Synopses & Reviews
The concept of symmetric space is of central importance in many branches of mathematics. Compactifications of these spaces have been studied from the points of view of representation theory, geometry, and random walks. This work is devoted to the study of the interrelationships among these various compactifications and, in particular, focuses on the martin compactifications. It is the first exposition to treat compactifications of symmetric spaces systematically and to uniformized the various points of view. The work is largely self-contained, with comprehensive references to the literature. It is an excellent resource for both researchers and graduate students.
Table of Contents
Preface.-Introduction.-Subalgebras and Parabolic Subgroups.-Geometrical Constructions of Compactifications.-The Satake-Furstenberg Compactifications.-The Karpelevic Compactification.-Martin compactifications.-The Martin Compactification.- The Martin Compactification.-An intrinsic approach to the boundaries of X.-Compactification via the Ground State.-Harnack Inequality, Martin's Method and the Positive spectrum for Random Walks.-the Furstenberg Boundary and Bounded Harmonic Functions.-Integral Representation of Positive Eigenfunctions of Convolution Operators.-Random Walks and Ground State Properties.-Extension to Semisimple Algebraic Groups Defined over a Local Field.-Appendix A.-Appendix B.-Bibliography.-List of Symbols.-Index.