Elliptic Operators over C*-Algebras

Abstract

Just as in the usual elliptic theory, in the theory of nonlocal elliptic operators we need not restrict ourselves to individual operators but can also consider families of nonlocal elliptic operators. The index of such a family is already not a number but an element of the K-group of the parameter space (or, which is the same, the K-group of the algebra of continuous functions on the parameter space.) In the classical elliptic theory, there is a long- and well-known construction that permits studying both the problem on the index of a single operator and the problem on the index of families, as well as several other problems (for more detail, see the literature cited below) in the framework of a unified approach related to studying the elliptic operators over an arbitrary C*-algebra Λ. (The problem on the index of a single operator is obtained for Λ = ℂ, and the problem on the index of families is obtained for Λ = C₀(X), where X is the parameter space.) In the present chapter, we generalize the theory of elliptic operators over C*-algebras to the case of nonlocal elliptic operators.