Bounded Approximate Identities in Ternary Banach Algebras

and Applied Analysis 3 Step 1. Let F {a} be singleton. Then, there are u ∈ U and v ∈ V such that ‖uv‖ < M, and ‖ u, v, a − a‖ < M 1 . 2.2 Letting w uv ◦ uv, then ‖ uv ◦ uv, a − a‖ ‖ u, v, u, v, a − a − u, v, a − a ‖ < . 2.3 Step 2. Let F {a1, a2}. There is a u1, v1 ∈ U × V such that ‖ u1, v1, a1 − a1‖ < / 1 M , and for u1, v1, a2 − a2 ∈ A there is a u2, v2 ∈ U × V such that ‖ u2, v2, u1, v1, a2 − a2 − u1, v1, a2 − a2 ‖ < . 2.4 Put w1 u1v1 and w2 u2v2. Then ‖ w2 ◦w1, ai − ai‖ ‖ u2, v2, ai u1, v1, ai − u2, v2, u1, v1, ai − ai‖ ‖ u2, v2, ai − u1, v1, ai − ai − u1, v1, ai ‖ < , 2.5