dc.contributor.advisor SHEU, Albert J.L. dc.contributor.author Kang, Su Chen dc.date.accessioned 2018-04-20T22:14:11Z dc.date.available 2018-04-20T22:14:11Z dc.date.issued 2017-05-31 dc.date.submitted 2017 dc.identifier.other http://dissertations.umi.com/ku:15335 dc.identifier.uri http://hdl.handle.net/1808/26334 dc.description.abstract This dissertation investigates the theory of quantum families of maps, which formulates a non-commutative topological way to study quantum analogs of spaces of continuous mappings, classical objects of interest from general topology. The fundamental element of non-commutative topology is a C$^*$-algebra. In the theory of C$^*$-algebras, Gelfand Theorem says that every commutative C$^*$-algebra is C$^*$-isomorphic to $C_0(X)$, where $X$ is a locally compact Hausdorff space. Extending this Gelfand duality conceptually to all C$^*$-algebras (not only those commutative ones), the non-commutative or quantum topology views any C$^*$-algebra $A$ as the function algebra of a corresponding "virtual" space $\mathscr{QS}(A)$, called a quantum space. Piotr So{\l}tan defined a quantum concept of the family of all maps from a quantum space $\mathscr{QS}(M)$ to another quantum space $\mathscr{QS}(B)$, and established some general properties of related objects, extending classical results on such families of mappings. However, a lot of his results carry the assumption that $M$ is finite dimensional (and $B$ is finitely generated), only under which the quantum family of all maps was proved to exist (in a unique way). In this dissertation, we study the most fundamental and important question about the existence (and uniqueness) of the quantum space of all maps for infinite-dimensional cases, and solve it for the fundamental case of $M=C(\mathbb{N}\cup\{\bm\infty\})$ where $\mathbb{N}\cup\{\bm\infty\}$ is the one-point compactification of $\mathbb{N}$. We find that new structures outside purely C$^*$-algebraic framework are needed from the von Neumann algebra theory in order to handle such a new situation. This opens up a new direction of research in quantizing spaces of maps betweem more general quantum spaces. dc.format.extent 100 pages dc.language.iso en dc.publisher University of Kansas dc.rights Copyright held by the author. dc.subject Mathematics dc.subject C*-algebras dc.subject compact quantum semigroup dc.subject non-commutative topology dc.subject quantum families of all maps dc.subject quantum space dc.subject quantum space of all maps dc.title QUANTUM FAMILIES OF MAPS dc.type Dissertation dc.contributor.cmtemember PORTER, JACK dc.contributor.cmtemember RALSTON, JOHN dc.contributor.cmtemember SHAO, SHUANGLIN dc.contributor.cmtemember STANISLAVOVA, MILENA dc.thesis.degreeDiscipline Mathematics dc.thesis.degreeLevel Ph.D. dc.identifier.orcid dc.rights.accessrights openAccess
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