Index group
In operator theory, a branch of mathematics, every Banach algebra can be associated with a group called its abstract index group.
Definition
[edit]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
- G0,
or in other words the connected component containing the identity 1 of A; G0 is a normal subgroup of G. The quotient group
Λ A = G/G0
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.
Examples
[edit]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,
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 G0 consists of elements homotopic, in G, to the identity in G, the constant function 1. One can choose the functions fn(z) = zn as representatives in G of distinct homotopy classes of maps T→T. Thus the index group
The Calkin algebra K is the quotient C*-algebra of L(H) with respect to the compact operators. Suppose