Proceedings of the Estonian Academy of Sciences, 2018, 67, 3, 271–281 https://doi.org/10.3176/proc.2018.3.05 Available online at www.eap.ee/proceedings Topological spectrum of elements in topological algebras Mati Abel ∗ and Yuliana de Jesús Zárate-Rodríguez Institute of Mathematics and Statistics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia; zayuri@xanum.uam.mx Received 15 September 2017, accepted 16 January 2018, available online 28 June 2018 c ⃝ 2018 Authors. This is an Open Access article distributed under the terms and conditions of the Creative Commons Attribution- NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/). Abstract. Properties of the sets of left, right, and two-sided topologically quasi-invertible elements, topological spectra, and topological spectral radii of elements in (not necessarily unital or commutative) topological algebras are studied. We prove the spectral mapping theorem for the topological spectrum of elements in commutative complex (not necessarily unital) topological algebras and show that the topological spectral radius (as a map) is a submultiplicative seminorm in a topological algebra with a functional topological spectrum. Key words: topological algebra, topological spectrum of an element, F -algebra, continuity of quasi-invertion, functional topological spectrum. 1. INTRODUCTION First of all we introduce all the notions that will be used later on. 1.1. A topological algebra A is a topological linear space over the field K (where K is R or C) with an associative separately continuous multiplication that turns A into an algebra over K. An element x ∈ A is left (right) quasi-invertible in A if there exists an element y ∈ A (respectively, z ∈ A) such that y ◦ x = y + x − yx = θ A (respectively, x ◦ z = x + z − xz = θ A ). Here and later on, we denote the zero element in A by θ A . An element x ∈ A is quasi-invertible if it is left and right quasi-invertible. The set of all left (right) quasi-invertible elements in A is denoted by Qinv ℓ (A) (respectively, by Qinv r (A)) and the set of all quasi-invertible elements in A by Qinv(A). An element x ∈ A is topologically left (right) quasi-invertible in A if there exists a net (y λ ) λ ∈Λ (respectively, (z μ ) μ ∈Δ ) of elements of A such that (y λ ◦ x) λ ∈Λ (respectively, (x ◦ z μ ) μ ∈Δ ) converges to zero in A. An element x ∈ A is topologically quasi-invertible if it is topologically left and right quasi-invertible. The set of all topologically left (right) quasi-invertible elements in A is denoted by Tqinv ℓ (A) (respectively, by Tqinv r (A)) and the set of all topologically quasi-invertible elements in A by Tqinv(A). Let A be a topological algebra with unit e. The set of all invertible elements in A is denoted by Inv(A). Then {e}− Qinv(A)= Inv(A). Similar equalities hold for left and right invertible elements, quasi-invertible elements, topologically invertible elements, and so on. ∗ Corresponding author, mati.abel@ut.ee