arXiv:2202.01766v1 [math.FA] 3 Feb 2022 CONTINUOUS BILINEAR MAPS ON BANACH ⋆-ALGEBRAS B. FADAEE Abstract. Let A be a unital Banach ⋆-algebra with unity 1, X be a Banach space and φ : A × A → X be a continuous bilinear map. We characterize the structure of φ where it satisfies any of the following properties: a, b ∈ A, ab ⋆ = z (a ⋆ b = z) ⇒ φ(a, b ⋆ )= φ(z, 1) (φ(a ⋆ ,b)= φ(z, 1)); a, b ∈ A, ab ⋆ = z (a ⋆ b = z) ⇒ φ(a, b ⋆ )= φ(1,z)(φ(a ⋆ ,b)= φ(1,z)), where z ∈ A is fixed. 1. Introduction In recent years, several authors studied the linear (additive) maps that behave like homomorphisms, derivations or right (left) centalizers when acting on spe- cial products (for instance, see [1, 3, 4, 6, 7, 8, 9, 10, 16, 17] and the references therein). The above questions and the question of characterizing linear maps that preserve special products on algebras can be solved by considering bilinear maps that preserve certain product properties. Motivated by these reasons, Breˇsar et al. [5] introduced the concept of zero product (resp., Jordan product, Lie product) determined algebras. In the continuation of this discussion, the problem of charac- terizing bilinear maps at specific products was considered. We refer the reader to [2, 11, 12, 13, 14, 15, 19] and references therein for results concerning characterizing bilinear maps through special products. With the idea of the above, in this article we will characterize continuous bilinear maps on Banach ⋆-algebras through special products based on the action of the involution. Our results can be useful in study- ing the structure of Banach ⋆-algebras. Proving our main result is also technical and it is based on complex analysis. In the second section, some preliminaires and necessary tools are presented. The third section contains the main results of the article. 2. Preliminaires Let A be a Banach ⋆-algebra. In this article, we will consider the following sets, which are defined based on specific multiplications. S r⋆ A (z )= {(a, b) ∈ A × A : ab ⋆ = z }, S l⋆ A (z )= {(a, b) ∈ A × A : a ⋆ b = z }, where z ∈ A is a fixed point. In order to prove our results we need the following lemmas from the complex analysis, see [18]. MSC(2020): 46K05, 47B48, 15A86. Keywords : Banach ⋆-algebra,bilinear map. 1