Universal Coefficient Theorem Research Articles

C*-algebras generated by representations of virtually nilpotent groups

We show that a C*-algebra generated by an irreducible representation of a finitely generated virtually nilpotent group satisfies the universal coefficient theorem and has real rank 0. This combines with previous joint work with Gillaspy and McKenney to show these C*-algebras are classified by their Elliott invariants. When we further assume the group is nilpotent we build explicit Cartan subalgebras that are closely related to the group and representation, although the Cartan subalgebras are generally not C*-diagonals.

Anomaly cancellation by generalised cohomology

Supersymmetric states in M-theory are mapped after compactification to perturbatively non-supersymmetric states in type IIA string theory, with the supersymmetric parts being encoded in the non-perturbative section of the string theory. An observer unable to recognise certain topological features of string theory will not detect supersymmetry. Such relativity of symmetry can also be derived in the context of Theorem 3 in Patrascu (Phys Rev D 90:045018, 2014). The tool of choice in this context is the universal coefficient theorem linking cohomology theories with coefficients that reveal respectively hide certain topological features. As a consequence of these observations, it is shown that the same theorem is capable of linking perturbative with non-perturbative string theoretical domains. A discussion of inflow anomaly cancellation is also included in the context of universal coefficient theorems.

A CATEGORICAL APPROACH TO THE BAUM–CONNES CONJECTURE FOR ÉTALE GROUPOIDS

Abstract We consider the equivariant Kasparov category associated to an étale groupoid, and by leveraging its triangulated structure we study its localization at the ‘weakly contractible’ objects, extending previous work by R. Meyer and R. Nest. We prove the subcategory of weakly contractible objects is complementary to the localizing subcategory of projective objects, which are defined in terms of ‘compactly induced’ algebras with respect to certain proper subgroupoids related to isotropy. The resulting ‘strong’ Baum–Connes conjecture implies the classical one, and its formulation clarifies several permanence properties and other functorial statements. We present multiple applications, including consequences for the Universal Coefficient Theorem, a generalized ‘going-down’ principle, injectivity results for groupoids that are amenable at infinity, the Baum–Connes conjecture for group bundles, and a result about the invariance of K-groups of twisted groupoid $C^*$ -algebras under homotopy of twists.

K-Theory and the Universal Coefficient Theorem for Simple Separable Exact C*-Algebras Not Isomorphic to Their Opposites

Abstract We construct uncountably many mutually nonisomorphic simple separable stably finite unital exact C*-algebras that are not isomorphic to their opposite algebras. In particular, we prove that there are uncountably many possibilities for the $K_0$-group, the $K_1$-group, and the tracial state space of such an algebra. We show that these C*-algebras satisfy the Universal Coefficient Theorem, which is new even for the already known example of an exact C*-algebra nonisomorphic to its opposite algebra produced in an earlier work.

Colored isomorphism of classifiable C-algebras

AbstractIt is shown that the colored isomorphism class of a unital, simple, $\mathcal {Z}$ -stable, separable amenable C $^*$ -algebra satisfying the universal coefficient theorem is determined by its tracial simplex.

Gorenstein duality and universal coefficient theorems

We describe a duality phenomenon for cohomology theories with the character of Gorenstein rings.

Oriented Borel–Moore homologies of toric varieties

We generalize the well known Künneth formula for Chow groups to an arbitrary oriented Borel–Moore homology theory satisfying localization and descent (e.g. algebraic bordism) when taking a product with a toric variety. As a corollary we obtain a universal coefficient theorem for the operational cohomology rings. We also give a new, homological, description for the homology groups of smooth toric varieties, which allows us to compute the algebraic bordism groups of some singular toric varieties.

The UCT problem for nuclear C*-algebras

In recent years, a large class of nuclear C∗-algebras have been classified, modulo an assumption on the universal coefficient theorem (UCT). We think this assumption is redundant and propose a strategy for proving it. Indeed, following the original proof of the classification theorem, we propose bridging the gap between reduction theorems and examples. While many such bridges are possible, various approximate ideal structures appear quite promising.

Extensions of C*-Algebras by a Small Ideal

Abstract We classify all essential extensions of the form $$ \begin{align*} &0 \rightarrow {\mathcal{W}} \rightarrow {D} \rightarrow A \rightarrow 0,\end{align*}$$where ${\mathcal {W}}$ is the unique separable simple C*-algebra with a unique tracial state, which is $KK$-contractible and has finite nuclear dimension, and $A$ is a separable amenable ${\mathcal {W}}$-embeddable C*-algebra, which satisfies the Universal Coefficient Theorem (UCT). We actually prove more general results. We also classify a class of amenable $C^*$-algebras, which have only one proper closed ideal ${\mathcal {W}}.$

The cyclic homology of k[x 1, x 2,…, x d ]/(x 1, x 2,…, x d )2

The Hochschild homology of the ring has been known and calculated several ways. This paper uses those calculations to calculate cyclic, negative cyclic, and periodic cyclic homology of over k for These calculations have been known for or fields containing it, but this paper studies the torsion which one gets for and through it, via the universal coefficient theorem, for general k. The computations are related to homology, cohomology, and Tate cohomology of cyclic groups.

The Haagerup property for twisted groupoid dynamical systems

We introduce the Haagerup property for twisted groupoid C⁎-dynamical systems in terms of naturally defined positive-definite operator-valued multipliers. By developing a version of ‘the Haagerup trick’ we prove that this property is equivalent to the Haagerup property of the reduced crossed product C⁎-algebra with respect to the canonical conditional expectation E. This extends a theorem of Dong and Ruan for discrete group actions, and implies that a given Cartan inclusion of separable C⁎-algebras has the Haagerup property if and only if the associated Weyl groupoid has the Haagerup property in the sense of Tu. We use the latter statement to prove that every separable C⁎-algebra which has the Haagerup property with respect to some Cartan subalgebra satisfies the Universal Coefficient Theorem. This generalises a recent result of Barlak and Li on the UCT for nuclear Cartan pairs.

The completion theorem in twisted equivariant K-theory for proper actions

We compare different algebraic structures in twisted equivariant K-theory for proper actions of discrete groups. After the construction of a module structure over untwisted equivariant K-theory, we prove a completion Theorem of Atiyah–Segal type for twisted equivariant K-theory. Using a universal coefficient theorem, we prove a cocompletion Theorem for twisted Borel K-homology for discrete groups.

Twisted Morava K–theory and connective covers of Lie groups

Twisted Morava K-theory, along with computational techniques, including a universal coefficient theorem and an Atiyah-Hirzebruch spectral sequence, was introduced by Craig Westerland and the first author. We employ these techniques to compute twisted Morava K-theory of all connective covers of the stable orthogonal group and stable unitary group, and their classifying spaces, as well as spheres and Eilenberg-MacLane spaces. This extends to the twisted case some of the results of Ravenel and Wilson and of Kitchloo, Laures, and Wilson for Morava K-theory. This also generalizes to all chromatic levels computations by Khorami (and in part those of Douglas) at chromatic level one, i.e. for the case of twisted K-theory. We establish that for natural twists in all cases, there are only two possibilities: either that the twisted Morava homology vanishes, or that it is isomorphic to untwisted homology. We also provide a variant on the twist of Morava K-theory, with mod 2 cohomology in place of integral cohomology.

Equivariant KK-Theory and the Continuous Rokhlin Property

Abstract We introduce and study the continuous Rokhlin property for actions of compact groups on $C^*$-algebras. An important technical result is a characterization of the continuous Rokhlin property in terms of asymptotic retracts. As a consequence, we derive strong $KK$-theoretical obstructions to the continuous Rokhlin property. Using these, we show that the Universal Coefficient Theorem (UCT) is preserved under formation of crossed products and passage to fixed point algebras by such actions, even in the absence of nuclearity. Our analysis of the $KK$-theory of the crossed product allows us to prove a ${{\mathbb {T}}}$-equivariant version of Kirchberg–Phillips: two circle actions with the continuous Rokhlin property on Kirchberg algebras are conjugate whenever they are $KK^{{{\mathbb {T}}}}$-equivalent. In the presence of the UCT, this is equivalent to having isomorphic equivariant $K$-theory. We moreover characterize the $KK^{{{\mathbb {T}}}}$-theoretical invariants that arise in this way. Finally, we identify a $KK^{{{\mathbb {T}}}}$-theoretic obstruction to the continuous property, which is shown to be the only obstruction in the setting of Kirchberg algebras. We show by means of explicit examples that the Rokhlin property is strictly weaker than the continuous Rokhlin property.

Homological Algebra for Persistence Modules

We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module and sheaf tensor product and Hom bifunctors as well as their derived functors, Tor and Ext, and give explicit computations for interval modules. We give a classification of injective, projective, and flat interval modules. We state Kunneth theorems and universal coefficient theorems for the homology and cohomology of chain complexes of persistence modules in both the sheaf and graded modules settings and show how these theorems can be applied to persistence modules arising from filtered cell complexes. We also give a Gabriel-Popescu theorem for persistence modules. Finally, we examine categories enriched over persistence modules. We show that the graded module point of view produces a closed symmetric monoidal category that is enriched over itself.

On classification of non-unital amenable simple C*-algebras, II

We present a classification theorem for separable amenable simple stably projectionless C∗-algebras with finite nuclear dimension whose K0 vanish on traces which satisfy the Universal Coefficient Theorem. One of C∗-algebras in the class is denoted by Z0 which has a unique tracial state, K0(Z0)=Z and K1(Z0)={0}. Let A and B be two separable amenable simple C∗-algebras satisfying the UCT. We show that A⊗Z0≅B⊗Z0 if and only if Ell(A⊗Z0)=Ell(B⊗Z0). A class of simple separable C∗-algebras which are approximately sub-homogeneous whose spectra having bounded dimension is shown to exhaust all possible Elliott invariant for C∗-algebras of the form A⊗Z0, where A is any finite separable simple amenable C∗-algebras.

Subalgebras of simple AF-algebras

It is shown that if $A$ is a separable, exact $C^*$-algebra which satisfies the Universal Coefficient Theorem (UCT) and has a faithful, amenable trace, then A admits a trace-preserving embedding into a simple, unital AF-algebra with a unique trace. Modulo the UCT, this provides an abstract characterization of $C^*$-sub­algebras of simple, unital AF-algebras. As a consequence, for a countable, discrete, amenable group $G$ acting on a second countable, locally compact, Hausdorff space $X$, $C_0(X) \rtimes_r G$ embeds into a simple, unital AF-algebra if, and only if, $X$ admits a faithful, invariant, Borel, probability measure. Also, for any countable, discrete, amenable group $G$, the reduced group $C^*$-algebra $C^*_r(G)$ admits a trace-preserving embedding into the universal UHF-algebra.

Cartan subalgebras and the UCT problem, II

We study the connection between the UCT problem and Cartan subalgebras in C*-algebras. The UCT problem asks whether every separable nuclear C*-algebra satisfies the UCT, i.e., a noncommutative analogue of the classical universal coefficient theorem from algebraic topology. This UCT problem is one of the remaining major open questions in the structure and classification theory of simple nuclear C*-algebras. Since the class of separable nuclear C*-algebras is closed under crossed products by finite groups, it is a natural and important task to understand the behaviour of the UCT under such crossed products. We make a contribution towards a better understanding by showing that for certain approximately inner actions of finite cyclic groups on UCT Kirchberg algebras, the crossed products satisfy the UCT if and only if we can find Cartan subalgebras which are invariant under the actions of our finite cyclic groups. We also show that the class of actions we are able to treat is big enough to characterize the UCT problem, in the sense that every such action (even on a particular Kirchberg algebra, namely the Cuntz algebra mathcal O_2) leads to a crossed product satisfying the UCT if and only if every separable nuclear C*-algebra satisfies the UCT. Our results rely on a new construction of Cartan subalgebras in certain inductive limit C*-algebras. This new tool turns out to be of independent interest. For instance, among other things, the second author has used it to construct Cartan subalgebras in all classifiable unital stably finite C*-algebras.

Rokhlin dimension: duality, tracial properties, and crossed products

We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of$C^{\ast }$-algebras, including:$D$-absorbing$C^{\ast }$-algebras, where$D$is a strongly self-absorbing$C^{\ast }$-algebra; stable$C^{\ast }$-algebras;$C^{\ast }$-algebras with finite nuclear dimension (or decomposition rank);$C^{\ast }$-algebras with finite stable rank (or real rank); and$C^{\ast }$-algebras whose$K$-theory is either trivial, rational, or$n$-divisible for$n\in \mathbb{N}$. The combination of nuclearity and the universal coefficient theorem (UCT) is also shown to be preserved by these actions. Some of these results are new even in the well-studied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property. At the core of our arguments is a certain local approximation of the crossed product by a continuous$C(X)$-algebra with fibers that are stably isomorphic to the underlying algebra. The space$X$is computed in some cases of interest, and we use its description to construct a$\mathbb{Z}_{2}$-action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.

Self-similark-Graph C*-Algebras

AbstractIn this paper, we introduce a notion of a self-similar action of a group $G$ on a $k$-graph $\Lambda $ and associate it a universal C$^\ast $-algebra ${{\mathcal{O}}}_{G,\Lambda }$. We prove that ${{\mathcal{O}}}_{G,\Lambda }$ can be realized as the Cuntz–Pimsner algebra of a product system. If $G$ is amenable and the action is pseudo free, then ${{\mathcal{O}}}_{G,\Lambda }$ is shown to be isomorphic to a “path-like” groupoid C$^\ast $-algebra. This facilitates studying the properties of ${{\mathcal{O}}}_{G,\Lambda }$. We show that ${{\mathcal{O}}}_{G,\Lambda }$ is always nuclear and satisfies the universal coefficient theorem; we characterize the simplicity of ${{\mathcal{O}}}_{G,\Lambda }$ in terms of the underlying action, and we prove that, whenever ${{\mathcal{O}}}_{G,\Lambda }$ is simple, there is a dichotomy: it is either stably finite or purely infinite, depending on whether $\Lambda $ has nonzero graph traces or not. Our main results generalize the recent work of Exel and Pardo on self-similar graphs.