Abstract
Let V1 and V2 be two -Banach algebras and Ri be the right operator Banach algebra and Li be the left operator Banach algebra of Vi(i=1,2). We give a characterization of the Jacobson radical for the projective tensor product V1rV2 in terms of the Jacobson radical for R1rL2. If V1 and V2 are isomorphic, then we show that this characterization can also be given in terms of the Jacobson radical for R2rL1.
Highlights
If V1 and V2 are isomorphic, we show that this characterization can be given in terms of the Jacobson radical for R2 L1
In [1,2], using the right quasi regularity property, Kyuno and Coppage and Luh gave a characterization of Jacobson radical in -rings
An element x of a -Banach algebra V is right quasi regular if and only if for all, x is right quasi regular in the right operator Banach algebra R of V. Extending this result to the projective tensor product of -Banach algebras, we prove, Lemma 2.2
Summary
In [1,2], using the right quasi regularity property, Kyuno and Coppage and Luh gave a characterization of Jacobson radical in -rings. Definition 1.3 A subset I of a -Banach algebra V is said to be a right (left) -ideal of V if 1) I is a subspace of V (in the vector space sense); 2) x y I y x I x I , ; y V i.e., I V I V I I . Definition 1.7 Let V be a -Banach algebra. An element x V is called right quasi regular if for any , there exist i , vi V , i 1, 2, , n such that n n v x v ivi v x ivi 0 v V
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