- Title
- Applications of complex functions on problems in Banach algebras
- Creator
- Snyman, Jandré Dillon
- Subject
- Banach algebras
- Subject
- Functions of several complex variables
- Date Issued
- 2019
- Date
- 2019
- Type
- Thesis
- Type
- Masters
- Type
- MSc
- Identifier
- http://hdl.handle.net/10948/48643
- Identifier
- vital:41055
- Description
- In this dissertation, we provide applications of complex function theory to problems in Banach algebras. We discuss the structure of analytic functions f : D → A, where D is a domain of C and A is a Banach algebra as given by Aupetit in [3]: either the set {λ ∈ D : Sp(f(λ)) is finite} is of capacity zero, or there exists an integer n such that Sp(f(λ)) has exactly n elements, for every λ, except on a closed, discrete set of capacity zero, where the spectrum has at most n−1 elements. This deep result, which describes the structure of Sp(f(λ)) for all λ ∈ D, relies heavily on subharmonic techniques, which are also included in the dissertation. Let A and B be Banach algebras. A linear mapping φ : A → B is called a Jordan homomorphism if and only if φ(xy + yx) = φ(x)φ(y) + φ(y)φ(x) for every x, y ∈ A. This is equivalent to saying that φ(x 2 ) = φ(x) 2 for every x ∈ A. The following problem, due to I. Kaplansky, is still unsolved for the general Banach algebra case: Let A and B be unital Banach algebras and φ : A → B a unital, invertibility preserving linear mapping. Under what conditions of A and B is φ a Jordan homomorphism? The author’s honours project [24] served as an exposition of the GleasonKahane-Żelazko Theorem [7, Theorem 4], which provides an answer to Kaplansky’s problem in the case where B = C. In this dissertation we look at other special cases of Kaplansky’s problem, such as the case where A and B are von Neumann algberas, as solved by Aupetit [4, Theorem 1.3] and remark that his result holds for the more general case where A is any C ∗ -algebra that has the property that every self-adjoint element is the limit of a sequence of linear combinations of orthogonal idempotents in A, and B is a semi-simple Banach algebra. This result relies heavily on complex function theory, spectral theory and holomorphic functional calculus. We also provide detailed expositions of the work of Taylor [27] in which an operator calculus on undounded, closed linear operators is developed, and the work of Allan [1] in which a holomorphic functional calculus is defined for locally convex algebras.
- Format
- iv, 119 leaves
- Format
- Publisher
- Nelson Mandela University
- Publisher
- Faculty of Science
- Language
- English
- Rights
- Nelson Mandela University
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Collections
NMU School of Computer Science, Mathematics, Physics and Statistics
| Thumbnail | File | Description | Size | Format | |||
|---|---|---|---|---|---|---|---|
| View Details Download | SOURCE1 | Dissertation Dillon Snyman FINAL.pdf.pdf | 730 KB | Adobe Acrobat PDF | View Details Download |