In this project the principal investigator will study the structure of unital, simple, amenable C*-algebras and homomorphisms from one such C*-algebra to another. From a theorem of Gelfand we know that a unital commutative C*-algebra is isomorphic to the algebra of continuous functions on some compact Hausdorrf space. The structure of a commutative C*-algebra is thus completely determined by the underlying space, or the topological structure of the space. A homomorphism from one commutative C*-algebra to another is induced and determined by a continuous map from one underlying space to another. Thus, in the noncommutative setting, this project comes down to a study of noncommutative topology. The central goals of the project are (1) to use K-theory-related data to classify separable, simple, amenable C*-algebras, (2) to determine approximate unitary equivalence classes of homomorphisms, and (3) to find applications to the study of noncommutative topology and topological dynamical systems.
The simplest C*-algebra is the system of all complex numbers. The next most simple C*-algebras are systems of matrices of complex numbers. In general, C*-algebras are systems of operators (which can be thought of as generalizations of matrices). For example, differentiation and integration are operators on certain function spaces. Operators can also be used, for example, as models for observables for the microscopic physical world. A system of operators has the structure of addition and multiplication, just like the system of numbers. Unlike the system of numbers, where two times three is the same as three times two, in a general C*-algebra the product A times B may not be the same as B times A. This noncommutativity reflects the reality of quantum physics and corresponds to the famous Heisenberg uncertainty principle. C*-algebras arise in many diverse areas of science and engineering, of which quantum mechanics is just one important example. For purposes of application, as well as for theoretical reasons, it is important to understand the structure of C*-algebras, or the structure of systems of operators formed from different applications. The aim of this project is to find the simplest essential data that determine both the structure of a C*-algebra and the relations that exist between C*-algebras so that applications become possible. To be useful, the data should be easy to obtain and relatively easy to compute. Furthermore, if two C*-algebras give rise to the same set of data, then the algebras should be identical for purposes of all applications. The prinicipal investigator has to search these data and invent tools to provide a proof that such data can indeed be used to determine completely the structure of the corresponding C*-algebras. The expected immediate applications will be to the study of dynamical systems. However, a long-term impact should be felt in many other areas of mathematics (e.g., linear algebra, operator theory, group representations, noncommutative topology, noncommutative geometry). In the last few years, some related research involved the training of several Ph.D. students. This project will also include both graduate student training and the mentoring of postdoctoral researchers.
This project studied $C^*$-algebras. The set of all complex numbers is a $C^*$-algebra. The set of all $n imes n$ matrices is another $C^*$-algebra. It is common for $C^*$-algebras to have infinite dimension and have non-commutative multiplication. Many mathematical problems that arise from different fields of science may be solved through studying certain relevant $C^*$-algebras. The main goals of this project was to understand the structure of certain commonly appearing $C^*$-algebras and maps between them. A great number of unital simple $C^*$-algebras appeared with the property of being locally AH. To be exploit this in applications, one needs to know how to distinguish and identify algebras with this property. Based on some previously published results, the PI found that just by computing a small amount of data, also known as the Elliott invariant, he was able to completely determine the structure of these $C^*$-algebras. This discovery provides important new ways to apply $C^*$-algebra theory to other fields. For example, in the study of minimal dynamical systems, a crucial sub algebra could be easily determined to have local AH property which eventually led the classification of certain dynamical systems up to the isomorphism of the associated crossed products. A complex number of absolute value 1 can be connected to 1 along a path of complex numbers of absolute value 1 whose emph{length} does not exceed half of the circumference of the unit circle, namely $pi.$ In the matrix algebras, a unitary matrix can also be connected via a path of unitary matrices to the identity matrix. This can be done by using the matrix logarithm. Such a path can also be constructed with length not exceeding $pi.$ In a infinite dimensional $C^*$-algebras, we may not be able to find a path of desirable length. In this world, there is a peculiar $C^*$-algebra ${cal Z}$ called the Jiang-Su algebra. It is an infinite dimensional simple $C^*$-algebra that looks very much like complex numbers. One of the successes of this project was the discovery that in ${cal Z}$ a unitary path to the identity can be found with length not exceeding $3pi.$ Since ${cal Z}$ has played great and important roles in the recent study of $C^*$-algebras, this finding is important for understanding all $C^*$-algebras. For example, from this finding, the PI and another researcher, found that the corona algebra of the stablished Jiang-Su algebra has real rank zero. %who understands this last sentence? Another achievement of this project was the establishment of criteria for determining when two maps between a given pair $C^*$-algebras are approximately unitarily equivalent. PI proved that such a criteria works for the cases that the source $C^*$-algebras are AH and the target $C^*$-algebras are in a classifiable class. The criteria is in terms of $K$-theory related data. We also included a description of which maps can be approximately diagonalized. This project has so far revealed a great deal of important structure in $C^*$-algebras and paves the way for applications in the study of topological dynamical systems, group representations, non-commutative geometry, and in quantum mechanics. The methods developed in this project have been adopted by other researchers in related fields. Outcomes of this research have been reported in several international conferences and workshops as well as summer and winter schools. There are several postdoctoral fellows whose research projects have been influenced by these outcomes. Students from many diverse backgrounds will carry on these ideas in their study and eventually in their teaching of mathematics everywhere.
