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Speaker: Ezio Vasselli, Roma
Title: \(KK\)-theory for \(C^*\)-precosheaves and holonomy-equivariance
Abstract: Let \(X\) be a space with a good base \(\Delta\) ordered under inclusion. Then any \(C^*\)-precosheaf over \(\Delta\) defines a \(C^*\)-dynamical system for the fundamental group \(\pi_1(X)\), called the holonomy \(C^*\)-system. We define Kasparov cycles for \(C^*\)-precosheaves \(A,B\) and describe them in terms of holonomy-equivariant cycles carrying an additional filtration structure. This leads to the notion of holonomy-equivariant \(KK\)-theory, that we denote by \(KK^\Delta(A,B)\). When the \(C^*\)-precosheaves are those defined by ideal structures of \(C^*\)-algebras, \(KK^\Delta\) can be regarded as a twist of Kasparov-Kirchberg \(KK_X\)-theory, where \(X\) is the space acting on the Jacobson spectra: if \(\pi_1(X)\) is trivial, then there is a natural transformation \(KK_X \to KK^\Delta\). Joint works with Giuseppe Ruzzi.
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