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| Speaker: |
Ezio Vasselli, Roma
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| Title: |
\(KK\)-theory for \(C^*\)-precosheaves and holonomy-equivariance
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| 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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