I ntrnat. J. Math. & Mh. Sci.
(1978)307-317
Vol.
307
ON CERTAIN QUASI-COMPLEMENTED
AND COMPLEMENTED BANACH ALGEBRAS
PAK.KEN WONG
Department of Mathematics
Seton Hall University
South Orange, New Jersey 07079 U.S.A.
(Received February 27, 1978)
ABSTRACT.
In this paper, we continue the study of quasl-complemented
algebras and complemented algebras.
The former are generalizations of the
latter and were introduced in [4] and studied in [4] and [ii].
Some re-
sults are proved.
KEY WORDS AND PHRASES.
Quasi-complemented and complemented Banach gebras.
AMS(MOS) SUBJECT CLASSIFICATION (1970) CODES.
I.
INTRODUCTION.
Quasi-complemented algebras, which are generalizations of complemented
algebras, were introduced in [4] and studied in [4] and [11].
In this paper,
we continue the study of these two classes of algebras.
In Section 3, we introduce the concept of continuous quasl-complementor
on a seml-simple annihilator Banach algebra.
This is similar to the concept
P. Wong
308
annihilator Banach algebra such that
ClA(XA)
x
A be a simple
Let
of continuous complementor given by Alexander in [i].
for all
x
in
A.
A
is infinite dimensional, we show that every quasi-complementor on
q
we obtain that a quasi-complementor
E
the set
A
This result is not true if
tinuous.
A
is continuous if and only if
By using these
A.
of all q-projections is closed and bounded in
q
is con-
In this case,
is finite dimensional.
on
A
If
results, we give a characterization of continuous quasl-complementors (Theorem
Section 4 is devoted to the study of uniformly continuous quasi-complemen-
A
Let
tors.
for all
x
be a semi-simple annihilator Banach algebra in which
A
in
and
q
A.
Suppose that
A
has no
Then we show that
A
is a
a quasi-complementor on
minimal left ideals of dimension less than three.
B
dense subalgebra of some dual B*-algebra
R
right ideals
A.
o.f
R
and
ClA(XA)
x
q
(R)*
Also every continuous complementor on
A
for all closed
A
is uniformly
continuous.
2.
NOTATION AND PRELIMINARIES.
For any subset
S
in an algebra
left and right annihilators of
algebra.
J
ideal
and
A(R)
rA(A(R))
R,
then
B
(0)
R,
A
we have
A.
R
if and only if
A be a Banach
Let
rA(J)
(O)
If
A(rA(J))
if and
(resp.
S
of
A).
A, cl(S)
Also
for the norm on
A
J
is called a dual algebra.
(resp.
(S)
and
note the left and right annihilators of
l-II
respectively.
denote the
A be a Banach algebra which is a subalgebra of a Banach algebra
For each subset
in
A,
rA(S)
and
is called an annihilator algebra, if for every closed left
A
Let
S
A
in
let
and for every closed right ideal
J
only if
and
Then
S
A(S)
A,
and
I’I
cl
A(s))
S
will denote the closure of
(resp.
r(S)
in
B
B.
EA(S)
(resp.
for the norm on
B.
and
A).
rA(S))
We write
de-
309
QUASI-COMPLEMENTED AND COMPLEMENTED BANACH ALGEBRAS
A be a Banach algebra and let
Let
ideals in
L
r
be the set of all closed right
Following [4], we shall say that
A.
mented algebra if there exists a mapping
q
R
having the following properties:
R
R
R
R
I
Hence
+ Rq
Rq
R2,
R
L
of
r
into itself
(2.1)
(R e L r)
(2.2)
Rq
(R I, R
e
2
Lr )"
(2.3)
A.
(right) quasi-complementor on
q
A, A
is always dense in
(0)
-
is a (right) quasi-comple-
q
(R e L r
R
then
is called a
q
The mapping
R
(0)
(Rq)q
if
that
q
A
(0)
(0)
and
q
A
We know
(see [4]).
R-- A.
if and only if
A quasi-complemented algebra
A
is called a
(right) complemented algebra
if it satisfies:
R
+ Rq
In this case, the mapping
(2.4)
(R e L r ).
A
is called a (right) complementor on
q
A
(see [6, p. 651, Definition i]).
Let
A be a semi-slmple Banach algebra with a quasi-complementor
f
minimal idempotent
in
A
is called a q-projection if
The set of all q-projection in
every non-zero right ideal of
A
A
is denoted by
E
q
(fA) q
(i
q.
A
f)A.
By Lemma 3.1 in [Ii],
contains a q-projection.
In this paper, all algebras and linear spaces under consideration are over
Definitions not explicitly given are taken from Rickart’s
the complex field.
book [5].
We end the section with two new examples of complemented and quasi-complemented algebras.
EXAMPLE I.
A be a dual B*-algebra and
Let
Then the algebra
A
the complementor
q
0)
#
a symmetric norming function.
given in [I0, p. 293] is a complemented algebra with
R
+
A0)
(R)*
(Theorem 3 4 in [ii]).
P. Wong
310
EXAMPLE 2.
be an infinite compact group with the Haar measure and
G
Let
the algebra of all continuous functions on
absolute value and
L
L
I(G)
in an auxiliary norm.
(resp.
It is well known that
A
and
R
/
ELl(G) (R)*)
is a quasi-complementor on
EA(R)*
L I(G)). However,
R
q
It is easy to see that the mapping
by Theorem 3.4 in [ii],
(resp.
A
+
is not a complementor.
q
CONTINUOUS QUAS I-COMPLEMENTORS
3.
Let A
MA,
Then
M
A
PR
q
+ (i
fA
R-- fA
f)A.
Let
For each
A.
for some q-projection
f
in
be the projection on
R
along
PR
There-
R
q.
is continuous.
DEFINITION.
complementor
then
A.
the set of all minimal right ideals of
by Lemma 3.1 in [ii],
R + R
fore,
be a semi-simple annihilator Banach algebra with a quasi-comple-
and
q
mentor
ao,
the group algebra.
normed by the maximum of the
are dual A*-algebras which are not two-sided ideals of their completions
I(G)
R e
G,
q
PaAn
REMARK.
Suppose
on
A
a
n
e A
a A e M
A
n
with
(n
0, i, 2, ...).
is’said to be continuous if whenever
converges to
Pa0
A
a
n
A quasi-
converges to
uniformly on any minimal left ideal of
A.
This is similar to the definition of continuous complementor intro-
duced by Alexander (see [I, p. 387, Definition]).
Let
such that
A
be a semi-simple annihilator quasi-complemented Banach algebra
x e
ClA(XA)
for all
x
in
all minimal closed two-sided ideals of
for all closed right ideals
R
is the direct topological sum of
mentor on
1%.
Let
H%
of
I%.
{1%
A
A.
and
{1%
Define
% e A}
B%
Rq/ I%
q% by R
Then by [4, p. 144, Theorem 3.6]
% e A}
and
be a minimal left ideal of
q%
1%.
Hilbert space under some equivalent inner product norm by
Let
the family of
A
is a quasi-comple-
Then
H%
is a
[4, p. 145, Lemma 4.2].
be the algebra of all completely continuous linear operators on
A
QUASI-COMPLEMENTED AND COMPLEMENTED BANACH ALGEBRAS
Then by the proof of [4, p. 146, Theorem 4.3],
B
II" If
such that
Lemma 5.1],
B
LEMMA 3.1.
each
A
0’
By the proof of [8, p. 442,
have the same socle.
A
A quasl-complementor
R e M
A
Let
with
is a q-projection in
@
I.
on
q
A
on
is continuous if and only if
(0)
I01
then
R
for some
I0
Hence, for all x
I0.
PR(X)
and so
0
continuous if and only if each
ClA(XA)
PROOF.
for all
x
q
H
Let
B,
I-ll
Also
in
on
A
A.
A
If
B;
p
M
M
majorizes
A] q)
I"
[3, p. 471,
H,
Using
q
is
Theorem 6.8].
is continuous.
A.
As observed before,
x(H)
M / H
q
A
H
and
H
is a minimal left ideal of
M
B.
x
M.
in
Since
II’II
and
H
[4, p. 145, Lemma 4.1] that
(M A)/ H-- (M/’ A)H.
MPH
cl([M / A] q)
H
[M /’
A]q/’
(3.1)
H.
Then
p
on
that
"I are
Therefore, we have
S(Mp)
B.
H.
We show that
be the smallest closed subspace of
for all
MH
of
is
is a dense dual sub-
can be extended to a quasi-complementor
S(M)
it follows from
S(M)
A
on
for all closed right ideals
In fact, let
contains the range
alent on
I.
in
the algebra of all completely continuous linear operators on
cl([M
(M)*.
If
is infinite dimensional, then every
be a minimal left ideal of
by [4, p. 148, Theorem 5.4],
p
fx.
where
is continuous.
q%
a Hilbert space under some equivalent inner product and
algebra of
fA,
A be a simple annihilator Banach algebra in which
Let
quasl-complementor
x
R
Then
PR(X)
A,
in
for all
The following result is a generalization of
LEMMA 3.2.
e A.
0
[I, p. 387, Theorem 2.2], we can show that
this fact and the proof of
x e
is a dense subalgebra of
is continuous.
qA
PROOF.
f
I
and
A
I
majorlzes
I
311
(3.2)
equiv-
P. WONG
312
By the proof of [4, p. 145, Lemma 4.2],
(see [4, p. 148] for the last equality).
ClA((M
A
M
’
A)HA).
A
Since
is infinite dimensional, by
[4, p. 145,
Theorem 4.2 (iii)] and (3.1)
[ClA(S(M)A) ]q/
[MA]q/ H.
S(M)
S(M
Therefore, by (3.2),
A)HA)) ]q/’ H
[elA((M
H
s(MP).
Hence it follows from [3, p. 464, Lemma
4.1] and [3, p. 465, Theorem 4.2] that
M
p
In particular,
(M)*.
p
is con-
tinuous by [i, p. 388, Theorem 2.4].
nM
"I
(n
a minimal left ideal of
B
and
Let
f
a A e
n
Suppose
Hence
a
/
a A
n
a B
n
Then
P
in
a^
n
for all
aA
(x)
f x
n
n
P
converges to
fore
a0A
A
such that
L
be a minimal left ideal of
II’II
a
with
I’I
and
n
+ a
II" II
in
0
A.
L
Then
L;
are equivalent on
also
be a (unique) q-projection contained in
n
x
in
A.
Since
p
P
is continuous,
is
a A.
n
aA
n
L
uniformly on
for all
II’II
Suppose
E
I"
in
and
I1" II
hence in
x
There-
A which is a dense subalgebra of a
in
I’I
majorizes
on
A.
By [8, p. 442, Lemma
B
of all hermitian minimal idempotents of
A
in the socle of
one
Let
for all
ClA(XA)
B.
5.1], the set
Q(e) e E
and so
q
E
A.
such that
Q(e)A
E
E
one mapping from
Let
E
q
is contained
be the set of all q-projections
by [4, p. 149, Lemma 6.4], there exists a unique
e e E,
For each
element
2
be a seml-slmple annihilator quasi-complemented Banach algebra
x e
B*-algebra
A.
1
is continuous and this completes the proof.
q
Let
in
n.
...)
0
onto
q
eA;
the mapping
Q
e
-
Q(e)
is a
and is called the q-derlved mapping
(see [3] and [4]).
As shown in [3, p. 475], Lemma 3.2 is not true in general, if the algebra
A
is finite dimensional.
In this case, we have the following result:
QUASI-COMPLEMENTED AND COMPLEMENTED BANACH ALGEBRAS
LEMMA 3.3.
A be a simple finite dimensional annihilator Banach
Let
algebra with a quasi-complementor
in
A.
q
Then
E
and
q
the set of all q-projections
q
E
is continuous if and only if
is a closed and bounded
q
A.
subset of
By [4, p. 143, Corollay 3.2],
PROOF.
A.
be a minimal left ideal of
H
Then
q-derived mapping.
q
A
is a Hilbert space and
By [1, p. 388, Theorem 2.4],
is continuous.
A.
is a complementor on
q
H.
taken as the B*-algebra of all linear operators on
if
313
Q
Let
Q
H
Let
can be
be the
is continuous if and only
Now Lemma 3.3 follows from Lemma 4.1 in [ii].
We have the main result of this section.
THEOREM 3.4.
Let
A
Banach algebra such that
A
0
{%
E
A
I%
be a semi-simple annihilator quasi-complemented
x
%
q
E
%
q
A
in
and let
q
Then a quasi-complementor
on
A 0,
%
A
where
I%.
This follows from Lemma 3.1, 3.2 and 3.3.
UNIFORMLY CONTINUOUS QUASI-COMPLEMENTORS.
A
In this section, we assume that
algebra with a quasi-complementor
A.
Once again,
E
the set of all q-projections in
q
x
is closed and bounded for each
is the set of all q-projections in
PROOF.
4.
for all
is finite dimensional}.
is continuous if and only if
E
ClA(XA)
as in 3.
tinuous if
{efA
B%
A.
x
f E E }
q
Also let
ClA(XA)
I%, HA,
x
for all
q%
and
A
B%
in
and
be
I’I
is denoted by
A quasi-complementor
the operator bound norm of
REMARK.
such that
q
M will be the set of all minimal right ideals of
A
The norm on
DEFINITION.
is a semi-simple annihilator Banach
q
on
A
is said to be uniformly con-
is closed and bounded with respect to
IIPfAII
PfA"
A uniformly continuous quasi-complementor
fact, by Theorem 3.4, we can assume that
A
q
is continuous.
In
is simple and finite dimensional.
P. WONG
314
Let
be a minimal left ideal of
H
H.
taken as the B*-algebra of all linear operators on
Theorem 4],
E
is bounded.
q
,,,.llPfAll
we have
lfll
Hence by Theorem 3.4,
If
and
u
Let
A.
three.
q
R
(R)* A
Theorem 5.4],
will denote the
v
(h, v)u
v)(h)
(U
x
in which
q
for all closed right ideals
We know that
PROOF.
H, u
for all
ClA(XA)
g
R
in
x
_
on
p%
BE.
[4, p. 143, Corollary 3.2], q%
and so by the proof of Theorem 4.3 in [ii],
for all closed right ideals
J%
BE.
in
By [4, p. 148,
If
H%
is
is a complementor
has the form
p%
If
and
e A).
q% (%
is continuous and so is
q
finite dimensional, then by
is infinite dimensional, this
H%
is also true by the proof of Lemma 3.2.
We show that there exists a constant
Ihll
_<
lhl
_<
M
lhll
(h e
M
HE,
such that
/I lhll
Suppose (4.1) does not hold.
I
and
IXnl kn > n.
II z n II
I,
IZnl
<
and
f
n
Xn’
e E
q
x’)
n
and
0.
Put
Write
HE,
It can be assumed that
(4.2)
% e A).
Then there exists
lXnl
/.
(h e
(4.1)
% e A).
We follow the argument in [i, p. 393, Lemma 4.3].
lhll_< lhl
H.
A.
of
induces a quasi-complementor
q%
h
for all
B
is a dense subalgebra of some dual B*-algebra
A
Then
is closed.
q
has no minimal left ideals of dimension less than
A
Suppose that
E
be a semi-simple annihilator Banach algebra with a
A
uniformly continuous quasi-complementor
in
and
is continuous.
q
defined by the relation
THEOREM 4.1.
h e H
It follows now that
q
are elements of a Hilbert space
v
H
operator on
f e E
for all
can be
[7, p. 259,
Then by
sup{l [fhl[
[lfll
Since
A
By the proof of Lemma 3.3,
A.
x
n
in
By (4.2), we can find
z
n
Yn k-i
n Xn
x
n n
+ x’n
Xn’
and
+
with
H
such that
n
in
Zn
n
fn (Yn
e C,
x’n
Hn
e H
such that
n
Yn )/(yn’ Yn )"
Then
QUASI-COMPLEMENTED AND COMPLEMENTED BANACH ALGEBRAS
{[
Hence
PfnAl
yn Yn
x II
n
II
(x)II
f A n
n
lynl
(Yn Yn
(Yn’ Yn
is unbounded and this contradicts the uniform continuity of
Therefore (4.1) holds.
q.
l(xn’
315
Now by using the argument in Theorem 4.3, in [ii],
we can complete the proof.
Theorem 4.1 shows that there is essentially one type of uniformly con-
A.
tinuous quasi-complementors on
The following result generalizes [4, p. 153, Theorem 7.6].
COROLLARY 4.2.
mentor on
PROOF.
A
A
Let
THEOREM 4.3.
B.
is a left ideal of
A
Let
is a complementor, then we have:
q
be a semi-simple annihilator Banach algebra such that
has no minimal left ideal of dimension less than three.
q
tinuous complementor
PROOF.
A
on
By [6, p. 655, Theorem 4],
and dual algebra.
Let
{1%
By [i, p. 390, Theorem 3.2],
[i, p. 391, Theorem 3.3],
J%
in
and
H%
q%,
q%
p%
Then every con-
is uniformly continuous.
minimal closed two-sided ideals
ideals
is a comple-
q
Then
This follows from Theorem 4.1 and Theorem 3.4 in [ii].
On the other hand, if
A
A
if and only if
be as in Theorem 4.1.
B
and
A
is the direct topological sum of its
% e A}
B%
each of which is a complemented
"I
be as before and
induces a complementor
P%
has the form
(J%)*
J%
p%
the norm on
on
and by
B%
for all closed right
By [i, p. 393, Lemma 4.3], there exists a constant
B%.
B%.
M
such
that
Ilhll
Let
E
A
q
B
!
Ihl
!
be the B*()-sum of
MIIhll
{B%
(h e
% e A}.
X
HX,
Then
(4.3)
A).
B
is a dual B*-algebra and
B.
coincides with the set of all hermitian minimal idempotents in
is a left ideal of
such that
B,
lba]] e k]bl
Since
it is well-known that there exists a constant
[]a]]
for all
b
in
B
and
a
in
A.
Then
k
P. WONG
316
llfxll
llPfA(X) ll
{PfA
Hence
{Pf nA
klf
<
f e E
llxll
kllxll
f
be a Cauchy sequence, where
f
large enough,
f
and
m
1% n
two-sided ideal
A
Ifnhnl 1/2
1/2
Ifnhnl
>
<
{Pf n A
that
f
f
and
1% n
with
lhnl
h
If nn
-f
Since
f
i.
Since
<_
{f
[ii],
f
f
/
n
ll(PfnA- PfA )(x)ll
x
in
A,
completes the proof.
P# nA
/
q
Let
n
-f h
mn
h
H% n
n
e
H% n
by (4.3) we have
II
lhnl MI IPfnA- PfmA I"
Therefore, we can assume
Hence
I) n
H%n
and
Since
E
h e
Ihl
< }
MI lPfnA- Pf A II
m
is a Cauchy sequence in
n
Theorem 4.2, in
for all
fm)hl
[l (f n
sup
(0),
fml%n
h
hi< Mllfnn
Let
we can choose
i
n
mn
belong to the same
fml
E
in
and
m
fm I% n
but
is a Cauchy sequence; a contradiction.
If n
and so
We show that, for
q
fn e 1%n
--< MI IPf nA- PfmA II
But
f
and
Then there exists some minimal closed
such that
be the minimal left ideal in
such that
A
in
are contained in the same minimal closed two-
n
of
e E
n
Suppose this is not so.
sided ideal.
x
It remains to show that it is closed.
is bounded.
q
for all
P
fA
in
I’I
l" I.
for some
llfnx- fxll
and so
{PfA
<
f
in
fl
klf n
f e
is closed in
q
Eq}
E
q
by
Since
llxl[
is closed.
This
QUASI-COMPLEMENTED AND COMPLEMENTED BANACH ALGEBRAS
317
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