Contemporary Mathematics

Volume 645, 2015

http://dx.doi.org/10.1090/conm/645/12929

On algebraic properties of the spectrum and spectral radius

of elements in a unital algebra

Mart Abel

Abstract. The present paper generalizes some results about spectra and

spectral radii of elements of a unital algebra from a Banach algebra case to the

case of a complex algebra, providing the necessary conditions in an algebraic

form. We mainly consider the following questions: 1) when an element of an

algebra belongs to its center; 2) when two elements of an algebra coincide;

and 3) how to describe a circle on a complex plane, which would contain the

spectrum of an element.

1. Introduction

In many problems in mathematics one needs to know whether some element

belongs to the center of an algebra or whether two elements of the algebra are

equal. It happens that for obtaining answers to these questions, it is enough to

have certain information about the spectra or spectral radii of elements of the

algebra under consideration. In [2], [3] and [4], some results about these problems

were obtained in the case of unital semisimple Banach algebras. Our aim is, first,

to generalize this type of results for more general cases without using the topology

and, second, to characterize the properties, which are actually needed in the proofs,

algebraically. As an addition, we generalize some results, which characterize some

circles on the complex plane, which contain the spectrum of an element.

Throughout the whole paper, let A be an algebra over the field C of complex

numbers and let A have a unit eA. Denote by Inv(A) the set of invertible elements

of A and by Z(A) the center of A. For every a ∈ A, let

σ(a) = {λ ∈ C : a − λeA ∈ Inv(A)}

be the spectrum of an element a ∈ A and

ρ(a) = sup{|λ|: λ ∈ σ(a)}

the spectral radius of a.

2010 Mathematics Subject Classification. Primary 17C20; Secondary 16N20, 47Lxx.

Key words and phrases. Spectrum, spectral radius, Jacobson radical.

The research was supported by institutional research funding IUT20-57 of the Estonian Min-

istry of Education and Research. The author is grateful to the Professors Mati Abel, Maria

Fragoloupoulou, Marina Haralampidou and late professor Anastasion Mallios for the fruitful dis-

cussions and encouraging ideas.

c 2015 American Mathematical Society

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