## International Conference on $p$-ADIC MATHEMATICAL PHYSICS AND ITS APPLICATIONS $p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

﻿ Khodr Shamseddine
(Joint work with Jose Aguayo and Miguel Nova)

### Characterization of Compact and Self-adjoint Operators, and Study of Positive Operators on a Banach Space over the Complex Levi-Civita Field

Abstract

Let $c_0$ denote the space of all null sequences of elements of the complex Levi-Civita field $\mathcal{C}$. We define a natural inner product on $c_0$ which induces the sup-norm of $c_0$. Unlike classical Hilbert spaces, $c_0$ is not orthomodular with respect to this inner product, so we characterize those closed subspaces of $c_0$ that have orthonormal complements. We also present characterizations of normal projections, adjoint and self-adjoint operators, and compact operators on $c_0$. Then we work on some $B*$-algebras of operators on $c_0$, including those mentioned above; and we define an inner product on such algebras that induces the usual norm of operators. Finally, we study the properties of positive operators, which we then use to introduce a partial order on compact and self-adjoint operators on $c_0$.