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The Path to Gelfand Duality & Spectral Theory

Vats, Abhijeet LU (2023) In Bachelor’s Theses in Mathematical Sciences MATK11 20231
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
We provide an introduction to Banach Algebras and C*-algebras. The discussion of C*-algebras leads to the two central theorems of this thesis: Gelfand Duality & the Gelfand-Naimark-Segal (GNS) Theorem. The latter tells us that every C*-algebra may be treated as some (special) subalgebra of bounded endomorphisms of a C-Hilbert space, while the former tells us that every commutative C*-algebra can be realized as some space of continuous functions. Gelfand Duality is used, alongside some von Neumann algebra theory, as a stepping stone to the proof of the Spectral Theorem for Bounded Normal Operators. In particular, the proof depends on the crucial fact that the Gelfand spectrum of an abelian von Neumann algebra is a Stonian space.
Please use this url to cite or link to this publication:
author
Vats, Abhijeet LU
supervisor
organization
course
MATK11 20231
year
type
M2 - Bachelor Degree
subject
publication/series
Bachelor’s Theses in Mathematical Sciences
report number
LUNFMA-4147-2023
ISSN
1654-6229
other publication id
2023:K18
language
English
id
9129834
date added to LUP
2025-06-27 15:56:19
date last changed
2025-06-27 15:56:19
@misc{9129834,
  abstract     = {{We provide an introduction to Banach Algebras and C*-algebras. The discussion of C*-algebras leads to the two central theorems of this thesis: Gelfand Duality & the Gelfand-Naimark-Segal (GNS) Theorem. The latter tells us that every C*-algebra may be treated as some (special) subalgebra of bounded endomorphisms of a C-Hilbert space, while the former tells us that every commutative C*-algebra can be realized as some space of continuous functions. Gelfand Duality is used, alongside some von Neumann algebra theory, as a stepping stone to the proof of the Spectral Theorem for Bounded Normal Operators. In particular, the proof depends on the crucial fact that the Gelfand spectrum of an abelian von Neumann algebra is a Stonian space.}},
  author       = {{Vats, Abhijeet}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor’s Theses in Mathematical Sciences}},
  title        = {{The Path to Gelfand Duality & Spectral Theory}},
  year         = {{2023}},
}