Related papers: The Range of United K-Theory

We provide a homotopy theorist's point of view on $KK$- and $E$-theory for $C^{*}$-algebras. We construct stable $\infty$-categories representing these theories through a sequence of Dwyer-Kan localizations of the category of…

We give an example of a unital C*-algebra $\mathbf{A}$ with a computable presentation and for which neither $K_0(\mathbf{A})$ nor $K_1(\mathbf{A})$ has a computable presentation.

Using the Evans spectral sequence and its counter-part for real $K$-theory, we compute both the real and complex $K$-theory of several infinite families of $C^*$-algebras based on higher-rank graphs of rank $3$ and $4$. The higher-rank…

We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell-Jones conjecture…

We prove that a simple, separable, nuclear, purely infinite classifiable $C^*$-algebra is weakly semiprojective if and only if its $K$-groups are direct sums of cyclic groups.

We decompose the K-theory space of a Waldhausen category in terms of its Dwyer-Kan simplicial localization. This leads to a criterion for functors to induce equivalences of K-theory spectra that generalizes and explains many of the criteria…

K-theory and Ext are computed for the C*-algebra C*(E) of any countable directed graph E. The results generalize the K-theory computations of Raeburn and Szymanski and the Ext computations of Tomforde for row-finite graphs. As a…

A category is described to which the Cuntz semigroup belongs and as a functor into which it preserves inductive limits.

Following Elliott's earlier work, we show that the Elliott invariant of any finite separable simple $C^*$-algebra with finite nuclear dimension can always be described as a scaled simple ordered group pairing together with a countable…

We show that bounded type implies finite type for a constructible subcategory of the module category of a finitely generated algebra over a field, which is a variant of the first Brauer-Thrall conjecture. A full subcategory is constructible…

We introduce a general framework to unify several variants of twisted topological $K$-theory. We focus on the role of finite dimensional real simple algebras with a product-preserving involution, showing that Grothendieck-Witt groups…

We study the range of a classifiable class ${\cal A}$ of unital separable simple amenable $C^*$-algebras which satisfy the Universal Coefficient Theorem. The class ${\cal A}$ contains all unital simple AH-algebras. We show that all unital…

We begin the study of unitary representations of Hecke algebras of complex reflections groups. We obtain a complete classification for the Hecke algebra of the symmetric group $\mathfrak{S}_n$ over the complex numbers. Interestingly, the…

It is proved that the assembly maps in algebraic K- and L-theory with respect to the family of finite subgroups is injective for groups with finite asymptotic dimension that admit a finite model for the classifying space for proper actions.…

We introduce the Cuntz-Thomsen picture of $\mathcal{C}$-equivariant Kasparov theory, denoted $\mathrm{KK}^\mathcal{C}$, for a unitary tensor category $\mathcal{C}$ with countably many isomorphism classes of simple objects. We use this…

We show that if two rings have equivalent derived categories then they have the same algebraic K-theory. Similar results are given for G-theory, and for a large class of abelian categories.

We classify graph C*-algebras, namely, Cuntz-Krieger algebras associated to the Bass-Hashimoto edge incidence operator of a finite graph. This is done by a purely graph theoretical calculation of the K-theory and the position of the unit…

A universal coefficient theorem in the setting of Kirchberg's ideal-related KK-theory was obtained in the fundamental case of a C*-algebra with one specified ideal by Bonkat and proved there to split, unnaturally, under certain conditions.…

Let $A$ be a finite-dimensional algebra over a field $k$. We define $A$ to be $\mathbf{C}$-dichotomic if it has the dichotomy property of the representation type on complexes of projective $A$-modules. $\mathbf{C}$-dichotomy implies the…

By the Telescope Conjecture for Module Categories, we mean the following claim: "Let R be any ring and (A, B) be a hereditary cotorsion pair in Mod-R with A and B closed under direct limits. Then (A, B) is of finite type." We prove a…