This course, Advanced Analysis, contains some lectures which I have not written up as posts. For the topic of Banach algebras and C*-algebras the lectures I give in class follow pretty closely Arveson’s presentation from “A Short Course in Spectral Theory” (except that we do more examples in class). But there is one topic – the holomorphic functional calculus -for which I decided to take a slightly different route, and for the students’ reference I am writing up my point of view.
Throughout this lecture we fix a unital Banach algebra $latex A$. By “unital Banach algebra” we mean that $latex A$ is a Banach algebra with normalised unit $latex 1_A$. For a complex number $latex t in mathbb{C}$ we write $latex t$ for $latex t cdot 1_A$; in particular $latex 1 = 1_A$. The spectrum $latex sigma(a)$ of an element $latex a in A$ is the set
$latex sigma(a) = {t in mathbb{C} : a- t textrm{ is not invertible in } A}. $
The resolvent set of $latex a$, $latex rho(a)$, is defined to be the complement of the spectrum,
$latex rho(a) = mathbb{C} setminus sigma(a)$.