QUANTUM FAMILIES OF MAPS

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.