In operator theory, every Banach algebra can be associated with a group called its abstract index group.
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 identity component
or in other words the connected component containing the identity 1 of A; G0 is a normal subgroup of G. The quotient group
is the abstract index group of A. Because G0, being the component of an open set, is both open and closed in G, the index group is a discrete group.
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 in G, the constant function 1. 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 the quotient C*-algebra of L(H) with respect to the compact operators. Suppose π is the quotient map. By Atkinson's theorem, an invertible elements in K is of the form π(T) where T is a Fredholm operators. The index group ΛK is again a countable discrete group. In fact, ΛK is isomorphic to the additive group of integers Z, via the Fredholm index. In other words, for Fredholm operators, the two notions of index coincide.