ABSTRACT

Let and be two G-Banach algebras and be the right operator Banach algebra and be the left operator Banach algebra of. We give a characterization of the Jacobson radical for the projective tensor product in terms of the Jacobson radical for. If and are isomorphic, then we show that this characterization can also be given in terms of the Jacobson radical for.

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 R_i be the right operator Banach algebra and L_i be the left operator Banach algebra of. We give a characterization of Jacobson radical in terms of

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)

,

.

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)

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:

,

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

Definition 1.6

Let X and Y be two normed spaces. The projective tensor norm on is defined as: 

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:

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 

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

Now,

(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.

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:

where, and

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 V_i be a G-Banach algebra (over F) with right operator Banach algebra R_i 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,.

Let the G-Banach algebras V₁ and V₂ are isomorphic. In that case, we have the following result.

Theorem 2.5

Let V_i be a G-Banach algebra (over F) with right operator Banach algebra R_i 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

where,.

Let (The dual space of R₁).

We define by, where.

Then.

Similarly, for, we can define by

.

Now, let

where

In particular, taking, , we get,

where, and.

But and are arbitrary. So, Thus is well defined.

Now, Let. Then

where, and.

Again,

(2.2)

We have, , ÎV₂. So, there exist, such that,.

Now, and

So, the expression (2.2) is equal to

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,

[since and]

Butand

, 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.

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 R_B 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?

REFERENCES