Bott periodicity for C*-algebras

Builds on K0 of a C*-algebra K1 of a C*-algebra Unitary element C*-algebra Topological K-theory

Statement

Let $A$ be a C*-algebra with unitization $\widetilde{A}$ . Let $SA$ be its suspension, viewed as the C*-algebra of continuous functions $f \colon \mathbb{T} \to A$ with $f(1) = 0$ , where $\mathbb{T}$ is the unit circle. Every element of $K_0(A)$ has the form $[p] - [s(p)]$ , where $p$ is a projection in some $M_n(\widetilde{A})$ . Here $s(p)$ is the scalar projection obtained by applying the quotient map $\widetilde{A} \to \C$ entrywise. For a projection $p \in M_n(\widetilde{A})$ , let $f_p$ be the loop of unitaries $f_p(z) = z p + (1_n - p)$ , $z \in \mathbb{T}$ . Then the formula $\beta_A\big([p] - [s(p)]\big) = \big[f_p f_{s(p)}^*\big]$ defines an isomorphism of abelian groups $\beta_A \colon K_0(A) \to K_1(SA)$ , natural in $A$ , called the Bott map. Together with the natural isomorphism $K_1(A) \cong K_0(SA)$ , it gives natural isomorphisms $K_0(S^2 A) \cong K_0(A), \qquad K_1(S^2 A) \cong K_1(A).$

Remarks

The loop $f_p f_{s(p)}^*$ equals $1_n$ at $z = 1$ , so it is a unitary in matrices over the unitization of $SA$ whose class lies in $K_1(SA)$ . One proof, due to Cuntz, uses the Toeplitz extension $0 \to \mathcal{K} \to \mathcal{T} \to C(\mathbb{T}) \to 0$ tensored with $A$ . Defining $K_n(A) = K_0(S^n A)$ for $n \ge 0$ , the theorem says $K_{n+2}(A) \cong K_n(A)$ , so operator K-theory has only two distinct groups. For $A = C(X)$ with $X$ compact Hausdorff, it recovers Bott periodicity in topological K-theory. Example: for $A = \C$ it gives $K_1(C_0(\R)) \cong K_0(\C) \cong \Z$ , and then $K_0(C_0(\R^2)) \cong \Z$ .

Going deeper

For a student

Bott periodicity is what lets the long exact sequence of an extension of C*-algebras close up into the six-term exact sequence. The connecting map $K_0(B) \to K_1(I)$ of that sequence, the exponential map, is built from the Bott map and the index map. For commutative algebras it contains the classical periodicity $\widetilde{K}^0(S^2 X) \cong \widetilde{K}^0(X)$ for compact spaces $X$ .

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C*-algebra concept
K0 of a C*-algebra concept
K1 of a C*-algebra concept
Topological K-theory concept
Unitary element concept
Bott periodicity for C*-algebras statement

Where this leads

3 pages build on this one.

Tag	`2VHFR8XFWK`
URI	`m2://2VHFR8XFWK`
Kind	statement, theorem
Writing	W0 stub
MSC	46L80, 19K99
Status	This copy of the site has no build data. Run `m2 status 2VHFR8XFWK`.
About	K0 of a C-algebra, K1 of a C-algebra
Uses	K0 of a C-algebra, K1 of a C-algebra, Unitary element, C*-algebra, Topological K-theory
Changed	3 November 2026

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