Clifford Algebras and Bott Periodicity: The first part of the proof

3.2 The first part of the proof

We can now begin to prove Theorem A. In the remainder of this section, we shall reduce the problem to something easier via a chain of rather straightforward homotopy equivalences, and then in the next section, we shall do some work to finish it off.

The representation theory of Clifford algebras was easier to study with graded modules, but analysis is easier on ungraded things. For $k \geq 1$ , we let

\tilde{\mathcal{F}}^k(H^0) = \{ B \in \mathcal{F}(H^0): B^* = -B,\; B J_i + J_i B = 0\text{ for }i = 1, \ldots , k - 1\} ,

and whenever $k \equiv -1\pmod4$ , we set

\tilde{\mathcal{F}}^k_*(H^0) = \{ B \in \tilde{\mathcal{F}}^k_*(H^0): J_1 J_2 \cdots J_kB \text{ is not essentially definite}\} .

Otherwise, we set $\tilde{\mathcal{F}}^k_*(H^0) = \tilde{\mathcal{F}}^k(H^0)$ . Then $B \mapsto BJ_k$ gives a homeomorphism $\tilde{\mathcal{F}}^k_*(H^0) \cong \mathcal{F}^k_*(H)$ . Similarly, we set $\tilde{\mathcal{F}}^0_*(H^0) = \mathcal{F}(H^0)$ . Thus, from now on, we will write $\mathcal{F}^k_*$ to mean $\tilde{\mathcal{F}}^k_*(H^0)$ .

In this notation, we want to prove

Theorem A'

For $k \geq 1$ , there is a homotopy equivalence

\mathcal{F}^k_* \to \Omega (\mathcal{F}^{k - 1}_*).

Note that our loops start from

J_{k - 1}

-J_{k - 1}

, but that doesn't matter.

To prove the theorem, it is convenient to consider the subspace of unitary elements in our operator algebras. We make the following huge list of definitions. Let $\mathcal{A} = \operatorname{End}(H)$ , and $U$ denote the unitary elements in a Banach algebra. Then define

$\mathcal{A}^k = \{ T \in \mathcal{A} : T^* = -T,\; T J_i + J_i T = 0\text{ for }i = 1, \ldots , k - 1\}$
$\mathcal{K} =$ compact operators in $\mathcal{A}$	$\mathcal{K}^k = \mathcal{A} \cap \mathcal{A}^k$
$\mathcal{B} = \mathcal{A}/\mathcal{K}$	$\mathcal{B}^k = \mathcal{A}^k/\mathcal{K}^k$
$\mathcal{G} = \mathcal{B}^\times$	$\mathcal{G}^k = \mathcal{G}\cap \mathcal{B}^k$	$\mathcal{G}^k_* = \mathcal{G}^k \cap p(\mathcal{F}^k_*)$
$G^k_* = \mathcal{G}^k_* \cap U$	$F^k_* = p^{-1} G^k_* \cap U$
$\mathcal{L} = \mathcal{A}^\times = \mathrm{GL}(H)$	$L = \mathcal{L} \cap U$	$L^k = \mathcal{L} \cap \mathcal{A}^k$
$\Omega _k = \{ T \in \mathcal{A}^k: T^*T = 1,\; T \equiv J_{k - 1} \pmod{\mathcal{K}}\}$

where $p: \mathcal{A} \to \mathcal{A}/\mathcal{K}$ is the natural projection (recall that $\mathcal{F} = p^{-1}(\mathcal{G})$ ).

We need a theorem of Bartle and Graves.

Theorem (Bartle–Graves, [artlegraves52)

Let $U, B$ be Banach spaces and $\pi : U \to B$ be a bounded linear surjection. Then $\pi$ has a (not necessarily linear) section.

Lemma

All the arrows marked as $\sim$ in the diagram below are homotopy equivalences:

$\begin{useimager} \[ \begin{tikzcd}[row sep=small, column sep=tiny] F^k_* \ar[rrrr, bend left, red] \ar[rr, "\alpha"] \ar[dd, "\sim"] \ar[rd, "\sim"] && P(L^{k - 1}_*, \Omega_{k - 1}) \ar[rd, dash, dashed, gray] \ar[dd, green!50!black] \ar[rr, green!50!black] & & \Omega_{k - 1}\\ & \mathcal{F}^k_* \ar[rr, crossing over, dash, dashed] && \Omega \mathcal{F}^{k - 1}_* \ar[dd, "\sim"]\\ G^k_* \ar[rr, "\beta", pos=0.6] \ar[rd, "\sim"] & & \Omega G^{k - 1}_* \ar[rd, "\sim"]\\ & \mathcal{G}^k_* \ar[from=uu, crossing over, "\sim"', pos=0.3] \ar[rr, dash, gray, dashed] & & \Omega \mathcal{G}^{k - 1}_* \end{tikzcd} \] \end{useimager}$

where $\alpha (T)_t = J_{k - 1} \exp (\pi t TJ_{k - 1})$ , and $\beta$ is defined by the same formula. Note that by assumption, since elements $T \in G_*^k$ satisfy $T^2 = -1$ and anti-commute with $J_{k - 1}$ , it follows that $(TJ_{k - 1})^2 = -1$ as well. So we can equivalently write

\beta (T)_t = J_{k - 1}\cos \pi t + A \sin \pi t,

which makes it clearer that the maps indeed land in where we claim they do.

The dashed lines are there to make the diagram look like a cube, as opposed to indicating the existence of certain morphisms. The theorem would then follow once we prove that the red and green arrows are homotopy equivalences.

Proof

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$\mathcal{F}^k_* \to \mathcal{G}_*^k$ : Pick a section of $\mathcal{A} \to \mathcal{B}$ and average to give a local section, hence this is a fiber bundle with contractible fiber $\mathcal{K}^k_*$ .
$\Omega \mathcal{F}_*^{k - 1} \to \Omega \mathcal{G}_*^{k - 1}$ : Follows from the previous result (for a different $k$ ).
$G_*^k \to \mathcal{G}^k_*$ : Use the unitary retraction $x_t = x ((1 - t)\sqrt{x^* x}^{-1} + tI)$ , noting that $x^* x$ commutes with all $J_i$ .
$F_*^k \to \mathcal{F}_*^k$ : The elements of $F_*^k$ consist of the elements $x \in \mathcal{F}_*^k$ that satisfy the addition hypothesis that $\operatorname{ess.spec}x = \{ \pm i\}$ , $\| x\| = 1$ . Noting that $\operatorname{ess.spec}$ is continuous in $A$ , we can construct a deformation retract of $\mathcal{F}_*^k$ into $F_*^k$ by first scaling so that $\inf |\operatorname{ess.spec}x| = 1$ , then choose a symmetric deformation retract of the imaginary axis onto the closed interval $[-i, +i]$ and apply spectral deformation. Since such a function is odd, it preserves the matrices that commute and anti-commute with $B$ , and hence the whole process lives inside $\mathcal{F}_*^k$ .

Proof

□