Noncommutative Differential Calculi

Until now, our noncommutative callisthenics have involved generalizations of topology and linear algebra, and a new integral. We are now ready to cross the Rubicon into differential calculus. The first, and crucial, step is to introduce first-order differential forms on a noncommutative space defined by a (complex) pre-C*-algebra a . We say crucial because, in most developed differential calculi (e.g., the usual de Rham complex of differential forms on a manifold), specification of what is to be understood by a space of 1-forms is effectively enough to introduce the full calculus. We shall start by indicating the simplest thing that one can do barehanded with just the algebra, that is, introduce the module of universal 1-forms. The construction is actually simpler if a is assumed to be noncommutative. It has a rather abstract appearance, but, as we shall eventually see, has something to do with noncommutative geometry proper. Along this road, near the end, we arrive at the Hochschild-Kostant-Rosenberg-Connes theorem, which amounts to a homological construction of differential forms. This is one of the key results in this book, and in the whole of noncommutative geometry. Along the way, we prove the Chern isomorphism theorem. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.