• B-names
• boolean valued models
• C*-algebras
• complex numbers
• functional analysis
• set theory

In my thesis I study how a commutative unital C*-algebra with extremely disconnected spectrum can be identified with the B-names for complex numbers in the boolean valued model for set theory V^B (B a complete boolean algebra).
I analyze C^B the set of B-names for complex numbers in the boolean model V^B. This family of objects is shown to be isomorphic (in the sense of B-boolean valued models) to C^+(St(B),C), which is the set of all continuous functions from the Stone space of B with image in the one point compactification of C such that the preimage of the infinite point is meager.
The boolean isomorphism between C^B and C^+(St(B),C) might be an interesting tool to translate ideas and results arising in set theory to ideas and results arising in the study of commutative C*-algebras and conversely.
This is the case since commutative C*-algebras can be studied in this context appealing to Gelfand Transform: given a commutative unital C*-algebra A with extremely disconnected spectrum, there is an isomorphism (which can be defined using the Gelfand Transform) of the C*-algebras A and C(St(B),C), where B is the boolean algebra given by clopen sets in the weak* topology on the spectrum of A. By means of this isomorphism A can be therefore embedded in C^B, the set of B-names for complex numbers in the boolean model V^B.
Another part of the thesis is dedicated to Shoenfield absoluteness theorem and how this can be used to carry properties from the theory of C*-algebras, seen as boolean valued models, to the first order theory of complex numbers and vice versa.