I nternat. J. Math. & Math. Si.
Vol. 5 No. 2 (i982) 275-280
275
THE COMPACTUM AND FINITE
DIMENSIONALITY IN BANACH ALGEBRAS
ABDULLAH H. AL-MOAJIL
Gulf Organization for Industrial Consulting
P.O. Box 5114
Doha, Qatar
(Received March 2, 1981 and in revised form September 23, 1981)
ABSTRACT.
Given a Banach algebra
of elements x
g
A, the compactum of A
A such that the operator a
is defined to be the set
xax is compact.
General properties
of the compactum and its relation to the socle of A are discussed.
Characteriza-
tions of finite dimensionality of a seml-slmple Banach algebra are given in terms
of the compactum and the socle of A.
KEY WORDS AND PHRASES.
Compa operaor, socle, minimal idempotent, speum.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
47B05.
i.
Primay, 46H05, 46H10; Secondary
INTRODUCTION.
Familiarity with Banach algebras is assumed.
area is Rickart
[5], which
An elementary survey of this
is accessible to anyone with a basic knowledge of mea-
sure theory.
Let A be a Banach algebra.
Define the compactum of A to be the set C(A)
Tx(a)_
xax.
pact
operator}.
Banach algebra.
{x e A:
T E is a com-
In [i], J.C. Alexander investigated the properties of a Banach
algebra A which satisfies A
[2].
For x e A let T be the operator on A defined by
x
C(A).
He called this type of algebra a compact
The concept was also considered by Erdos, Giotopoulos and Lambrou
It has its origin in a result by Vala which states that if X is a Banach
space and T and
T’
are non-zero elements of B(X), then the operator S
compact on B(X) if and only if both T and
T’
are compact on X [3].
TST’
is
276
A.H. AL-MOAJIL
The purpose of this paper is to look at some general properties of C(A) and to
give a characterization of finite dimensionality for a semi-simple Banach algebra
A, using C(A).
2.
The latter result generalizes a theorem of A.W. Tullo
[4].
GENERAL PROPERTIES.
If A is a Banach algebra with minimal left and right ideals, and if the sume of
the minimal left ideals coincides with the sum of the minimal right ideals, then the
resulting ideal is called the socle of A.
We let S(A) denote the socle of A.
If
X is a Banach space then B(X) will stand for the algebra of bounded operators on
X and K(X) will denote the subalgebra of B(X)’consisting of compact operators.
If
A is an algebra and x e A, then (X) will denote the spectrum of x in A.
Our terminology is consist’ent with that of [5], and all algebras considered
We recall again that a Banach algebra A
are over the field of complex numbers C.
is compact if A
C(A).
Let A be a Banach algebra.
PROPOSITION I.
Then,
a)
C(A) is a closed set.
b)
If x g C(A) then the ideals xA and Ax are both contained in C(A).
c)
If B is a closed subalgebra of A such that B
C(A), then B is a
compact Banach algebra.
PROOF.
A we have
a
n)
Let (X
[l all (llx[I + [[
II (rx
be a sequence in C(A) which converges to x.
rx
(a)
n
Xnl [) IIx Xnl I.
compact for all n, we get T
ay, and since T
a
Hence
x ax
n n
ITx
x
is compact.
II < IIxax
rx
axll + II xn ax
(llx[l+[IXnll) llx- Xnl[
x
x ax
n n
where
[’1
is
conver’ges to T x and since T
x
n
Therefore, x g C(A). This proves (a).
x
n
is compact it follows that T
xy
/
ya, T
x
is compact, i.e., xy g
and
C(A).
Part (c) follows from the definition of
compact Banach algebra and the fact that if x g B
x
x
n
A, then Txy is the composition of the maps a
Similarly yx e C(A), which proves (b).
T
<
n
It follows that T
denotes the operator norm.
If x g C(A) and y
II xax
Then for each
C(A), then the restriction of
to B is still compact.
PROPOSITION 2.
(a)
If A is finite dimensional then C(A)
A.
(b)
If X is
COMPACTUM AND FINITE DIMENSIONALITY IN BANACH ALGEBRAS
a Banach space then C(B(X))
K(X).
Part (a) is obvious, while part (b) follows by Vala’s theorem which
PROOF.
states that
277
T, T’
g
K(X) if and only if the map S
TST’
is compact on
B(X) [3].
We state them here without proofs.
The next two lemmas appear in [i].
LEMMA i.
/
If A is a compact Banach algebra which is not a radical algebra,
then A contains an idempotent e such that eAe is finite dimensional [i, 4.3].
LEMMA 2.
Let A be a semi-slmple Banach algebra and x
and x g S(A) if and only if T
PROPOSITION 3.
A.
g
Then S(A) exists
has finite rank (i.e., xAx is finite dimensional).
x
Then C(A) is nonzero
Let A be a semi-slmple Banach algebra.
if and only if S(A) is nonzero, and in this case S(A) c C(A).
PROOF.
Suppose that C(A) is nonzero.
Choose x g C(A).
xA is contained in C(A), by Proposition l(b).
J
Then the right ideal
Let J be the closure of xA.
Then
Therefore, by Proposition l(c), J is a compact
C(A) since C(A) is closed.
Banach algebra, and hence, by Lema i, it contains an idempotent e such that eJe is
finite dimensional.
But since eA
J, we have eAe
It follows, by Lema 2, that e
is finite dimensional.
eJe, and hence eAe
e(eA) e
g
S(A).
If S(A) is nonzero then, applying Lema 2, we get S(A)
PROPOSITION 4.
If A is a Banach algebra and x
accumulation point for (x).
commuting with x such that
PROOF.
Moreover, if %
E
CA),
C(A).
then 0 is the only
0 then there exists y g A
(x), %
Tx (y) 2y.
First note that if A has no identity and A
I
is the Banach algebra
obtained by adjoining an identity to A in the usual manner, then the
perator
T
x
on A
as an element of A or as an element of A
I
I is still compact. Moreover, (x)
is the same. Therefore, we may assume that A has an identity.
Let C be a maximal commutative subalgebra containing x and let T be the
restriction of T
x
to C.
I, i.e., ()t- T) Sy
(T), then there exists S
If %
y for all y
C.
we choose y g C invertible, then (Sy) y
%
(x2).
2)
This says that @(x
g
B(C) such that (%
This is equivalent to
-i
(T).
is an inverse for %
()t- x2) Sy
x
2
y.
T)S=
If
That is
Since T is compact, the conclusion follows
from the general theory for compact operators and the spectral mapping theorem.
278
3.
A. H
AL-MOAJIL
CHARACTERIZATION OF FINITE DIMENSIONALITY.
For the remainder of this paper we will give a characterization of finite dlmensionality for semi-simple Banach algebras.
Our result generalizes a theorem of
A. W. Tullo which states that a semi-simple Banach algebra A which satisfies A=S(A)
is finite dimensional
0.
Before stating and proving our main theorem we need two
Let A be a semi-slmple Banach algebra such that xC(A) =(0)implles
LEMMA 3.
x
[4].
Then J contains an
Suppose that J # (0) is a closed right ideal in A.
idempotent e such that e e S(A).
Choose x e J and y g C(A) such that xy
PROOF.
xyA
We also have xyA
C(A).
Lemma 2 that e
Since A is semi-simple, it follows that
But e e xyA implies that eA
exyAe is finite dimensional.
e (eA) e
Then, by Proposition l(b),
xyA is
Hence, by Lemma i, xyA contains an idempotent e such that
not a radical algebra.
eAe
J.
# 0.
exyAe.
Wherefore, eAe
c
xyA, and hence
is finite dimensional, and it follows from
S(A).
The next lemma appears in [6], and we include it here for the sake of completeness.
We recall that an idempotent e in an algebra A is minimal if eAe is a divisThis is equivalent to saying that eA (Ae) is a minimal right (left)
ion algebra.
ideal [5].
By an idempotent we always mean a non-zero one.
LEMMA 4.
If e and f are minimal
Let A be a semi-simple normed algebra.
idempotents in A then eAf is at most 1-dimenslonal.
PROOF.
Suppose that eAf # 0 and choose x such that exf # 0.
by minimality of eA.
Moreover, by the Gelfand-Mazur theorem, fAr
the field of complex numbers.
exfAf
It follows that eAf
exfC.
Then exfA
eA
fC where C is
That is eAf is
1-dimensional.
THEOREM.
x
Let A be a semi-simple Banach algebra such that xC(A) =(0)implles
0, then the following statements are equivalent,
a)
A is finite dimensional.
b)
(A. W. Tullo) A
c)
S(A)
d)
S(A) is closed.
C(A).
S(A).
COMPACTUM AND FINITE DIMENSIONALITY IN BANACH ALGEBRAS
If (a) holds then T
PROOF.
A
279
has finite rank for every x E A.
x
Therefore,
If (b) holds then(c) follows from Proposition 3.
S(A) by Lemma 2.
that C(A) is closed shows that (c) implies (d).
The fact
It remains to show that (d) implies
(a).
Suppose that S(A) is closed.
We first show that A cannot contain an infinite
set of pairwise orthogonal idempotents.
Suppose that to the contrary such a set
{en}= I
is a non-zero closed right ideal for each
i.
Ji
A.
ei
Then
It follows by Lemma 3, that each
Let gi
Then
fi ei"
gig S(A),
{gi}i=l
idempotent and
wise
Let
exists.
contains an idempotent
Ji
and since
fi ei fi
Now let
0.
fi
gl
Since S(A) is closed It follows that y
iim
n-o
2illgifly
gi y
Therefore, the set
gi"
{gi
gi
"
Yn
/
Yn
such that
2
Then
i
2illgill
i=l
figS(A).
gi is an
Note that gi # 0, other-
n
2
fi
it follows that
is a pairwise orthogonal family.
fi fi ei fi
fi
Ji
is contained in
S(A).
Yn
S(A)
E
But we have
yAy, and since it is an
infinite set and linearly independent by the orthogonality of its elements we have
This contradicts the fact that y g S(A), by Lemma 2.
yAy is infinite dimensional.
Hence, A contains at most a finite set of pairwise orthogonal idempotents.
Now let
{el,
e
possible cardinality.
mal, then
n}
be a set of pairwise orthogona’l idempotents of maximal
Then each e
i
is minimal.
Otherwise if
Thus
elA properly
contains a closed right ideal I which, by Lemma 3,
contains an idempotent f, necessarily different from e
I
fel,
e
2
say, is not mini-
is not a minimal ideal, hence it is not a minimal closed ideal
elA
[5; 2.1.10].
e
el,
,... en}
I
and f
elf.
Then
{fel,
is a pairwise orthogonal family of idempotents which contra-
dicts the maximality of n.
Now let f
e
I +
+
e
n
We claim that f is an identity for A.
If (i
f)
A # 0, then since it is a closed ideal, Lemma 3 implies that it contains an idempotent g
(i
f) g.
gonal to e. for i
1
g
2
g(l
f) g
i,
hg
g(l
Let h
n.
f).
Then h is an idempotent which is ortho-
Then by maximality of n, we get h
0 which is a contradiction.
Hence (I
f) A
Hence g
0.
0.
Noting
that the conclusion of Lemma 3 holds for closed left ideals as well, a similar
280
A.H. AL-MOAJIL
argument as the above gives A(I
follows that A--fAf
f)
0.
Therefore f is an identity of A.
(= el)A (= ei)= ,3lei
i
e
i
Ae.3
is at most 1-dimensional for i,j
i,
n.
Ae..
J
It
But, by Lemma 4
Hence, A is finite dimensional.
This shows that (d) implies (a) and concludes the proof of the theorem.
ACKNOWLEDGEMENT.
The author would like to thank the International Center for Theoretical
Physics (ICTP) in Trieste, Italy for their encouragement and hospitality during the
work on the paper.
REFERENCES
i.
ALEXANDER, J.C. Compact Banach Algebras, Proc. London Math. Soc. (3) i_8,
(1968), pp. 1-18.
2.
ERDOS, J., GIOTOPOULOS, S., and LAMBROU, M. Rank one elements of Banach
Algebras, Mathematika 24 (1977), pp. 178-181.
3.
VALA, K.
On compact sets of compact operators, Ann. Acad. Sci. Fenn.. Ser. AI,
No. 351 (1964).
4.
TULLO, A.. Conditions on Banach algebras which imply finite dimensionality,
Proc. Edinburgh Math. Soc. (2) 2_0 (1976), pp. i-5.
5.
RICKART, C.E. General theory of Banach algebras.
N.J., MR 22 No. 5903 (1960).
6.
AL-MOAJIL, A.H.
Von Nostrand, Princeton,
Finite dimensionality of a normed algebra which is equal to
its socle, Mathematics Seminar Notes, Kobe Univ., Japan, Vol. 7 (1979),
pp. 133-137.