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 = τ of a unital Banach algebra 𝒜 is principal. (b) The closed ideal ℐ=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× _θℬ with θ -Lau product to the corresponding ones on 𝒜 and ℬ .