I/p
(3.2)
I/p < 2 so
I/p + 21/P(I
2
t) I/p is continuous on [0,I] and F(O)
F(t) < 2 for all 0 _< t <_ 6. For 6 _< t <_ I,
+ (I t) I/p < 2. By (3.1) and (3.2)
M.
above, II(I + B) ll < 2. Contradiction! Hence B
Obviously, F(t)
t
Similarly C
adjoint of I
+ B.
Z
i=l
Hence
fCi)
lll+C[l
Since M is an algebra,
li+RN’
I e M. Since
1S
e
M (use
g(i)
CA
c, (G)
(CG)
< 2).
is+BN
I
I is compact,
O
llcll
CB
S
M.
Thus for all i
I e M.
O
This proves M
B(X).
I + C is the
C.M. CHONG
516
If M is an M-ideal in B(X) and there exists an isometry
with (all)i>
B(X) so that (M) contains an A
i
COROLLARY 2.
Y.aljeje
B(X)
PROOF.
Since
then M
E
tljej
B(X).
Suppose T e M and T is not compact.
PROOF.
mk + nk
mk
ij=mk+
k
rk
k=Zl Tk,
+
<
(x)
X,
k
Let B
Z e
k_>[
(yk)
X
where m
ke
+ N
n
N, and
k
k
@
z
X allwlthsupports
(z i)
k
{l: m
inok=
k
< i
+I
B=
emk+l
.@ej,
C
E
C
k_>l
have norm one and DB
Bk,
k
yjej )
emk+l
Dk
kE>ICk and D
E
k_>
1Dk"
jZ>l_
k
zej@emk+l
r.
Then all of these operators
A
CTB
Let P be the matrix obtained from the identity matrix I by interchanging
column and (m
k
in X since n
g
Choose
7
B(X)
,
fact, if
+ n k + j)-th column
If’B
"
c
J _< 8).
Then P is an isometry
o
so that
.c_ E
II@I[
((gmk+nk+l,
by (G)
then
M
B(X).
M, then M
attains its norm at B
G
for all k and j(l _<
mk+l)k I)
e B(X)
"(1
M.
"(I
E
G--
wher
defined by
Hence
)).
((I,I,I,I
and
Egijej@ e i.
O((I))k+l
"(l()
II"(
Define norm one functional
+ Y
II--
2.
I(Y +
k
Sp [{z k + emk+nk+l{{(Zk+
21/P’
where
1
P
+
=
I.
emk+nk+
E
X
{{GI[
A
M. In
has norm one and
But for any norm one
--l(gmk+nk+l,mk+l + JkE zjgj,mk+l)kl
YI)(G)I
Then
:+l
B(X), we have
<_
(mk+J)-th
8N.
k
CLAI,
_< mk + nk
l=Zk(Xk)"
E
I Xjemk
k
mk+l
we may assume
has norm one, there exists norm one vectors
Yk
Yk(TkXk)
so that
tlj ej e i, fITkll
Wlog
mk+l
Since each T
A=
B(X).
(M), by the lemma (M)
g
If M is an M-ideal in B(X) and contains a noncompact T
THEOREM 3.
k
(M) is an M-ideal in B(X) and A
B(X)
Hence M
x
o
B(X).
then M
T
_.cw
I)
,
M-IDEALS IN B(X)
o
II/II_< /P’
# 0.
s.t. y(G)
So
P(G) belongs to
e M
if
(gmk+nk+l,mk+l)_>l
,c o.
I,
then since
c o.
e
i
B(X)*
e
B(X).
.
This proves the claim.
M
(((Cmk+l,mk+nk+l)kK I)
by I(G)
(((CGB)mk+l,mk+l)k_>l)
is not in
+
III
SUPlk (CG)I,_
Imk
mk
+
and so
II
Indeed,
has norm one
defined
I
and attains its norm at T e M,
517
hGM
M
The sequence of the diagonal entries of
Thus by oorollary 2, M
defined by (G)
e B(X)
Next
Thus Mt S=
=onr=on,
But for any norm one
2.
G e B(X), we have
I( +
<
i )(G)
_<
.
M
Thus
There is e >
G
k
that
JY’eokg’3’
DIGD 2
.n__
k
Spllxk + ek+nk+lll
21/P
7.
So there is G
7.
(CG)
JeOk
mk+l
j
x[l
IIcGII
since CG e B(X),
contradiction
gije j e ieM
llgkl >
mk+nk+lemk+nk+I e..3
such that
((CG)mk+l mk+nk+l)k>
for infinitely many k, where
0 such that
and D
We can choose diagonal matrices D
has the same form as B in the claim above.
Since
DIGD 2
e M, M
2
in B(X) so
B(X).
REFERENCES
i.
ALFSEN, E. and EFFROS, E.
98-173.
2.
SMITH, R. and WARD, J. M-idegl
27(1978), 337-349.
3.
SMITH, R. and WARD, J. Applications of convexity and M-ideal
Banach algebras, 9uart. J. Math. 30(1979), 365-384.
4.
FLINN, P. A characterization of M-ideals in B(
P
98(1982), 73-80.
5.
SMITH, R. and WARD, J.
6.
BONSALL and DUNCAN Numerical Ranse of Operators on Normed spaq, London Math. Soc.
Lecture Note Series 2, Cambridge (1971).
7.
BEHRENDS, E. M-structure and the Banach-stone Theorem, Lecture notes in Mathematics
736, Springer-Verlag (1979).
8.
BONSALL and DUNCAN Complete Normed Algebra, Ergebnisse der Math., 80, SpringerVerlag (1973).
9.
CHO, Chong-Man and JOHNSON, W.B. A characterization of subspace of
K(X) is an M-ideal is L(X), Proc. Amer. Math. Soc. Vol.
i0.
11.
Structure in real Banach space, Ann. of Math. 96(1972),
sructure
M-ideas in
B(p),
in Bgnach algebras, J Functional Analysis
for
< p <
theory to quotient
=,
Pacific J. Math.
Pacific J. Math 81(1979), 227-237.
93(1985)
for which
466-470.
LIMA, A. Intersection properties of balls and subspaces of Banach spaces, Trans.
Amer. Math. Soc. 227(1977), 1-62.
LIMA, A. M-ideals of compact operators in classical Banach spaces, Math. Scand.
44(1979), 207-217.