Arens-irregularity of tensor product of Banach algebras

We introduce Banach algebras arising from tensor norms. By these Banach algebras, we make Arens regular Banach algebras such that \alpha the tensor product becomes irregular, where \alpha is tensor norm. We illustrate injective tensor product, does not preserve bounded approximate identity and it is not algebra norm.

Keywords:

Arens products , Arens regularity , compact operators , approximable operators , nuclear operators , tensor norm , approximate identity , approximation property