Trivolutive Banach Algebras and Their Closed Ideals

In this paper, we give a characterization of the kernel of τ in a trivolutive Banach algebra (A, τ). The main results (under a certain condition) are: (a) The closed ideal I = ker τ of a unital Banach algebra A is principal. (b) The closed ideal I = ker τ has a bounded approximate identity if A has a bounded approximate identity. Also we give some concrete example which indicates that the necessary condition must occur. Moreover, we relate trivolutions on the algebra A ×θ B with θ-Lau product to the corresponding ones on A and B.