# 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 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*as representatives in G of distinct homotopy classes of maps

^{n}**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.

## References