K_0 of a C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a unital $C^*$ -algebra. Form the commutative monoid $V(A)$ of Murray–von Neumann equivalence classes of projections in $M_\infty(A)=\bigcup_n M_n(A)$ , with addition induced by block sum. The group $K_0(A)$ is the Grothendieck group of $V(A)$ . Thus its elements are formal differences $[p]-[q]$ , modulo stabilization and projection equivalence. If $A$ is nonunital, define

K_0(A)=\ker\bigl(K_0(\widetilde A)\to K_0(\mathbb C)\bigr),

where $\widetilde A$ is the unitization and the map comes from the scalar quotient.

Representative calculus

Two formal differences represent the same class precisely after adding suitable auxiliary projections and passing to Murray–von Neumann equivalence. A $*$ -homomorphism $f:A\to B$ sends a projection $p$ to its matrix amplification $f^{(n)}(p)$ , inducing a homomorphism $K_0(f):K_0(A)\to K_0(B)$ . The construction and its nonunital normalization are developed in Blackadar, Chapter III.

Basic properties

The functor $K_0$ is invariant under homotopy, matrix stabilization, and strong Morita equivalence. For the compact operators $\mathcal K(H)$ on an infinite-dimensional separable Hilbert space, finite-rank projections give

K_0(\mathcal K(H))\cong\mathbb Z.

For a unital algebra, $K_0(A)$ also carries a positive cone and the distinguished order-unit class $[1_A]$ , data often used in classification.

Conventions and scope

Some authors begin with stable homotopy classes of projections rather than Murray–von Neumann classes; the resulting group is canonically isomorphic. The symbol $K_0$ here denotes operator $K$ -theory of complex $C^*$ -algebras. It is not algebraic $K_0$ , although the two constructions share the Grothendieck-group pattern.

References

Bruce Blackadar, K-Theory for Operator Algebras, 2nd ed., Cambridge University Press, 1998. DOI record. Relevant: Chapter III on $K_0$ , order, and projection representatives.
N. E. Wegge-Olsen, K-Theory and C-Algebras: A Friendly Approach*, Oxford University Press, 1993. DOI record. Relevant: Chapters 5–6 on projections and $K_0$ .