Date of Award
Doctor of Philosophy (PhD)
The Borsuk-Ulam theorem in algebraic topology shows that there are significant restrictions on how any topological sphere interacts with the antipodal action of reflection through the origin (which maps x to -x). For example, any map f from a sphere to itself which is continuous and odd (f(-x) = -f(x)) must be homotopically nontrivial. We consider various equivalent forms of the theorem in terms of the function algebras on spheres and examine which forms generalize to certain noncommutative Banach and C*-algebras with finite group actions.
Chapter 1 contains background material on C*-algebras, K-theory, and group actions. Next, in Chapter 2, we examine statements related to the Borsuk-Ulam theorem that may be applied on Banach algebras with actions of the two element group; this work indicates when roots of elements do not exist and is motivated by the results of Ali Taghavi. We see that a variant of the Borsuk-Ulam theorem on the function algebra of a sphere, written in terms of individual odd elements in the algebra, does not extend to the noncommutative setting. In Chapter 3, we show that antipodally equivariant maps between theta-deformed spheres of the same dimension are nontrivial on K-theory. This generalizes the commutative case and parallels the work of Makoto Yamashita on the q-spheres, although our methods are quite different. Finally, Chapter 4 concerns a conjecture of Ludwik Dabrowski that seeks to generalize noncommutative Borsuk-Ulam theory to arbitrary C*-algebras through the use of unreduced suspensions. We prove Dabrowski's conjecture and propose a new direction for continued study.
Chair and Committee
John Xiang . McCarthy Tang
Renato Feres, Alex Seidel, John Shareshian, Ari Stern,
Passer, Benjamin, "Noncommutative Borsuk-Ulam Theorems" (2016). Arts & Sciences Electronic Theses and Dissertations. 775.