##### Abstract

This thesis surveys a pair of topics which both depend on holomorphic functions and Banach Algebras.
Firstly, the prerequisite background knowledge common to both such as holomorphic functions, Banach spaces and
algebras, and module theory is provided.
Secondly, KMS states arising on Cuntz-Krieger algebras are described.
A Cuntz-Krieger graph algebra A is the universal C^* algebra satisfying certain defining relations between its partial isometries p_v which are derived from the directed graph. It becomes a C^* dynamical system when equipped with a gauge action a_t
defined on partial isometries by
a_t(p_v) = e^{it}p_v.
Its KMS states can now be studied. (The KMS
condition arises in physics in which it is
a local equilibrium condition for the states
of the operator algebra generated by local observables with the action of conjugation
by the (time) evolution operator U_t = e^{it H}.)
Examples including KMS states on matrix algebras and the generalization of Cuntz-Krieger algebras to Cuntz-Pimsner algebras are provided.
Thirdly, algebras of bounded holomorphic functions are discussed. The maximal ideal space for an algebra of bounded holomorphic functions on a Riemann surface R is described.
(In particular, this holds for domains in the complex plane C.) By the correspondence between maximal ideals and algebra homomorphism to $\mathbb{C}$, the maximal ideal
space may be equipped with its induced weak*
topology which is known as the Gelfand topology.
A couple of interesting problems are the corona problem and the complement problem.
The corona problem concerns whether the point evaluations (functionals induced by the canonical map R \to H^(R)*) are weak-star
dense in the algebra of continous functions on the maximal ideal space and the complement problem concerns
whether a nonsquare matrix in the algebra H^(R) can be augmented to become a unipotent matrix of the algebra. Wolff's proof
of the corona theorem for the open unit disk D is given.