Theorem of the month – Bott Periodicity for operator K-theory

In this post we review the proof of the Bott periodicity theorem for C*-algebras, generalizing the original Bott periodicity theorem from homotopy theory (and topological K-theory).

Bott periodicity is on of the major theorems of K-theory. Let’s start by introducing the needed definitions. First of all, given a unital C*-algebra $A$ the group $K_0$ can be treated as the algebraic K-group of the C*-algebra $A$ , which is just the quotient of the collection of idempotent matrices in $M_{\infty}(A) = \lim M_n(A)$ by the equivalence relation of similarity meaning two idempotents $A_n\in M_n(A)$ and $B_m \in M_m(A)$ are similar if there exists numbers $s, t\in \N$ such that $n + s = m+t$ and an invertible $V\in GL_{n+s}(A)$ such that

$V(A_n\oplus 0_s)V^{-1} = B_m\oplus 0_t \in M_{n+s}(A).$

We define the group $K_1(A) = K_0(SA)$ (which is different from the algebraic $K_1$ group of $A$ ) where

$SA = C_0((0,1), A)$

is called the suspension of $A$ . The $K_1$ group of $A$ is most often defined as the quotients

$K_1(A) = U_\infty(A)/U_\infty(A)_0 = GL_\infty(A)/GL_\infty(A)_0$

where the subscript $0$ denotes the connected component of the identity. Here we see more clearly how the topology of the algebra comes into play. See Theorem 8.2.2 of [3] for the proof of the equivalence of these two definitions.

The construction of the $K$ groups are (covariantly) functorial, by letting a morphism act on the element of the matrix representing the idempotent, and if $A$ happens to be non-unital then we define the groups $K_n(A)$ as the kernel of the lift

$\phi_\star : K_n(A_1)\to \C$

of the unique unital *-homomorphism from $A_1$ (the unitization of $A$ ) to $\C$ .

There are two important benefits to using the suspension construction to define the higher K-groups. For starters it can be used just like the spectrum of a generalized cohomology theory in topology, to produce a family of functors by the assignment

$K_n(A) = K_0(S^nA).$

Which yields for every short exact sequence of C*-algebras

$0\to B\to C\to A\to 0$

a long exact sequence of abelian groups

$... \to K_0S^n(B)\to K _0 S^n(C) \to K _0 S^n(A) \to K _0 S^{n-1}(B) \to ... \to K _0 (A)$

see theorem 11.1.12 of [1] for the definition of the connecting morphisms $\delta: KS^n A \to KS^{n-1}B$ (spoiler, the last one is simply $\delta(x) = [wp_n w^\star] - [p_n]$ where $w$ is the lift of the two unitaries $diag(u, v) \in U_{s+t}^+(C/A)$ with $u \in U_{s} (C/A)$ representing $x$ in $K_1(C/J)$ and $v$ some unitary in $U_{t} (C/A)$ for which $diag(v, u)$ becomes homotopic to the identity matrix). Note that such a unitary $v$ always exists and that $diag(u,v)$ can always be lifted, since any surjective morphism of C*-algebras induces a bijection on the connected component of the identity of the unitaries (see resp. Corollary 4.3.5 and Corollary 4.3.3 of [1]).

The other benefit of using the suspension notation is that one can more clearly see what the second K-theory groups should look like, since the split extension

$0\to SA\to C(\mathbb{T}^0, A)\to A\to 0$

(here $\mathbb{T}^0$ is the circle) yields a split extension

$0\to K_1(SA)\to K_1(C( \mathbb{T}^0 , A))\to K_1(A)\to 0.$

With the identifications

$K_1(C( \mathbb{T}^0 , A)) = GL_\infty( C( \mathbb{T}^0 , A)) / GL_\infty( C( \mathbb{T}^0 , A))_0 \simeq C( \mathbb{T}^0 , GL_\infty(A))/ C(\mathbb{T}^0, GL_\infty(A) )_0$

we see that $K_1(C( \mathbb{T}^0 , A))$ is nothing but the (groupoid of) homotopy classes of loops with arbitrary base points. The lift of the quotient map is just the evaluation map at $1 \in \mathbb{T}^0 \simeq [0,1]/0\sim 1$ , so we get that the kernel of this quotient map is just

$K_2(A) = K_1(SA) = K_0(S^2A) \simeq \pi_1(GL_\infty(A)).$

For example, if $A = \C$ the fundamental group is $\pi_1(GL_\infty(\C)) = \Z$ .

We will need to know that the K-theory functor is stable (meaning $K_\star(A\otimes \mathbb{K}) = K_\star(A)$ , half exact (meaning its sends short exact sequences to sequences that are exact in the midle) and homotopy invariant. An elementary proof of these facts can be found in most book on operator theory like [2] and/or [1]. We note also that the condition of homotopy invariance is implied by split exactness and stability (see for instance Theorem 46 of this paper), and that the above conditions imply the functors $K$ is split exact and additive (Theorem 11.1.7 of [1]).

The remarkable fact is that these properties alone assures us that the functor $K_\star$ satisfies Bott periodicity theorem, that is, the infinite sequence of groups $K_\star(A) = (K_n(A))$ is determined by its first two entries. More precisely

Theorem: Bott periodicity (for C*-algebras) – Let $A$ be a C*-algebra, then

$K_{i}(A) \simeq K_{i+2}(A) \qquad \text{for all } i$

There are many proofs of this theorem. In [3] the author shows that the map

$\beta_A : K_0(A) \to K_2(A)$

defined by sending kan idempotent $[e] \in GL_\infty(A)$ to the loop $f_e \in C(\mathbb{T}^0, GL_\infty(A))$ determined by

$f_e(z) = ze +(1-e)$

is an isomorphism. Here we will repeat the proof of Cuntz, that can be found in [1]. For that we will need to introduce the Toepliz algebra $\mathcal{T}$ defined as the C*-subalgebra of $B(l^2(\N))$ generated by $I$ and the unilateral shift operator

$S(e_1, e_2, ...) = (0, e_1, e_2, ...).$

Note that $S^\star(e_1, e_2, ...) = (e_2, ...)$ , so $SS^\star(e_1, e_2, .. ) = (0, e_2, ...)$ hence

$(S^{n-1}(S^\star)^{n-1} - S^n(S^\star)^n)(e_1, e_2, ...) = (0, ..., 0, e_n, 0, ...).$

From this one can deduce (after some work) that the algebra generated by $S$ and $I$ contains all the rank 1 projections, which in turn span the finite rank operators (Theorem 2.4.6 of [2]). Since these are dense in the compact operators we conclude that $\mathbb{K}\subset \mathcal{T}$ . Since

$(S^\star S - SS^\star)(e_1, e_2, e_3, ...) = (e_1, 0, 0, ...)$

is a compact operator, we see that $S$ descends to a unitary operator on $\mathcal{T}/\mathbb{K}$ , from which we conclude that $Spec(\mathcal{T}/\mathbb{K})\subset C(\mathbb{T}).$ Note now that $(S-\lambda I)$ is not invertible for any $\lambda \in \mathbb{T}$ . One can show it is not surjective, but it is just as easy to see that it is not bounded away from 0 (meaning there exists a sequence of unit lenght vectors $e_i$ such that $Te_i \to 0$ . To see this, take $e_1$ to be a complex number such that $\sum^n |e_1|^2 = 1$ and let $v = (e_1, \lambda^-1 e_1, ...,, \lambda^{-(n-1)}e_1, 0, 0, ... )$ with $\lambda \in \mathbb{T}$ , then $||v||_2 = 1$ and

$|| (S - \lambda I)v|| = 2 |e_1| \to 0.$

A similar argument can be used to show that $(S - \lambda I) + F$ is not invertible for any finite rank operator hence, using that the invertible operators form an open set, we conclude that $(S - \lambda I) + K$ is not invertible for any compact operator. Once the argument is repeated for $(S^\star - \lambda I)$ we can conclude that $Spec(\mathcal{T}/\mathbb{K}) = C(\mathbb{T})$ .

The subalgebra of functions in $C( \mathbb{T} )$ vanishing at 1 is generated by the the polynomials centered at 1 (Weierstrass theorem), hence by the function $f(z) = 1 - z$ on $\mathbb{T}$ , which under functional calculus correspond to the operator $I- S$ in $\mathcal{T}/\mathbb{K}$ . Denote by $\mathcal{T}_0\subset \mathcal{T}$ the subalgebra generated by $I- S$ . We have

$\mathcal{T}_0/\mathbb{K}\simeq C_0(0,1)$

Now by tensoring with the exact sequence $0\to \mathbb{K}\to \mathcal{T}_0\to C_0(0,1)\to 0$ with $A$ we get an exact sequence

$0\to A\otimes \mathbb{K} \to A\otimes \mathcal{T}_0\to SA\to 0$

Recall that tensoring with a fixed C*-algebra $A$ is always exact on exact sequences of nuclear C*-algebras (just as with flat modules). Wrinting $K$ for the $K_0$ functor, we get from this a long exact sequence of K-theory groups

$... \to KS(A\otimes \mathbb{K} )\to K(SA\otimes \mathcal{T}_0) \to KS^2A\to K(A\otimes \mathbb{K} )\to K(A\otimes \mathcal{T}_0) \to KSA.$

The goal now (and a result of independent interest) is to show that $K(B\otimes \mathcal{T}_0) = 0$ for any $B$ , this would imply that the map $KS^2A \to K(A\otimes \mathbb{K}) = K(A)$ is an isomorphism. Here we use the fact that $K$ is stable. The sequence

$0\to \mathcal{T}_0\to \mathcal{T}\xrightarrow[]{p} \C \to 0$

is split by the unital morphism $j: \C\to \mathcal{T}$ since $p\circ j = id_\C$ , hence using the split exactness of $K$ we get a new split exact sequence

$0\to K\mathcal{T}_0 \to K\mathcal{T} \xrightarrow[]{p_\star} K\C\to 0$

By split exacness, $p_\star \circ j_\star = id$ . We will show that $j_\star\circ p_\star = id$ yielding that $K\mathcal{T}_0 = 0$ since $p_\star$ is an isomorphism of abelian groups and the sequence is exact).

Given a minimal projection $p$ (like $I- S S^\star$ ) in a C*-algebra $B$ there the standard way to construct an imbedding

$B\to B\otimes \mathbb{K} \qquad b\mapsto b\otimes p$

which induces an isomorphism in $K$ -theory. By letting $\sigma: K\mathcal{T}\to K(\mathcal{T}\otimes \mathbb{K})$ be the induced map using the (rank 1) projection $p = I-SS^\star$ we get an isomorphism of their respective $K$ -groups, so we may as well work with the latter.

From here on out, things will get a little more sketchy, but the idea is to use a larger C*-algebra $\hat{\mathcal{T}} = C^*(\mathbb{K}\otimes \mathcal{T}, \mathcal{T}\otimes 1) \subset \mathcal{T}\otimes \mathcal{T}$ to produce a homotopy between $id$ and $j_\star\circ p_\star$ , then using the homotopy invariance of $K$ . From the above C*-algebra one gets exact sequences

Rendered by QuickLaTeX.com

where $q_1$ is the quotient map by $\mathbb{K}\otimes \mathcal{T}$ and $q_2$ is the quotient map by $\mathbb{K}$ , and $\overline{\mathcal{T}}$ is the fibered broduct (or pullback) in the category of C*-agebras (see this article by Pedersen for a good overview of pullback and pushout constructions of C*-algebras). Note that the full subcategory of separable C*-algebras is closed under pullbacks,

The upper sequence turns out to be split exact, hence since we assume $K$ is split exact we get that $\iota_\star: K(\mathbb{K}\otimes \mathcal{T} \to K\overline{\mathcal{T}}$ is injective.

Homotopies in operator theory seem to be a wellspring of convoluted formulas and the next one is no exception. Let

$v = S\otimes 1 \quad w = (1-SS^\star)\otimes S \quad e = (1-SS^\star)\otimes (1- SS^\star).$

$u_0 = v^2v^{\star 2} - w v^\star + vw^\star + e$

and define the two self-adjoints idempotents

$u_1 = v^2v^{\star 2} + (1 - vv^\star) v^\star + v(1 - vv^\star).$

Using the fact that any self adjoint unitary $u$ is homotopic to the identity (by the path of unitaries $t\mapsto exp(it\pi(I-u)/2)$ ), we can conclude that $u_0$ and $u_1$ are homotopic.

Let

$\alpha_t: \mathcal{T}\to \hat{\mathcal{T}} \qquad \alpha_t(S) = u_t(S\otimes 1)$

note that this determines $\alpha_t$ since $S$ and $S^\star$ generate $\mathcal{T}$ and that $\alpha_0$ is clearly homotopic to $\alpha_1$ . Now set

$\beta_t : \mathcal{T} \to \overline{\mathcal{T}} \qquad \beta_t(S) = \alpha_t(S)\oplus S.$

Denote by $\gamam$ the function $\gamma(S) = v^2v^\star \oplus S$ and using the additivity of $K$ we get

$\beta_{0\star} -\gamma \star= \iota_\star \sigma_\star$

and

$\beta_{1\star} - \gamma= \iota_\star\sigma j_\star p_\star$

from homotopy invariance of $K$ we conclude that

$\iota_\star \sigma_\star = \iota_\star \sigma_\star \j_\star p_\star$

but since $\iota_\star \sigma_\star$ is injective hence a left cancellable monomorphism, we conclude that $j_\star p_\star = id$ , which concludes the proof.

Bibliography

[1] – Wegge-Olsen: K-theory: A friendly approach
[2]- Murphy: C*-Algebras and Operator Theory
[3] – Blackadar – K-theory for operator algebras

Bibliography

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