Elements of Noncommutative Geometry, 1

Our purpose and main concern in writing this book is to illuminate classical concepts from the noncommutative viewpoint, to make the language and techniques of noncommutative geometry accessible and familiar to practi tioners of classical mathematics, and to benefit physicists interested in the uses of noncommutative spaces. Same may say that ours is a very "com mutative" way to deal with noncommutative matters; this charge we readily admit. Noncommutative geometry amounts to a program of unification of math ematics under the aegis of the quantum apparatus, i.e., the theory of ope rators and of C -algebras. Largely the creation of a single person, Alain Connes, noncommutative geometry is just coming of age as the new century opens. The bible of the subject is, and will remain, Connes' Noncommuta tive Geometry (1994), itself the "3.8-fold expansion" of the French Geome trie non commutative ( 1990). Theseare extraordinary books, a "tapestry" of physics and mathematics, in the words of Vaughan jones, and the work of a "poet of modern science," according to Daniel Kastler, replete with subtle knowledge and insights apt to inspire several generations.