Pimsner–Voiculescu exact sequence

Remarks

Pimsner and Voiculescu proved it with a Toeplitz-type algebra $\mathcal{T}_\alpha$ that fits into an extension $0 \to A \otimes \mathcal{K} \to \mathcal{T}_\alpha \to A \rtimes_\alpha \Z \to 0$. The main step shows that the inclusion $A \to \mathcal{T}_\alpha$ is an isomorphism on K-theory, and the six-term exact sequence of the extension then becomes the sequence above. Example: for $A = C(\mathbb{T})$ and $\alpha$ the rotation by an angle $2 \pi \theta$, the rotation is homotopic to the identity, so $\alpha_* = \id$ and $\id - \alpha_* = 0$. The sequence then splits into short exact sequences $0 \to K_i(C(\mathbb{T})) \to K_i(A_\theta) \to K_{1-i}(C(\mathbb{T})) \to 0$, which give $K_0(A_\theta) \cong K_1(A_\theta) \cong \Z^2$ for the noncommutative torus $A_\theta$. For the trivial automorphism, $A \rtimes \Z \cong A \otimes C(\mathbb{T})$, and the sequence gives short exact sequences $0 \to K_i(A) \to K_i(A \otimes C(\mathbb{T})) \to K_{1-i}(A) \to 0$.

Going deeper

For a student

Together with the Connes–Thom isomorphism for actions of $\R$, it is the basic tool for crossed products. Classical applications are the K-theory of the irrational rotation algebras and their embedding into AF algebras. It also gives the K-theory of the Cuntz algebras through a crossed product description.