A bivariant Chern character for \(p\)-summable quasihomomorphisms
Bivariant \(K\)-theory (\(KK\)-theory) has become by now the natural framework for generalized index theorems of Atiyah-Singer type. Similarly, the cohomological versions of such index theorems should find their natural framework in bivariant cyclic cohomology theory. The translation of index theory from the \(K\)-theoretic to the cohomological picture should be provided by a bivariant Chern character. One only can hope to construct such a character for a ``smooth'' version of bivariant \(K\)-theory (considering ``smooth'', holomorphically closed, dense subalgebras of \(C^\ast\)-algebras rather than \(C^\ast\)-algebras themselves is necessitated by the pathological behaviour of cyclic theories for \(C^\ast\)-algebras). The author achieves an important step towards a realization of this program: The construction of a natural homomorphism \[ ch_{\mathrm{biv}}: KK^\ast_{\mathrm{smooth}}(A,B)\to PHC^\ast(A^ \infty, B^ \infty) \] which yields the commutative diagram \[ \begin{tikzcd} K_i(A^\infty)\simeq K_i(A) \ar[r,"\otimes_A(x)"]\ar[d,"ch"'] & K_i(B)\simeq K_i(B^\infty)\ar[d,"ch"]\\ PHC^\ast(A^\infty) \ar[r,"ch_{\mathrm{biv}}(x)"'] & PHC^\ast(B^\infty)\end{tikzcd} \] The smooth bivariant \(K\)-groups \(KK^\ast_{smooth}(A,B)\) are described in terms of finitely (\(p\)-)summable quasihomomorphisms à la Cuntz. These are given by a pair of homomorphisms \[ (\varphi_ 0,\varphi_ 1): A^ \infty\to {\mathcal B}({\mathcal H})\otimes_ \pi B^ \infty \] of smooth subalgebras \(A^ \infty\subset A\), \(B^ \infty\subset B\) such that \(\varphi_ 0-\varphi_ 1\) maps \(A^ \infty\) to \(l^ p({\mathcal H})\otimes_ \pi B^ \infty\) with \(l^ p({\mathcal H})\), \(1\leq p<\infty\), the \(p\)th Schattenideal in \({\mathcal B}({\mathcal H})\). Equivalently, using the exact sequence \[ 0\to qA \to QA\simeq A * A@>id * id>> A\to 0 \] a \(p\)- summable quasihomomorphism is described by a homomorphism of pairs \[ \varphi= \varphi_ 0 *\varphi_ 1: (QA^ \infty,qA^ \infty)\to \bigl({\mathcal B}({\mathcal H})\otimes_ \pi B^ \infty, l^ p({\mathcal H})\otimes_ \pi B^ \infty\bigr). \] The operator trace defines a linear functional on the \(n\)th power of \(l^ p({\mathcal H})\) for \(n\geq p\). The maps \(i_ 0,i_ 1: A\to QA\) induce a map of complexes \[ i_{0*}- i_{1*}\in HC^ 0\bigl(A^{\infty b},F_{-1}(QA^ \infty,qA^ \infty)^ b\bigr). \] (Here \(A^ b\) denotes the total cyclic bicomplex of \(A\) and \(F_{-n}(A,I)^ b\), for \(I\subset A\) an ideal, denotes the subcomplex of elements of total \(I\)-adic valuation at least \(n\).) The author constructs a chain homotopy from \(S^ n(i_{0*}- i_{1*})\) (with \(S\) the Connes periodicity operator), to a chain map in \[ HC^{2n}\bigl(A^{\infty b}, F_{-n-1} (QA^ \infty,qA^ \infty)^ b\bigr). \] The proof uses a clever adaption of the Cartan homotopy formula for the action \({\mathcal L}_ \delta\) on the cyclic bicomplex \(C^ b\) induced by a derivation \(\delta\) on \(C\) to the analogous action \({\mathcal L}_ N\) induced by an arbitrary linear selfmap \(N\) of \(C\). (In our case the selfmap \(N\) of \(QA^ \infty\) corresponds to the number operator on the associated graded algebra \(GR_{qA^ \infty}(QA^ \infty)\).) This does not provide a nullhomotopy of \(S{\mathcal L}_ N\) (as in the case of derivations) but a chain homotopy from \(S\) to a chain map in \[ HC^ 2\bigl(F_{-k}(QA^ \infty,qA^ \infty)^ b,\;F_{-k-1}(QA^ \infty,qA^ \infty)^ b\bigr). \] Iterating these maps \(n\)-times and composing them with \(\varphi_*\) yields elements in \[ HC^{2n}\bigl(A^{\infty b},F_{-n-1} ({\mathcal B}({\mathcal H})\otimes_ \pi B^ \infty, l^ p({\mathcal H})\otimes_ \pi B^ \infty)^ b\bigr). \] Contracting finally with the trace (well defined for \(n\geq p\)), one ends up with elements \[ ch^{2n}(\varphi)\in HC^{2n} (A^ \infty,B^ \infty), \] which are natural in \((A^ \infty,B^ \infty)\) and compatible under the \(S\)-operation. Moreover, they are uniquely determined by these properties. The advantage of this Chern character over the bivariant Chern character of Jones and Kassel is that no use of excision properties of cyclic homology is made. Therefore no \(H\)-unitality assumption (a condition rarely met in practice) is necessary. Moreover this enables one to obtain explicit and natural formulas for the bivariant Chern character on the level of chain complexes.
scientific article; zbMATH DE number 54869
scientific article