A note on the \(L\)-theory of infinite product categories (Q2844269)

The author constructs an example where the functor \(L^{\langle-\infty\rangle}_*\) does not commute with infinite products. It was proven by \textit{G. Carlsson} and \textit{E. K. Pedersen} [Topology 34, No. 3, 731--758 (1995; Zbl 0838.55004)] that \(L^{\langle-\infty\rangle}_*\) commutes with products of families \((\mathcal{A}_i)_{i\in I}\) of small additive categories with involution satisfying that for a \(p_0\in \mathbb{N}\), \(K_p(\mathcal{A}_i)=0\) for all \(p\geq p_0\) and \(i\in I\). The author constructs a countable family of additive categories with involution using the \(C^*\)-algebras \(C(S^{2k})\), where \(k\) runs over the natural numbers, such that \(L^{\langle-\infty\rangle}_*\) does not commute with the product.