1. Introduction
In [1,2], using the right quasi regularity property, Kyuno and Coppage and Luh gave a characterization of Jacobson radical in G-rings. Many interesting results on the internal properties of Jacobson radical for G-rings were developed in [2-5] by different research workers. In [6], some of these results are extended to G-algebras. In this paper, we consider two G-Banach algebras V1 and V2 and consider their projective tensor product
. Let Ri be the right operator Banach algebra and Li be the left operator Banach algebra of
. We give a characterization of Jacobson radical
in terms of ![](https://www.scirp.org/html/10-5300232\f5fd8ce1-a029-472a-8648-3067291282df.jpg)
Before going to present our main results, we first give some basic terminologies (refer to [5-12]) which are needed in our discussion.
Definition 1.1
Let X be a ring having the unit element e. A new multiplication called the circle composition (refer to [5]) on X is defined by:
.This composition makes sense even when X does not have the unit element. An element x of X is said to be right quasi regular if it has a right quasi inverse w.r.t. this composition, i.e., there exists x¢ÎX such that
.
Definition 1.2
Let V and G be two linear spaces over a field F. V is said to be a G-algebra over F if, for x, y,
;
,
;
, the following conditions are satisfied:
1)
;
2)
;
3)
;
4) ![](https://www.scirp.org/html/10-5300232\b3566526-28c9-4adb-a830-896066876621.jpg)
,
.
The G-algebra is denoted by
. If V and G are normed linear spaces over F, then G-algebra
is called a G-normed algebra if conditions 1) to 4) hold and further 5)
holds.
A G-normed algebra
is called a G-Banach algebra if V is a Banach space. Any Banach algebra can be regarded as a G-Banach algebra by suitably choosing G.
Definition 1.3
A subset I of a G-Banach algebra V is said to be a right (left) G-ideal of V if
1) I is a subspace of V (in the vector space sense);
2) ![](https://www.scirp.org/html/10-5300232\c8627dcd-54e3-4712-8980-eba53387a291.jpg)
i.e.,
.
A right G-ideal, which is a left G-ideal as well, is called a two-sided G-ideal or simply a G-ideal.
Definition 1.4
Let V be a G-Banach algebra and let
. Then the mapping
defined by
is a right Banach space endomorphism of V. The collection R of all endomorphisms generated by
;
,
is a Banach algebra under the operations:
,
![](https://www.scirp.org/html/10-5300232\571ee816-19cf-4894-b8dd-75c5e26f9609.jpg)
![](https://www.scirp.org/html/10-5300232\20dac3c2-1f25-4f86-9704-d63150f6bc77.jpg)
where
,
and the norm:
.
This Banach algebra is termed as the right operator Banach algebra of G-Banach algebra V. We can similarly define the left operator Banach algebra L of V as the Banach algebra generated by the set of all left endomorphisms of V in the form
where
.
Definition 1.5
Let V and
be G-Banach algebras over F and
:
be a mapping. Then
is called a G-Banach algebra homomorphism if 1)
and 2)
for all
;
and ![](https://www.scirp.org/html/10-5300232\3f014174-94d3-4cc3-ae87-31b5a446ddda.jpg)
Definition 1.6
Let X and Y be two normed spaces. The projective tensor norm
on
is defined as:
![](https://www.scirp.org/html/10-5300232\dfa510a2-cae3-4473-a2d5-e95e73347bf0.jpg)
where the infimum is taken over all (finite) representations of u. The completion of
is called the projective tensor product of X and Y, and is denoted by
.
Let
and
be G-Banach algebras over F1 and F2 isomorphic to F. The projective tensor product
with the projective tensor norm is a
-Banach algebra over F, where a multiplication is defined by the formula:
![](https://www.scirp.org/html/10-5300232\28ae59dc-a3e8-4630-9fbc-7c6d8ac1e11d.jpg)
where x,
;
,
;
,
.
Definition 1.7
Let V be a G-Banach algebra. Let
. An element x in V is said to be
-right quasi regular with a-right quasi inverse y if
. x is said to be a right quasi regular element of V if it is a-right quasi regular for each
.
Equivalently, an element
is called right quasi regular if for any
, there exist
,
,
such that
![](https://www.scirp.org/html/10-5300232\3a6721c3-36bc-42b0-b64b-327bfc30bfbf.jpg)
An ideal I of V is said to be right quasi regular if each of its elements is right quasi regular.
We have, right quasi regularity is a radical property in an algebra. The maximal right quasi regular ideal is called the Jacobson radical of V and it is denoted by J(V).
2. Main Results
In [6], we have the following Lemma regarding right quasi regularity of a G-Banach algebra and its operator algebra.
Lemma 2.1
An element x of a G-Banach algebra V is right quasi regular if and only if for all
,
is right quasi regular in the right operator Banach algebra R of V.
Extending this result to the projective tensor product of G-Banach algebras, we prove,
Lemma 2.2
Let V and
be two G and
-Banach algebras respectively. Let R be the right operator Banach algebra of V and L be the left operator Banach algebra of
. If
is right quasi regular in
, then
is right quasi regular in
for
, and conversely.
Proof. Since
is right quasi regular in
, so, for any
, there exist
,
,
such that for any
,
(2.1)
Let
. We take ![](https://www.scirp.org/html/10-5300232\fb2a3f6f-b7f5-4044-84dd-0887c233b2c9.jpg)
Now,
![](https://www.scirp.org/html/10-5300232\d5c3522b-ee03-4a32-b516-51f3a258b2bc.jpg)
(by (2.1)).
But,
is arbitrary.
So, x + y - xy = 0. Thus, x, i.e.,
is right quasi regular in
.
The converse follows in the same way. ![](https://www.scirp.org/html/10-5300232\3db26ee1-dd44-4dda-8504-80d856eae7c5.jpg)
In [13], we have defined the following ideal for the projective tensor product of V and V¢.
Lemma 2.3
Let V and
be two G and
-Banach algebras respectively. Let R be the right operator Banach algebra of V and L be the left operator Banach algebra of
. Let J be an ideal of
. We define:
![](https://www.scirp.org/html/10-5300232\3a0ec766-7a75-4ab4-9116-322b5335d628.jpg)
where
, and ![](https://www.scirp.org/html/10-5300232\dba03d33-71d7-46cd-907d-e978834722ab.jpg)
Then
is an ideal of
.
Using the above defined ideal, now, we give the characterization of Jacobson radical for the projective tensor product of two G-Banach algebras
in terms of the Jacobson radical of the projective tensor product of corresponding right and left operator Banach algebras.
Theorem 2.4
Let Vi be a G-Banach algebra (over F) with right operator Banach algebra Ri and left operator Banach algebra
respectively. Then the Jacobson radical of
is given by:
.
Proof. Let
.
Then
is a right quasi regular element of
. By Lemma 2.2, for any a,
,
is a right quasi regular element of
, i.e.,
.
So,
.
Hence,
.
Thus,
.
Conversely, let
.
Then
.
So, for any a,
,
is a right quasi regular element of
. By Lemma 2.2,
is a right quasi regular element of
, i.e.
So,
.
Thus,
. ![](https://www.scirp.org/html/10-5300232\20dc8262-f0c3-444e-a9c2-20ef7baf26c7.jpg)
Let the G-Banach algebras V1 and V2 are isomorphic. In that case, we have the following result.
Theorem 2.5
Let Vi be a G-Banach algebra (over F) with right operator Banach algebra Ri and left operator Banach algebra
respectively. If there exists a G-Banach algebra isomorphism f from V1 onto V2, then
is a homomorphic image of
.
Proof. Let
, where
,
. We define
by
![](https://www.scirp.org/html/10-5300232\a2a6b1ad-10e8-4746-891b-3f85becf03fd.jpg)
where
,
.
Let
(The dual space of R1).
We define
by
, where
.
Then
.
Similarly, for
, we can define
by
.
Now, let
where
![](https://www.scirp.org/html/10-5300232\ffbb5af6-2ea6-4e20-86cb-82e5e0e3c1ae.jpg)
In particular, taking
,
, we get,
![](https://www.scirp.org/html/10-5300232\1530ace8-ef6f-4acc-ad45-93d9468ecf2a.jpg)
where
, and
.
![](https://www.scirp.org/html/10-5300232\1981d9e3-b1ce-4d69-8b4e-268e2a999d06.jpg)
But
and
are arbitrary. So,
Thus
is well defined.
Now, Let
. Then
![](https://www.scirp.org/html/10-5300232\a7668ebe-036f-4ffe-bdd4-3c67a83d0872.jpg)
where
, and
.
![](https://www.scirp.org/html/10-5300232\7d60e96d-9e8a-483e-abdf-fef017e90812.jpg)
Again,
(2.2)
We have,
,
ÎV2. So, there exist
,
such that
,
.
Now,
and
![](https://www.scirp.org/html/10-5300232\1bbaa467-d489-4a55-b316-98f0c2375e74.jpg)
So, the expression (2.2) is equal to
![](https://www.scirp.org/html/10-5300232\2b9fe5c2-2c0b-4b36-a614-6d187ab061f3.jpg)
So,
is a homomorphism.
Since f is onto, so,
is also onto. Also, it can be shown that
is one-one.
Thus,
.
Corollary 2.6
Let the G-Banach algebras V1 and V2, as defined in Theorem 2.4 are isomorphic. Then we have,
.
Remark 2.7
If the isomorphism f from V1 onto V2 is isometric, then we can show that
is also an isometry. So, in that case,
.
The notion of direct summand for G-rings is discussed in [10] by Booth. For a G-Banach algebra V, an ideal P is called direct summand if there exists a G-ideal Q of V such that every element v of V is uniquely expressible in the form v = p + q,
,
, and V is written as
. Clearly, if
, then for
,
,
.
Now, we prove:
Deduction 2.8
If P is the direct summand for the G-Banach algebra
, then
is the direct summand for
.
Proof. Let
Clearly,
.
Let
and x = p + q, where
,
.
Since x is right quasi regular in
, so, for any
, we have, there exists
such that
.
Let
, where
,
.
So,
![](https://www.scirp.org/html/10-5300232\f89d663c-0c86-4b1b-ad09-9659cc2c7065.jpg)
[since
and
]
But
and
, and
.
So,
and
, for any
.
Thus p is right quasi regular in P and q is right quasi regular in Q, i.e.,
and
.
Hence
. ![](https://www.scirp.org/html/10-5300232\95161865-195d-4cd1-9aa8-48f43655fcf7.jpg)
In [4], there is a characterization of Jacobson radical for G-rings in terms of maximal regular left ideals.
Lemma 2.9
Let X be a G-ring. Then
where the intersection is over all maximal regular left ideals M of X.
Considering this aspect, we can raise the following problem:
Let the structures of maximal regular left ideals of the operator Banach algebras R1 and L2 are given. Using this, can we obtain the structure of the Jacobson radical for
?
In [6], Behrens radical for G-Banach algebras is introduced which contains the Jacobson radical. Let P denote the class of all subdirectly irreducible G-Banach algebras V such that the intersection of all non-zero ideals of V contains a non-zero idempotent element. The upper radical RB determined by the class P is called the Behrens radical for V.
Lemma 2.10
For a simple G-Banach algebra V,
.
Now, another problem can be raised:
Can we derive analogous result as in Theorem 2.4 in case of the Behrens radical for
?