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Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let
$\alpha \colon G \to {\text{Aut}} (A)$ be an action of G on A which has the weak tracial Rokhlin property. Let
$A^{\alpha}$ be the fixed point algebra. Then the radius of comparison satisfies
${\text{rc}} (A^{\alpha }) \leq {\text{rc}} (A)$ and
${\text{rc}} ( C^* (G, A, \alpha ) ) \leq ({1}/{\text{card} (G))} \cdot {\text{rc}} (A)$. The inclusion of
$A^{\alpha }$ in A induces an isomorphism from the purely positive part of the Cuntz semigroup
${\text{Cu}} (A^{\alpha })$ to the fixed points of the purely positive part of
${\text{Cu}} (A)$, and the purely positive part of
${\text{Cu}} ( C^* (G, A, \alpha ) )$ is isomorphic to this semigroup. We construct an example in which
$G \,{=}\, {\mathbb {Z}} / 2 {\mathbb {Z}}$, A is a simple unital AH algebra,
$\alpha $ has the Rokhlin property,
${\text{rc}} (A)> 0$,
${\text{rc}} (A^{\alpha }) = {\text{rc}} (A)$, and
${\text{rc}} (C^* (G, A, \alpha ) = ( {1}/{2}) {\text{rc}} (A)$.
We construct a Baum–Connes assembly map localised at the unit element of a discrete group $\Gamma$. This morphism, called $\mu _\tau$, is defined in $KK$-theory with coefficients in $\mathbb {R}$ by means of the action of the idempotent $[\tau ]\in KK_{\mathbin {{\mathbb {R}}}}^\Gamma (\mathbb {C},\mathbb {C})$ canonically associated to the group trace of $\Gamma$. We show that the corresponding $\tau$-Baum–Connes conjecture is weaker than the classical version, but still implies the strong Novikov conjecture. The right-hand side of $\mu _\tau$ is functorial with respect to the group $\Gamma$.
We prove that Cuntz semigroups of C*-algebras satisfy Edwards' condition with respect to every quasitrace. This condition is a key ingredient in the study of the realization problem of functions on the cone of quasitraces as ranks of positive elements. In the course of our investigation, we identify additional structure of the Cuntz semigroup of an arbitrary C*-algebra and of the cone of quasitraces.
A category structure for ordered Bratteli diagrams is proposed in which isomorphism coincides with the notion of equivalence of Herman, Putnam, and Skau. It is shown that the natural one-to-one correspondence between the category of Cantor minimal systems and the category of simple properly ordered Bratteli diagrams is in fact an equivalence of categories. This gives a Bratteli–Vershik model for factor maps between Cantor minimal systems. We give a construction of factor maps between Cantor minimal systems in terms of suitable maps (called premorphisms) between the corresponding ordered Bratteli diagrams, and we show that every factor map between two Cantor minimal systems is obtained in this way. Moreover, solving a natural question, we are able to characterize Glasner and Weiss’s notion of weak orbit equivalence of Cantor minimal systems in terms of the corresponding C*-algebra crossed products.
We prove a $\unicode[STIX]{x1D6E4}$-equivariant version of the algebraic index theorem, where $\unicode[STIX]{x1D6E4}$ is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Moscovici for hypo-elliptic operators and the index theorem for the extension of the algebra of pseudodifferential operators by a group of diffeomorphisms of the underlying manifold due to A. Savin, B. Sternin, E. Schrohe and D. Perrot.
An odd Fredholm module for a given invertible operator on a Hilbert space is specified by an unbounded so-called Dirac operator with compact resolvent and bounded commutator with the given invertible. Associated with this is an index pairing in terms of a Fredholm operator with Noether index. Here it is shown by a spectral flow argument how this index can be calculated as the signature of a finite dimensional matrix called the spectral localizer.
Fix a von Neumann algebra ${\mathcal{N}}$ equipped with a suitable trace $\unicode[STIX]{x1D70F}$. For a path of self-adjoint Breuer–Fredholm operators, the spectral flow measures the net amount of spectrum that moves from negative to non-negative. We consider specifically the case of paths of bounded perturbations of a fixed unbounded self-adjoint Breuer–Fredholm operator affiliated with ${\mathcal{N}}$. If the unbounded operator is $p$-summable (that is, its resolvents are contained in the ideal $L^{p}$), then it is possible to obtain an integral formula that calculates spectral flow. This integral formula was first proved by Carey and Phillips, building on earlier approaches of Phillips. Their proof was based on first obtaining a formula for the larger class of $\unicode[STIX]{x1D703}$-summable operators, and then using Laplace transforms to obtain a $p$-summable formula. In this paper, we present a direct proof of the $p$-summable formula that is both shorter and simpler than theirs.
We provide sufficient conditions to factorise an equivariant spectral triple as a Kasparov product of unbounded classes constructed from the group action on the algebra and from the fixed point spectral triple. We show that if factorisation occurs, then the equivariant index of the spectral triple vanishes. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and $\unicode[STIX]{x1D703}$-deformations. In particular, we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. Combining this with our index result yields a special case of the Atiyah–Hirzebruch theorem. We also present an example that shows what goes wrong in the absence of our sufficient conditions (and how we get around it for this example).
For a C*-algebra A, determining the Cuntz semigroup Cu(A ⊗) in terms of Cu(A) is an important problem, which we approach from the point of view of semigroup tensor products in the category of abstract Cuntz semigroups by analysing the passage of significant properties from Cu(A) to Cu(A)⊗Cu Cu(). We describe the effect of the natural map Cu(A) → Cu(A)⊗Cu Cu() in the order of Cu(A), and show that if A has real rank 0 and no elementary subquotients, Cu(A)⊗Cu Cu() enjoys the corresponding property of having a dense set of (equivalence classes of) projections. In the simple, non-elementary, real rank 0 and stable rank 1 situation, our investigations lead us to identify almost unperforation for projections with the fact that tensoring with is inert at the level of the Cuntz semigroup.
We provide an equivariant extension of the bivariant Cuntz semigroup introduced in previous work for the case of compact group actions over C*-algebras. Its functoriality properties are explored, and some well-known classification results are retrieved. Connections with crossed products are investigated, and a concrete presentation of equivariant Cuntz homology is provided. The theory that is here developed can be used to define the equivariant Cuntz semigroup. We show that the object thus obtained coincides with the one recently proposed by Gardella [‘Regularity properties and Rokhlin dimension for compact group actions’, Houston J. Math.43(3) (2017), 861–889], and we complement their work by providing an open projection picture of it.
We prove that every incidence graph of a finite projective plane allows a partitioning into incident point-line pairs. This is used to determine the order of the identity in the K0-group of so-called polygonal algebras associated with cocompact group actions on Ã2-buildings with three orbits. These C*-algebras are classified by the K0-group and the class of the identity in K0. To be more precise, we show that 2(q − 1) = 0, where q is the order of the links of the building. Furthermore, if q = 22l−1 with l ∈ ℤ, then the order of is q − 1.
In this article, we begin by recalling the inversion formula for the convolution with the box spline. The equivariant cohomology and the equivariant $K$-theory with respect to a compact torus $G$ of various spaces associated to a linear action of $G$ in a vector space $M$ can both be described using some vector spaces of distributions, on the dual of the group $G$ or on the dual of its Lie algebra $\mathfrak{g}$. The morphism from $K$-theory to cohomology is analyzed, and multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semi-discrete convolution with a box spline. Finally, the multiplicities of the index of a $G$-transversally elliptic operator on $M$ are determined using the infinitesimal index of the symbol.
Vincent Lafforgue's bivariant K-theory for Banach algebras is invariant in the second variable under a rather general notion of Morita equivalence. In particular, the ordinary topological K-theory for Banach algebras is invariant under Morita equivalences.
The Kasparov groups are extended to the setting of inverse limits of G-C*-algebras, where G is assumed to be a locally compact group. The K K-product and other important features of the theory are generalized to this setting.
We adapt the Toeplitz operator proof of Bott periodicity to give a short direct proof of Bott periodicity for the representable K-theory of σ-C*-algebras. We further show how the use of this proof and the right definitions simplifies the derivation of the basic properties of representable K-theory.