Kirchberg–Phillips classification theorem

Builds on C*-algebra Nuclear C*-algebra K0 of a C*-algebra K1 of a C*-algebra KK-theory Cuntz algebra

Statement

A Kirchberg algebra is a C*-algebra that is separable, simple, nuclear and purely infinite. Here purely infinite means that every nonzero hereditary C*-subalgebra contains an infinite projection. Let $A$ and $B$ be Kirchberg algebras.

$A \otimes \mathcal{K} \cong B \otimes \mathcal{K}$ if and only if $A$ and $B$ are KK-equivalent, where $\mathcal{K}$ is the algebra of compact operators on a separable infinite-dimensional Hilbert space.
For unital $A$ and $B$ : $A \cong B$ if and only if some KK-equivalence between them induces a map on $K_0$ sending $[1_A]$ to $[1_B]$ .
Suppose moreover that $A$ and $B$ satisfy the universal coefficient theorem. Then $A \otimes \mathcal{K} \cong B \otimes \mathcal{K}$ if and only if $K_0(A) \cong K_0(B)$ and $K_1(A) \cong K_1(B)$ . If they are unital, then $A \cong B$ if and only if $K_1(A) \cong K_1(B)$ and some isomorphism $K_0(A) \cong K_0(B)$ sends $[1_A]$ to $[1_B]$ .

Remarks

The theorem was proved independently by Kirchberg and by Phillips in the mid-1990s. Two key inputs are Kirchberg's theorems $A \otimes \mathcal{O}_2 \cong \mathcal{O}_2$ and $A \otimes \mathcal{O}_\infty \cong A$ for every unital Kirchberg algebra $A$ , where $\mathcal{O}_n$ are the Cuntz algebras. Example: $\mathcal{O}_n$ is a unital Kirchberg algebra in the UCT class with $K_0(\mathcal{O}_n) \cong \Z/(n-1)\Z$ , $[1] \mapsto 1$ , and $K_1(\mathcal{O}_n) = 0$ . So every unital Kirchberg algebra in the UCT class with these invariants is isomorphic to $\mathcal{O}_n$ . The range of the invariant is known. For countable abelian groups $G_0$ , $G_1$ and $g \in G_0$ , some unital Kirchberg algebra in the UCT class has $(K_0, [1], K_1) \cong (G_0, g, G_1)$ . Whether every separable nuclear C*-algebra satisfies the universal coefficient theorem is an open problem, so the UCT hypothesis in part 3 cannot at present be removed.

Going deeper

For a student

It is the purely infinite half of the Elliott classification program, and it was settled long before the stably finite half. The strategy is to lift KK-equivalences to isomorphisms, using that these algebras absorb $\mathcal{O}_\infty$ tensorially. The UCT turns KK-equivalence into an isomorphism of K-groups, which is how part 3 follows from parts 1 and 2.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

C*-algebra concept
Cuntz algebra example
K0 of a C*-algebra concept
K1 of a C*-algebra concept
KK-theory concept
Nuclear C*-algebra concept
Kirchberg–Phillips classification theorem statement

Tag	`XF19H09NC9`
URI	`m2://XF19H09NC9`
Kind	statement, theorem
Writing	W0 stub
MSC	46L35, 46L80, 19K35
Status	This copy of the site has no build data. Run `m2 status XF19H09NC9`.
About	K0 of a C-algebra, K1 of a C-algebra
Uses	C-algebra, Nuclear C-algebra, K0 of a C-algebra, K1 of a C-algebra, KK-theory, Cuntz algebra
Changed	3 November 2026