arXiv subjects

Ulrich Pennig

Publications and source records attributed to Ulrich Pennig.

26 records · Page 2Linked to original sources

arXiv2015

Deformations of nilpotent groups and homotopy symmetric $C^*$-algebras

The homotopy symmetric $C^*$-algebras are those separable $C^*$-algebras for which one can unsuspend in E-theory. We find a new simple condition that characterizes homotopy symmetric nuclear $C^*$-algebras and use it to show that the property of being homotopy symmetric passes to nuclear $C^*$-subalgebras and it has a number of other significant permanence properties. As an application, we show that if $I(G)$ is the kernel of the trivial representation $ι:C^*(G)\to \mathbb{C}$ for a countable discrete torsion free nilpotent group $G$, then $I(G)$ is homotopy symmetric and hence the Kasparov group $KK(I(G),B)$ can be realized as the homotopy classes of asymptotic morphisms $[[I(G),B \otimes \mathcal{K}]]$ for any separable $C^*$-algebra $B$.

math.OA↗

arXiv2015

A Dixmier-Douady theory for strongly self-absorbing C*-algebras

We show that the Dixmier-Douady theory of continuous field $C^*$-algebras with compact operators $\mathbb{K}$ as fibers extends significantly to a more general theory of fields with fibers $A\otimes \mathbb{K}$ where $A$ is a strongly self-absorbing C*-algebra. The classification of the corresponding locally trivial fields involves a generalized cohomology theory which is computable via the Atiyah-Hirzebruch spectral sequence. An important feature of the general theory is the appearance of characteristic classes in higher dimensions. We also give a necessary and sufficient $K$-theoretical condition for local triviality of these continuous fields over spaces of finite covering dimension.

math.OA↗

arXiv2015

A Dixmier-Douady Theory for strongly self-absorbing C*-algebras II: the Brauer group

We have previously shown that the isomorphism classes of orientable locally trivial fields of $C^*$-algebras over a compact metrizable space $X$ with fiber $D\otimes \mathbb{K}$, where $D$ is a strongly self-absorbing $C^*$-algebra, form an abelian group under the operation of tensor product. Moreover this group is isomorphic to the first group $\bar{E}^1_D(X)$ of the (reduced) generalized cohomology theory associated to the unit spectrum of topological K-theory with coefficients in $D$. Here we show that all the torsion elements of the group $\bar{E}^1_D(X)$ arise from locally trivial fields with fiber $D \otimes M_n(\mathbb{C})$, $n\geq 1$, for all known examples of strongly self-absorbing $C^*$-algebras $D$. Moreover the Brauer group generated by locally trivial fields with fiber $D\otimes M_n(\mathbb{C})$, $n\geq 1$ is isomorphic to ${\rm Tor}(\bar{E}^1_D(X))$.

math.OA↗

arXiv2015

A noncommutative model for higher twisted K-Theory

We develop a operator algebraic model for twisted $K$-theory, which includes the most general twistings as a generalized cohomology theory (i.e. all those classified by the unit spectrum $bgl_1(KU)$). Our model is based on strongly self-absorbing $C^*$-algebras. We compare it with the known homotopy theoretic descriptions in the literature, which either use parametrized stable homotopy theory or $\infty$-categories. We derive a similar comparison of analytic twisted $K$-homology with its topological counterpart based on generalized Thom spectra. Our model also works for twisted versions of localizations of the $K$-theory spectrum, like $KU[1/n]$ or $KU_{\mathbb{Q}}$.

math.KT↗

arXiv2013

Twisted K-theory and obstructions against positive scalar curvature metrics

We decompose the twisted index obstruction $θ(M)$ against positive scalar curvature metrics for oriented manifolds with spin universal cover into a pairing of a twisted $K$-homology with a twisted $K$-theory class and prove that $θ(M)$ does not vanish if $M$ is an orientable enlargeable manifold with spin universal cover.

math.GT↗

arXiv2013

Unit spectra of K-theory from strongly self-absorbing C*-algebras

We give an operator algebraic model for the first group of the unit spectrum $gl_1(KU)$ of complex topological K-theory, i.e. $[X, BGL_1(KU)]$, by bundles of stabilized infinite Cuntz C*-algebras $O_{\infty} \otimes \K$. We develop similar models for the localizations of $KU$ at a prime $p$ and away from $p$. Our work is based on the $\mathcal{I}$-monoid model for the units of $K$-theory by Sagave and Schlichtkrull and it was motivated by the goal of finding connections between the infinite loop space structure of the classifying space of the automorphism group of stabilized strongly self-absorbing C*-algebras that arose in our generalization of the Dixmier-Douady theory and classical spectra from algebraic topology.

math.AT↗

arXiv2011

Twisted K-theory with coefficients in C*-algebras

We introduce a twisted version of $K$-theory with coefficients in a $C^*$-algebra $A$, where the twist is given by a new kind of gerbe, which we call Morita bundle gerbe. We use the description of twisted $K$-theory in the torsion case by bundle gerbe modules as a guideline for our noncommutative generalization. As it turns out, there is an analogue of the Dixmier-Douady class living in a nonabelian cohomology set and we give a description of the latter via stable equivalence classes of our gerbes. We also define the analogue of torsion elements inside this set and extend the description of twisted $K$-theory in terms of modules over these gerbes. In case $A$ is the infinite Cuntz algebra, this may lead to an interpretation of higher twists for $K$-theory.

math.KT↗

arXiv2010

Quasi-multipliers of Hilbert and Banach C*-bimodules

Quasi-multipliers for a Hilbert C*-bimodule V were introduced by Brown, Mingo and Shen 1994 as a certain subset of the Banach bidual module V**. We give another (equivalent) definition of quasi-multipliers for Hilbert C*-bimodules using the centralizer approach and then show that quasi-multipliers are, in fact, universal (maximal) objects of a certain category. We also introduce quasi-multipliers for bimodules in Kasparov's sense and even for Banach bimodules over C*-algebras, provided these C*-algebras act non-degenerately. A topological picture of quasi-multipliers via the quasi-strict topology is given. Finally, we describe quasi-multipliers in two main situations: for the standard Hilbert bimodule l_2(A) and for bimodules of sections of Hilbert C*-bimodule bundles over locally compact spaces.

math.OA↗