KYUNGPOOK Math. J. 54(2014), 189-195 http://dx.doi.org/10.5666/KMJ.2014.54.2.189 The Structure of Maximal Ideal Space of Certain Banach Al- gebras of Vector-valued Functions Abbas Ali Shokri ∗ Department of Mathematics, Ahar Branch, Islamic Azad University, Ahar, Iran e-mail : a-shokri@iau-ahar.ac.ir Ali Shokri Department of Mathematics, Faculty of Basic Science, University of Maragheh, Maragheh, Iran e-mail : shokri@maragheh.ac.ir Abstract. Let X be a compact metric space, B be a unital commutative Banach algebra and α ∈ (0, 1]. In this paper, we first define the vector-valued (B-valued) α-Lipschitz operator algebra Lip α (X, B) and then study its structure and characterize of its maximal ideal space. 1. Introduction Let (X, d) be a compact metric space with at least two elements and (B, ‖ . ‖) be a Banach space over the scaler field F (= R or C ). For a constant 0 <α ≤ 1 and an operator f : X → B, set p α (f ) := sup s=t ‖ f (t) − f (s) ‖ d α (s, t) ; (s, t ∈ X), which is called the Lipschitz constant of f . Define Lip α (X, B) := {f : X → B : p α (f ) < ∞}, and for 0 <α< 1 lip α (X, B) := {f : X → B : ‖f (t)−f (s)‖ d α (s,t) → 0 as d(s, t) → 0, s, t ∈ X, s = t}. The elements of Lip α (X, B) and lip α (X, B) are called big and little α-Lipschitz operators, respectively [1]. Let C(X, B) be the set of all continuous operators from * Corresponding Author. Received April 25, 2011; accepted March 27, 2013. 2010 Mathematics Subject Classification: 47B48, 46J10. Key words and phrases: Injective norm, Banach algebras, Isometrically isomorphic, Max- imal ideal space. 189