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    QUANTUM FAMILIES OF MAPS

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    Kang_ku_0099D_15335_DATA_1.pdf (472.6Kb)
    Issue Date
    2017-05-31
    Author
    Kang, Su Chen
    Publisher
    University of Kansas
    Format
    100 pages
    Type
    Dissertation
    Degree Level
    Ph.D.
    Discipline
    Mathematics
    Rights
    Copyright held by the author.
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    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.
    URI
    http://hdl.handle.net/1808/26334
    Collections
    • Mathematics Dissertations and Theses [180]
    • Dissertations [4473]

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    785-864-8983
    KU Libraries
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    Lawrence, KS 66045
    785-864-8983

    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
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    Contact KU ScholarWorks
    785-864-8983
    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    785-864-8983

    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    Image Credits
     

     

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