This volume contains the proceedings of an AMS Special Session held at the Joint Mathematics Meetings in San Antonio in January 1993 to celebrate the first fifty years of C]*-algebra theory. The book contains carefully written expository and research articles by leaders in the field. Also included is a reprinting of the original 1943 paper on C]* -algebras by Gelfand and Neumark, which has had such a profound influence on the field. The volume covers a broad spectrum of topics, including the Gelfand-Neumark theorems, C] *-algebras and quantization, projections in C]*-algebras, Mackey's theory of group representations and their relation to C]*-algebras, transformation group C]*-algebras, the influence of algebraic topology on C]*-algebras, K-theory and index theory in operator algebras, exponential rank in C]*-algebras, and a survey of the development of type III von Neumann algebras. With historical perspectives and up-to-date overviews to orient readers new to the field, this book will interest mathematicians, physicists, and mathematical historians.
On the imbedding of normed rings into the ring of operators in Hilbert space by I. M. Gelfand and M. A. Neumark Notes on the Gelfand-Neumark theorem by R. V. Kadison $C^\ast$-algebras and quantization by I. E. Segal Quantization and $C^\ast$-algebras by M. A. Rieffel Some quantizations and reflections inspired by the Gelfand-Naimark theorem by E. G. Effros The role of $C^\ast$-algebras in infinite dimensional numerical linear algebra by W. B. Arveson Projections in $C^\ast$-algebras by B. Blackadar $C^\ast$-algebras and Mackey's theory of group representations by J. Rosenberg Transformation group $C^\ast$-algebras: A selective survey by J. Packer Algebraic topology and $C^\ast$-algebras by C. Schochet Twenty-five years in the theory of type III von Neumann algebras by M. Takesaki Classifying space for proper actions and $K$-theory of group $C^\ast$-algebras by P. Baum, A. Connes, and N. Higson Odd index theorems and operator algebras by R. G. Douglas Index theory without symbols by H. Moscovici and F. Wu A survey of exponential rank by N. C. Phillips Appendix On the imbedding of normed rings into the ring of operators in Hilbert space by I. Gelfand and M. Neumark.