Tong (Nicole) Wu

Thesis Advisor: Dr. Konrad Aguilar (Pomona College - Department of Mathematics)

Second Reader: Dr. Alfonso Castro (Harvey Mudd College - Department of Mathematics)

Contact: nwu@hmc.edu

Nonstandard Analysis, Ultraproducts, and C*-Algebras

(Abstract) Nonstandard analysis was developed by mathematical logician Abraham Robinson in 1960, where he extended the real numbers to a nonstandard structure that allowed for a clear and mathematically rigorous definition of infinitesimals. Perhaps the most noticeable thing at once of this development is that it salvaged the notion of infinitesimals. Although used frequently be mathematicians in the era of Newton and Leibnitz, infinitesimals were questioned and criticized for their lack of rigor and precise definitions. Eventually, calculus turned to limits to replace the need for infinitely small objects. The Robinson's work allowed mathematicians to once again to accept infinitesimals, this time as well-defined mathematical objects. Although this thesis does not focus on nonstandard real analysis, starting with nonstandard constructions on real numbers would give a more intuitive understanding of the process of taking nonstandard extensions of mathematical objects. Later, we will extend to nonstandard models of arbitrary objects. Any nonstandard extension can be explicitly given by an ultraproduct in a similar way to constructing the nonstandard reals. More specifically, we will work with nonstandard models of C*-algebras. The use of ultraproducts in C*-algebras and functional analysis is not a new notion, and some generalized extensions, such as the nonstandard hull construction, also exist in addition to the above ultraproduct construction.