I'm trying to understand a statement Segal makes in this book:

Let $C_q$ be the real Clifford algebra associated to the standard negative definite form on $\mathbb{R^q}$ and let $\Phi_q(n)$ be the space of symmetric unitary (with respect to the automorphism of $C_q$ induced by $e_i \mapsto -e_i$) $n\times n$-matrices over $C_q$.

Segal then claims that $\Phi_q=\cup_n \Phi_q(n)$ represents $KO^q$ for $q\geq 1$ and that this is a reformulation of bott periodicity.

Can someone indicate how this works or is there any good reference for this ?

I'm trying to understand a statement Segal makes in this book:

Let $C_q$ be the real Clifford algebra associated to the standard negative definite form on $\mathbb{R^q}$ and let $\Phi_q(n)$ be the space of symmetric unitary (with respect to the automorphism of $C_q$ induced by $e_i \mapsto -e_i$) $n\times n$-matrices over $C_q$.

Segal then claims that $\Phi_q=\cup_n \Phi_q(n)$ represents $KO^q$ for $q\geq 1$ and that this is a reformulation of Bott periodicity.

Can someone indicate how this works or is there any good reference for this ?