Relation of the Van Est spectral sequence to \(K\)-theory and cyclic homology (Q1320948)

We study how the smooth cohomology of the infinite general linear group \(GLA\) for a Banach algebra \(A\) relates to cyclic cohomology, Lie algebra cohomology and Dennis trace. Our main result is as follows: The diagram \[ \begin{tikzcd} HH_c^\ast A \ar[r,"B"] \ar[d, "D_{sm}"'] & HC_ c^{\ast-1}A \ar[d,"\mathcal A"] \\ H_{sm}^\ast GLA \ar[r,"\lambda"'] & H_{\mathrm{Lie}}^\ast\mathfrak {gla} \end{tikzcd} \] is commutative. Here \(B\) is a boundary map in Connes' long exact sequence relating continuous cyclic cohomology to continuous Hochschild cohomology [\textit{A. Connes}, Publ. Math., Inst. Hautes Étud. Sci. 62, 41-144 (1985; Zbl 0592.46056)]. \({\mathcal A}\) denotes the dual of the alternation operation that induces an isomorphism between the primitive elements in the Lie algebra homology of \({\mathfrak {gla}}=MA\) and the cyclic homology of \(A\) [\textit{J. L. Loday} and \textit{D. Quillen}, Comment. Math. Helv. 59, 565-591 (1984; Zbl 0565.17006)]. \(\lambda\) is the classical map from the smooth cohomology of a group to its Lie algebra cohomology, which can be identified with one of the edge homomorphisms in the Van Est spectral sequence. The definition of \(D_{sm}\) rests on the observation that the dual of the Dennis trace map factors through the smooth group cohomology of \(GLA\).