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Register a new userGraded $K$-Theory, Filtered $K$-theory and the classification of graph algebras
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We prove that an isomorphism of graded Grothendieck groups $K^{gr}_0$ of two Leavitt path algebras induces an isomorphism of a certain quotient of algebraic filtered $K$-theory and consequently an isomorphism of filtered $K$-theory of their associated graph $C^*$-algebras. As an application, we show that, since for a finite graph $E$ with no sinks, $K^{gr}_0big(L(E)big)$ of the Leavitt path algebra $L(E)$ coincides with Kriegers dimension group of its adjacency matrix $A_E$, our result relates the shift equivalence of graphs to the filtered $K$-theory and consequently gives that two arbitrary shift equivalent matrices give stably isomorphic graph $C^*$-algebras. This result was only known for irreducible graphs.
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We develop methods for computing graded K-theory of C*-algebras as defined in terms of Kasparov theory. We establish grad
We establish exact sequences in $KK$-theory for graded relative Cuntz-Pimsner algebras associated to nondegenerate $C^*$-correspondences. We use this to calculate the graded $K$-theory and $K$-homology of relative Cuntz-Krieger algebras of directed graphs for gradings induced by ${0,1}$-valued labellings of their edge sets.
We initiate the study of real $C^*$-algebras associated to higher-rank graphs $Lambda$, with a focus on their $K$-theory. Following Kasparov and Evans, we identify a spectral sequence which computes the $mathcal{CR}$ $K$-theory of $C^*_{mathbb R} (Lambda, gamma)$ for any involution $gamma$ on $Lambda$, and show that the $E^2$ page of this spectral sequence can be straightforwardly computed from the combinatorial data of the $k$-graph $Lambda$ and the involution $gamma$. We provide a complete description of $K^{CR}(C^*_{mathbb R}(Lambda, gamma))$ for several examples of higher-rank graphs $Lambda$ with involution.
We show that hermitian K-theory is universal among generalized motivic cohomology theories with transfers along finite Gorenstein morphisms with trivialized dualizing sheaf. As an application, we obtain a Hilbert scheme model for hermitian K-theory as a motivic space. We also give an application to computational complexity: we prove that 1-generic minimal border rank tensors degenerate to the big Coppersmith-Winograd tensor.
Using a construction closely related to Waldhausens $S_bullet$-construction, we produce a spectrum $K(mathbf{Var}_{/k})$ whose components model the Grothendieck ring of varieties (over a field $k$) $K_0 (mathbf{Var}_{/k})$. We then produce liftings of various motivic measures to spectrum-level maps, including maps into Waldhausens $K$-theory of spaces $A(ast)$ and to $K(mathbf{Q})$. We end with a conjecture relating $K(mathbf{Var}_{/k})$ and the doubly-iterated $K$-theory of the sphere spectrum.
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