Index group
# Index group

In operator theory, a branch of mathematics, 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 from T to the complex numbers is a Banach algebra, with the topology of uniform convergence. A function in C(T) is invertible (meaning that it has a pointwise multiplicative inverse, not that it is an invertible function) if it does not map any element of T to zero. The group $G_0$ consists of elements homotopic, in G, to the identity in G, the constant function 1. One can choose the functions $f_n(z)$ = $z^n$ as representatives in G of distinct homotopy classes of maps T→T. Thus the index group $Λ_{C(T)}$ is the set of homotopy classes, indexed by the winding number of its members. Thus $Λ_{C(T)}$ is isomorphic to the fundamental group of T. It is a countable discrete group.

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.

