The promise made at the beginning of Chapter 6 that βX is to be used to characterize the maximal ideals in C(X) and in C*(X), will be fulfilled in this chapter. The key to the description of the maximal ideals in C*(X) has already been given: C*(X) is isomorphic with C(βX), and the maximal ideals in the latter ring are in one-one correspondence with the points of βX.

Characterization of Maximal Ideals

We dene a new type of spectrum, called � -approximate spectrum, of an element a in a complex unital Banach algebra A and show that the � -approximate spectrum � ( a ) of a is compact. The relation between the � -approximate spectrum and the usual spec- trum is investigated. Also , an analogue of the classical Gleason- Kahane- _ Z elazko theorem is established: For each ϵ > 0, there is � > 0 such that if ϕ is a linear functional with ϕ( a ) 2 � ( a ) for all a 2 A , then ϕ is ϵ-almost multiplicative. Finally, we use these ideas to provide a sufficient condition for a

http://bims.iranjournals.ir/article_596_bec59d33577ac7d73d2c66407e9eec46.pdf

Almost multiplicative linear functionals and approximate spectrum

In order to understand the behaviour of a square matrix or a bounded linear operator on a Banach space or more generally an element of a Banach algebra, some subsets of the complex plane are associated with such an object. Most popular among these sets is the spectrum $$\sigma (a) $$
 of an element a in a complex unital Banach algebra A with unit 1 defined as follows: $$\begin{aligned} \sigma (a) := \{\lambda \in \mathbb {C}: \lambda - a \,\, \text{ is } \text{ not } \text{ invertible } \text{ in }\, A\}. \end{aligned}$$
 Here and also in what follows, we identify $$\lambda .1$$
 with $$\lambda $$
 . Also quite popular is Numerical range V(a) of a. This is defined as follows: $$\begin{aligned} V(a) := \{ \phi (a): \phi \,\, \text{ is } \text{ a } \text{ continuous } \text{ linear } \text{ functional } \text{ on } \,\, A \, \, \text{ satisfying } \, \, \Vert \phi \Vert = 1 = \phi (1) \}. \end{aligned}$$
 Then there are many generalizations, modifications, approximations etc. of the spectrum. Let $$\epsilon > 0$$
 and n a nonnegative integer. These include $$\epsilon -$$
 condition spectrum $$\sigma _{\epsilon }(a)$$
 , $$\epsilon -$$
 pseudospectrum $$\Lambda _{\epsilon }(a)$$
 and $$(n, \epsilon )-$$
 pseudospectrum $$\Lambda _{n, \epsilon }(a)$$
 . These are defined as follows: $$\begin{aligned} \sigma _{\epsilon }(a) := \left\{ \lambda \in \mathbb {C}: \Vert \lambda - a\Vert \Vert (\lambda -a)^{-1}\Vert \ge \frac{1}{\epsilon } \right\} \end{aligned}$$
 In this and the following definitions we follow the convention : $$ \Vert (\lambda -a)^{-1}\Vert = \infty $$
 if $$\lambda - a$$
 is not invertible. $$\begin{aligned}&\Lambda _{\epsilon }(a) := \left\{ \lambda \in \mathbb {C}: \Vert (\lambda -a)^{-1}\Vert \ge \frac{1}{\epsilon } \right\} \\&\Lambda _{n, \epsilon }(a) := \left\{ \lambda \in \mathbb {C}: \Vert (\lambda -a)^{-2^n}\Vert ^{ 1/2^n } \ge \frac{1}{\epsilon } \right\}. \end{aligned}$$
 In this survey article, we shall review some basic properties of these sets, relations among these sets and also discuss the effects of perturbations on these sets and the question of determining the properties of the element a from the knowledge of these sets.

https://link.springer.com/content/pdf/10.1007/s41478-019-00214-z.pdf

Spectrum and related sets: a survey

We prove by elementary methods the following generalization of a theorem due to Gleason, Kahane, and Żelazko. Let A be a real algebra with unit 1 such that the spectrum of every element in A is bounded and let φ:A→ℂ be a linear map such that φ(1)=1 and (φ(a))2

/pdf/gleason-kahane-zelazko-theorem-for-spectrally-bounded-33vtb01kaf.pdf

Daniel Sukumar

Papers

Gleason-kahane-Żelazko theorem for spectrally bounded algebra