Index group
In operator theory, every Banach algebra can be associated with a group called the abstract index group.
Definition
Let A be a Banach algebra and G the group of invertible elements in A. The set G is open and a topological group. Consider the component G0 containing the identity 1 ∈ A. G0 is a normal subgroup of G. The quotient group ΛA = G/G0 is called the abstract index group of A. Because G0, being the component of a open set, is both open and closed in G, the index group is a discrete group.
Examples
Let L(H) be the Banach algebra of bounded operators on a Hilbert space. The set of invertible elements in L(H) is path connected. Therefore ΛL(H) is the trivial group.
Let T denote the unit circle in the complex plane. The algebra C(T) of continuous functions on T is a Banach algebra, with the topology of uniform convergence. An element of C(T) is invertible if its image does not contain 0. The group G0 consists of elements homotopic, in G, to the identity function f 1(z) = z. Thus the index group ΛC(T) is the set of homotopy classes, indexed by the winding number of its members. It is a countable discrete group. One can choose the functions fn(z) = zn as representatives of distinct homotopy classes. Thus ΛC(T) is isomorphic to the fundamental group of T.
The Calkin algebra K is quotient C*-algebra of L(H) with respect to the compact operators. Suppose π is the quotient map. An invertible elements in C(H) is of the form π(T) where T is a Fredholm operators. The index group ΛK is again a countable discrete group. It is isomorphic to the integers Z, via the Fredholm index. In other words, for Fredholm operators, the two notions of index coincide.