In this short paper we present a generalization of the Hahn–Banach extension theorem for A-linear operators. Some theoretical applications and results are given. | Turk J Math (2017) 41: 1360 – 1364 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article The Hahn–Banach theorem for A-linear operators ˙ IC ˙ I˙ Bahri TURAN∗, Fatma BIL Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, Ankara, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this short paper we present a generalization of the Hahn–Banach extension theorem for A -linear operators. Some theoretical applications and results are given. Key words: Riesz space, positive operator, A -linear operator, Hahn–Banach extension theorem 1. Introduction We assume the reader to be familiar with the elementary theory of Riesz spaces and order bounded operators. In this regard, we use [1, 3, 7] as sources of unexplained terminology and notation. Moreover, all Riesz spaces under consideration are assumed to be Archimedean. We denote by Lb (E, F ) the class of all order bounded operators from the Riesz space E into F and by Lb (E) the order bounded operators from E into itself. Recall that π ∈ Lb (E) is called an orthomorphism of E if x ⊥ y in E implies that π(x) ⊥ y . Orthomorphisms of E will be denoted by Orth(E). The principal order ideal generated by the identity operator I in Orth(E) is called the ideal center of E and is denoted by Z(E). Let A be a Riesz algebra (lattice ordered algebra), . A is a Riesz space that is simultaneously an associative algebra with the additional property that a, b ∈ A+ implies that a · b ∈ A+ . An f -algebra A is a Riesz algebra that satisfies the extra requirement that a ∧ b = 0 implies a · c ∧ b = c · a ∧ b = 0 for all c ∈ A+ . If A is an Archimedean f -algebra, then A is necessarily associative and commutative. The collections Orth(E) are, with respect to composition as multiplication, Archimedean f -algebras with the identity mapping I as a unit element. Another well-known .
