Classifying spaces of a K-functor associated with a \(C^*\)-algebra (Q1063288)

The author constructs the classifying space of Karoubi's K-functors \(K^{p,q}(X;B)\) [\textit{M. Karoubi}, Trans. Am. Math. Soc. 147, 75-115 (1970; Zbl 0194.242)] on the fibre bundles with the compact space X as base space and the projective modules of finite type over the \(C^*\)- algebra B as fibers. Using the notations from the same paper, these spaces are \(K^{p,q}(B)\times \tilde {\mathfrak F}^{p,q}(\bar H_ B)\) where \(\bar H_ B=C^{p,q+2}\otimes H_ B\) and \(\tilde{\mathfrak F}^{p,q}(\bar H_ B)\) is the connected component of the generator \(\epsilon_{q+2}\) of the complex Clifford algebra \(C^{p,q+2}\) in \({\mathfrak F}^{p,q}(\bar H_ B)=\{D\in {\mathfrak F}^*\!_ B(\bar H_ B)\); \(D^*=D\), \(De_ i=-e_ iD\), \(i=1,...,p\), \(D\epsilon_ j=- \epsilon_ jD\), \(j=1,...,q+1\}\).