K_1 of a C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a unital $C^*$ -algebra, and form

U_\infty(A)=\bigcup_{n\geq1}U(M_n(A))

using the inclusions $u\mapsto\operatorname{diag}(u,1)$ . The group $K_1(A)$ is the set of path components of $U_\infty(A)$ , with addition induced by block sum. Equivalently, its elements are stable homotopy classes of unitaries in matrix $C^*$ -algebras over $A$ . For nonunital $A$ , define

K_1(A)=\ker\bigl(K_1(\widetilde A)\to K_1(\mathbb C)\bigr)

using the scalar quotient from the unitization.

Representative calculus

A nonunital class can be represented by a unitary $u\in M_n(\widetilde A)$ whose scalar image is $1_n$ . Stabilization permits adjoining identity blocks, and homotopy is taken through unitaries after a common stabilization. Although block sum defines the group law, $[uv]=[u]+[v]$ for stabilized unitaries. These equivalent models are proved in Blackadar, Chapter IV.

Basic properties and examples

The functor $K_1$ is homotopy invariant, matrix stable, and strongly Morita invariant. One has $K_1(\mathbb C)=0$ , while the winding number gives

K_1(C(S^1))\cong\mathbb Z.

Continuous $*$ -homomorphisms carry unitary representatives to unitary representatives and therefore induce group homomorphisms on $K_1$ .

Conventions and scope

For a nonunital algebra, a $K_1$ -representative generally belongs to a matrix algebra over $\widetilde A$ , not to $A$ itself. Definitions using connected components of stable invertibles instead of stable unitaries give the same group by polar decomposition. The adjective “odd” reflects Bott periodicity and the odd index pairing; it is not an additional grading on $A$ .

References

Bruce Blackadar, K-Theory for Operator Algebras, 2nd ed., Cambridge University Press, 1998. DOI record. Relevant: Chapter IV on $K_1$ , stable unitary groups, and Bott periodicity.
N. E. Wegge-Olsen, K-Theory and C-Algebras: A Friendly Approach*, Oxford University Press, 1993. DOI record. Relevant: Chapter 7 on $K_1$ and suspensions.