The Euler characteristic and the Euler-Poincaré formula for \(C^\ast\)-algebras (Q2794970)

The Euler characteristic \(\chi(A)\) of a \(C^*\)-algebra \(A\) is defined as the difference between the ranks of \(K_0(A)\) and \(K_1(A)\). The long exact \(K\)-theory sequence implies that this is additive for extensions of \(C^*\)-algebras. If \(A\) is filtered with subquotients \(B_1,\dots,B_n\), then \(\chi(A)\) is the alternating sum of \(\chi(B_i)\). This is considered here as an analogue of the Euler-Poincaré formula. The Euler characteristic vanishes for crossed products by the group of integers, and it satisfies an additivity formula for pull-backs whenever the Mayer-Vietoris sequence applies, and similarly for amalgamated free products. If the Künneth formula holds, then the Euler characteristic is multiplicative for tensor products.NEWLINENEWLINESimilarly, the Euler characteristic \(\chi(A,B)\) for two \(C^*\)-algebras \(A\) and \(B\) is defined as the difference of the ranks of \(KK_i(A,B)\) for \(i=0,1\). This enjoys similar properties.NEWLINENEWLINEThe Euler characteristic is a special case of the Lefschetz number of an element in \(KK(A,A)\). These Lefschetz numbers appeared already in [\textit{H. Emerson} and \textit{R. Meyer}, Math. Ann. 334, No. 4, 853--904 (2006; Zbl 1092.19003)]. An analogue of the Lefschetz fixed point formula for them is proved in [\textit{I. Dell'Ambrogio} et al., Doc. Math., J. DMV 19, 141--193 (2014; Zbl 1315.46079)], using a comparison between a geometric model for \(KK\)-theory and Kasparov's analytic theory.