Abstract index group

In operator theory, every Banach algebra can be associated with a group called its 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 identity component

:"G"₀,

or in other words the connected component containing the identity 1 of "A"; "G"₀ is a normal subgroup of "G". The quotient group

:Λ_"A" = "G"/"G"₀

is the abstract index group of "A". Because "G"₀, being the component of an 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 Λ_"L"("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 "G"₀ consists of elements homotopic, in "G", to the identity function "f"₁("z") = "z". Thus the index group Λ_"C"(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 "f_n"("z") = "zⁿ" as representatives of distinct homotopy classes. Thus Λ_"C"(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 Λ_"K" is again a countable discrete group. In fact, Λ_"K" 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.