We construct a new bivariant theory, that we call $KE$-theory, which is intermediate between the $KK$-theory of G. G. Kasparov, and the $E$-theory of A. Connes and N. Higson. For each pair of separable graded $C^*$-algebras $A$ and $B$, acted upon by a locally compact $σ$-compact group $G$, we define an abelian group $KE_G(A,B)$. We show that there is an associative product $KE_G(A,D) \otimes KE_G(D,B) \to KE_G(A,B)$. Various functoriality properties of the $KE$-theory groups and of the product are presented. The new theory has a simpler product than $KK$-theory and there are natural transformations $KK_G \to KE_G$ and $KE_G \to E_G$. The complete description of these maps will form the substance of a second paper.
This survey article on relative homological algebra in bivariant K-thoery is mainly intended for readers with a background knowledge in triangulated categories. We briefly recall the general theory of relative homological algebra in triangulated categories and latter specialize it to the non-equivariant and the equivariant bivariant K-thoery, where the actions on C*-algebras is by a finite cyclic group. We conclude by the explicit computation of the universal abelian invariant for separable C*-algebras with the action of $\mathbb{Z}/4$ by automorphisms.
We show that abstract Cuntz semigroups form a closed symmetric monoidal category. Thus, given Cuntz semigroups $S$ and $T$, there is another Cuntz semigroup $[[S,T]]$ playing the role of morphisms from $S$ to $T$. Applied to C$^*$-algebras $A$ and $B$, the semigroup $[[\mathrm{Cu}(A),\mathrm{Cu}(B)]]$ should be considered as the target in analogues of the UCT for bivariant theories of Cuntz semigroups. Abstract bivariant Cuntz semigroups are computable in a number of interesting cases. We also show that order-zero maps between C$^*$-algebras naturally define elements in the respective bivarian…
We extend Cuntz-Quillen's excision theorem for algebras and pro-algebras in arbitrary Q-linear categories with tensor product.The excision theorems for the bivariant periodic cyclic cohomology of discrete,topological and bornological algebras and pro-algebras follow from this.
We introduce a bivariant version of the Cuntz semigroup as equivalence classes of order zero maps generalizing the ordinary Cuntz semigroup. The theory has many properties formally analogous to KK-theory including a composition product. We establish basic properties, like additivity, stability and continuity, and study categorical aspects in the setting of local C*-algebras. We determine the bivariant Cuntz semigroup for numerous examples such as when the second algebra is a Kirchberg algebra, and Cuntz homology for compact Hausdorff spaces which provides a complete invariant. Moreover, we est…
We introduce a notion of Morita equivalence for non-selfadjoint operator algebras equipped with a completely isometric involution (operator *-algebras). We then show that the unbounded Kasparov product by a Morita equivalence bimodule induces an isomorphism between equivalence classes of twisted spectral triples over Morita equivalent operator *-algebras. This leads to a tentative definition of unbounded bivariant K-theory and we prove that this bivariant theory is related to Kasparov's bivariant K-theory via the Baaj-Julg bounded transform. Moreover, the unbounded Kasparov product provides a …
We establish the continuous homotopy invariance of bivariant local cyclic homology on the category of all σ-C^*-algebras. The argument relies vitally on an isomorphism between the smooth and continuous cylinder constructions using a technical criterion due to Meyer. As a consequence we compute the local cyclic homology of the infinite sphere.
In this article, we generalize to the case of regular locally compact quantum groups, two important results concerning actions of compact quantum groups. Let $G_1$ and $G_2$ be two monoidally equivalent regular locally compact quantum groups in the sense of De Commer. We introduce an induction procedure and we build an equivalence of the categories ${A}^{G_1}$ and ${A}^{G_2}$ consisting of continuous actions of $G_1$ and $G_2$ on $C^*$-algebras. As an application of this result, we derive a canonical equivalence of the categories ${KK}^{G_1}$ and ${KK}^{G_2}$. We introduce and investigate a no…
We study the twisted K-theory and K-homology of some infinite dimensional spaces, like SU(\infty), in the bivariant setting. Using a general procedure due to Cuntz we construct a bivariant K-theory on the category of separable σ-C^*-algebras that generalizes both twisted K-theory and K-homology of (locally) compact spaces. We construct a bivariant Chern--Connes type character taking values in bivariant local cyclic homology. We analyse the structure of the dual Chern--Connes character from (analytic) K-homology to local cyclic cohomology under some reasonable hypotheses. We also investigate th…
Using the machinery of weak fibration categories due to Schlank and the first author, we construct a convenient model structure on the pro-category of separable $C^*$-algebras $\mathrm{Pro}(\mathtt{SC^*})$. The opposite of this model category models the $\infty$-category of pointed noncommutative spaces $\mathtt{N}\mathcal{S_*}$ defined by the third author. Our model structure on $\mathrm{Pro}(\mathtt{SC^*})$ extends the well-known category of fibrant objects structure on $\mathtt{SC^*}$. We show that the pro-category $\mathrm{Pro}(\mathtt{SC^*})$ also contains, as a full coreflective subcateg…
By introducing a notion of smooth connection for unbounded $KK$-cycles, we show that the Kasparov product of such cycles can be defined directly, by an algebraic formula. In order to achieve this it is necessary to develop a framework of smooth algebras and a notion of differentiable $C^{*}$-module. The theory of operator spaces provides the required tools. Finally, the above mentioned $KK$-cycles with connection can be viewed as the morphisms in a category whose objects are spectral triples.
This article is the first of two where we investigate to what extent homotopy invariant, excisive and matrix stable homology theories help one distinguish between the Leavitt path algebras $L(E)$ and $L(F)$ of graphs $E$ and $F$ over a commutative ground ring $\ell$. In this first article we consider Leavitt path algebras of general graphs over general ground rings; the second article will focus mostly on purely infinite simple unital Leavitt path algebras over a field. Bivariant algebraic $K$-theory $kk$ is the universal homology theory with the properties above; we prove a structure theorem …
Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal $\infty$-categorical models for separable $C^*$-algebras $\mathtt{SC^*_\infty}$ and noncommutative spectra $\mathtt{NSp}$ using the framework of Higher Algebra due to Lurie. We study smashing (co)localizations of $\mathtt{SC^*_\infty}$ and $\mathtt{NSp}$ with respect to strongly self-absorbing $C^*$-algebras. We analyse the homotopy categories of the localizations of $\mathtt{SC^*_\infty}$ and give universal characterizations thereof. We construct a stable $\infty$-categorical model for bivariant connect…
This paper is to construct unstable, Morita stable and stable bivariant algebraic Kasparov $K$-theory spectra of $k$-algebras. These are shown to be homotopy invariant, excisive in each variable $K$-theories. We prove that the spectra represent universal unstable, Morita stable and stable bivariant homology theories respectively.
We construct equivariant $KK$-theory with coefficients in $\mathbb{R}$ and $\mathbb{R}/\mathbb{Z}$ as suitable inductive limits over ${\rm II}_1$-factors. We show that the Kasparov product, together with its usual functorial properties, extends to $KK$-theory with real coefficients. Let $Γ$ be a group. We define a $Γ$-algebra $A$ to be $K$-theoretically free and proper (KFP) if the group trace ${\bf tr}$ of $Γ$ acts as the unit element in $KK^Γ_{\mathbb{R}}(A,A)$. We show that free and proper $Γ$-algebras (in the sense of Kasparov) have the (KFP) property. Moreover, if $Γ$ is torsion free and …
Let $G$ be a group and $\ell$ a commutative unital $\ast$-ring with an element $λ\in \ell$ such that $λ+ λ^\ast = 1$. We introduce variants of hermitian bivariant $K$-theory for $\ast$-algebras equipped with a $G$-action or a $G$-grading. For any graph $E$ with finitely many vertices and any weight function $ω\colon E^1 \to G$, a distinguished triangle for $L(E)=L_\ell(E)$ in the hermitian $G$-graded bivariant $K$-theory category $kk^h_{G_{\mathrm{gr}}}$ is obtained, describing $L(E)$ as a cone of a matrix with coefficients in $\mathbb{Z}[G]$ associated to the incidence matrix of $E$ and the w…
Let $Γ$ be a lattice in a locally compact group $G$. In earlier work, we used $KK$-theory to equip the $K$-groups of any $Γ$-$C^{*}$-algebra on which the commensurator of $Γ$ acts with Hecke operators. When $Γ$ is arithmetic, this gives Hecke operators on the $K$-theory of certain $C^{*}$-algebras that are naturally associated with $Γ$. In this paper, we first study the topological $K$-theory of the arithmetic manifold associated to $Γ$. We prove that the Chern character commutes with Hecke operators. Afterwards, we show that the Shimura product of double cosets naturally corresponds to the Ka…
In this paper we refine a version of bivariant $K$-theory developed by Cuntz to define symmetric spectra representing the $KK$-theory of $C^\ast$-categories and discrete groupoid $C^\ast$-algebras. In both cases, the Kasparov product can be expressed as a smash product of spectra.
We extend McClure's results on the restriction maps in equivariant $K$-theory to bivariant $K$-theory: Let $G$ be a compact Lie group and $A$ and $B$ be $G$-$C^*$-algebras. Suppose that $KK^{H}_{n}(A, B)$ is a finitely generated $R(G)$-module for every $H \le G$ closed and $n \in \Z$. Then, if $KK^{F}_{*}(A, B) = 0$ for all $F \le G$ {\em finite cyclic}, then $KK^{G}_{*}(A, B) = 0$.
We study uniform perturbations of intermediate C*-subalgebras of inclusions of simple C*-algebras. If a unital simple C*-algebra has a simple C*-subalgebra of finite index, then sufficiently close simple intermediate C*-subalgebras are unitarily equivalent. These C*-subalgebras need not to be nuclear. The unitary can be chosen in the relative commutant algebra. An imediate corollary is the following: If the relative commutant is trivial, then the set of intermediate C*-subagebras is a finite set.
Let $Γ$ be a discrete group satisfying the approximation property (AP). Let $X$, $Y$ be $Γ$-spaces and $π\colon Y \to X$ be a proper factor map which is injective on the non-free part. We prove the one-to-one correspondence between intermediate ${\rm C}^\ast$-algebras of $C_0(X) \rtimes_r Γ\subset C_0(Y) \rtimes Γ$ and intermediate $Γ$-${\rm C}^\ast$-algebras of $C_0(X) \subset C_0(Y)$. This is a generalization of Suzuki's theorem that proves the statement for free actions.
We prove simplicity of all intermediate $C^*$-algebras $C^*_{r}(Γ)\subseteq \mathcal{B} \subseteq Γ\ltimes_r C(X)$ in the case of minimal actions of $C^*$-simple groups $Γ$ on compact spaces $X$. For this, we use the notion of stationary states, recently introduced by Hartman and Kalantar (arXiv:1712.10133v2). We show that the Powers' averaging property holds for the reduced crossed product $Γ\ltimes_r \mathcal{A}$ for any action $Γ\curvearrowright \mathcal{A}$ of a $C^*$-simple group $Γ$ on a unital $C^*$-algebra $\mathcal{A}$, and use it to prove a one-to-one correspondence between stationar…
Let $A$ be a C$^*$-algebra and let $D$ be a Cartan subalgebra of $A$. We study the following question: if $B$ is a C$^*$-algebra such that $D \subseteq B \subseteq A$, is $D$ a Cartan subalgebra of $B$? We give a positive answer in two cases: the case when there is a faithful conditional expectation from $A$ onto $B$, and the case when $A$ is nuclear and $D$ is a C$^*$-diagonal of $A$. In both cases there is a one-to-one correspondence between the intermediate C$^*$-algebras $B$, and a class of open subgroupoids of the groupoid $G$, where $Σ\rightarrow G$ is the twist associated with the embed…
Let $G$ be a discrete group acting on a von Neumann algebra $M$ by properly outer $*$-automorphisms. In this paper we study the containment $M \subseteq M\rtimes_αG$ of $M$ inside the crossed product. We characterize the intermediate von Neumann algebras, extending earlier work of other authors in the factor case. We also determine the $M$-bimodules that are closed in the Bures topology and which coincide with the $w^*$-closed ones under a mild hypothesis on $G$. We use these results to obtain a general version of Mercer's theorem concerning the extension of certain isometric $w^*$-continuous …
We examine inclusions of $C^*$-algebras of the form $A^H \subseteq A \rtimes_{r} G$, where $G$ and $H$ are groups acting on a unital simple $C^*$-algebra $A$ by outer automorphisms and $H$ is finite. It follows from a theorem of Izumi that $A^H \subseteq A$ is $C^*$-irreducible, in the sense that all intermediate $C^*$-algebras are simple. We show that $A^H \subseteq A \rtimes_{r} G$ is $C^*$-irreducible for all $G$ and $H$ as above if and only if $G$ and $H$ have trivial intersection in the outer automorphisms of $A$, and we give a Galois type classification of all intermediate $C^*$-algebras…
We consider noncommuting pairs P,Q of intermediate subfactors of an irreducible, finite-index inclusion N in M of II_1 factors such that P and Q are supertransitive with Jones index less than 4 over N. We show that up to isomorphism of the standard invariant, there is a unique such pair corresponding to each even value [P:N]=4cos^2(pi/2n) but none for the odd values [P:N]=4cos^2 (pi/(2n+1)). We also classify the angle values which occur between pairs of intermediate subfactors with small index over their intersection: if [P:N] < 4, then the unique nontrivial angle value is always cos^-1 (1/([P…
In this paper we introduce a row version of Kadison and Kastler's metric on the set of C*-subalgebras of $\mathbb{B}(\mathcal{H})$. By showing C*-algebras have row length (in the sense of Pisier) of at most 2 we show that the row metric is equivalent to the original Kadison Kastler metric. Ino and Watatani have recently proved that in certain circumstances sufficiently close intermediate C*-algebras occur as small unitary perturbations. By adjusting their arguments to work with the row metric we are able to obtain universal constants independent of inclusions.
A universal coefficient theorem is proved for C*-algebras over an arbitrary finite T_0-space X which have vanishing boundary maps. Under bootstrap assumptions, this leads to a complete classification of unital/stable real-rank-zero Kirchberg X-algebras with intermediate cancellation. Range results are obtained for (unital) purely infinite graph C*-algebras with intermediate cancellation and Cuntz-Krieger algebras with intermediate cancellation. Permanence results for extensions of these classes follow.
Practically and intrinsically, inclusions of operator algebras are of fundamental interest. The subject of this paper is intermediate operator algebras of inclusions. There are two previously known theorems which naturally and completely describe all intermediate operator algebras: the Galois Correspondence Theorem and the Tensor Splitting Theorem. Here we establish the third, new complete description theorem which gives a canonical bijective correspondence between intermediate operator algebras and intermediate extensions of dynamical systems. One can also regard this theorem as a crossed pro…
We describe the subfactor planar algebra of an intermediate subfactor $N\subset Q \subset M$ of an extremal subfactor $N\subset M$ of finite Jones index which is not necessarily irreducible.
In this paper, we explicitly work out the subfactor planar algebra $P^{(N \subset Q)}$ for an intermediate subfactor $N \subset Q \subset M$ of an irreducible subfactor $N \subset M$ of finite index. We do this in terms of the subfactor planar algebra $P^{(N \subset M)}$ by showing that if $T$ is any planar tangle, the associated operator $Z^{(N \subset Q)}_T$ can be read off from $Z^{(N \subset M)}_T$ by a formula involving the so-called {\em biprojection} corresponding to the intermediate subfactor $N \subset Q \subset M$ and a scalar $α(T)$ carefully chosen so as to ensure that the formula …
Let $Γ$ be a discrete group. When $Γ$ is nonamenable, the reduced and full group $C$*-algebras differ and it is generally believed that there should be many intermediate $C$*-algebras, however few examples are known. In this paper we give new constructions and compare existing constructions of intermediate group $C$*-algebras for both generic and specific groups $Γ$.
We introduce concepts of intermediate rank for countable groups that "interpolate" between consecutive values of the classical (integer-valued) rank. Various classes of groups are proved to have intermediate rank behaviors. We are especially interested in interpolation between rank 1 and rank 2. For instance, we construct groups "of rank 7/4". Our setting is essentially that of non positively curved spaces, where concepts of intermediate rank include polynomial rank, local rank, and mesoscopic rank. The resulting framework has interesting connections to operator algebras. We prove property RD …
