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.
By Atkinson's theorem, an invertible elements in K is of the form π(T) where T is a Fredholm operators.
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.