Exponential Functors, R-Matrices and Twists

In this paper we show that each polynomial exponential functor on complex finite-dimensional inner product spaces is defined up to equivalence of monoidal functors by an involutive solution to the Yang-Baxter equation (an involutive $R$-matrix), which determines an extremal character on $S_{\infty}$. These characters are classified by Thoma parameters, and Thoma parameters resulting from polynomial exponential functors are of a special kind. Moreover, we show that each $R$-matrix with Thoma parameters of this kind yield a corresponding polynomial exponential functor. In the second part of the paper we use these functors to construct a higher twist over $SU(n)$ for a localisation of $K$-theory that generalises the one given by the basic gerbe. We compute the indecomposable part of the rational characteristic classes of these twists in terms of the Thoma parameters of their $R$-matrices.Comment: 40 pages (fixed a mistake in Sec. 3.2, which does not affect the main result of the paper, Lemma 3.3 has been isolated and moved to an appendix, this agrees (up to minor layout changes) with the version accepted for publication

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 $KU_{\mathbb{Q}}$.Comment: 28 page

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 ${\rm Tor}(\bar{E}^1_D(X))$.Comment: 14 pages, this version agrees with the one that will be published in the Journal of Noncommutative Geometr

Crossed module actions on continuous trace C∗C^*-algebras

We lift an action of a torus $\mathbb{T}^n$ on the spectrum of a continuous trace algebra to an action of a certain crossed module of Lie groups that is an extension of $\mathbb{R}^n$. We compute equivariant Brauer and Picard groups for this crossed module and describe the obstruction to the existence of an action of $\mathbb{R}^n$ in our framework.Comment: 27 pages, added background material about T-Duality, added references, extended section about non-associative C*-algebra

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.Comment: 26p, this version agrees with the one published in J. Reine Angew. Math. except for a minor correction and an extension of Corollary 4.8, both of which were done after publicatio

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.Comment: 19 pages v2: to appear in Math. Scand., small glitches in one example and with formulation of definition correcte

Connective C*-algebras

Connectivity is a homotopy invariant property of separable C*-algebras which has three notable consequences: absence of nontrivial projections, quasidiagonality and a more geometric realisation of KK-theory for nuclear C*-algebras using asymptotic morphisms. The purpose of this paper is to further explore the class of connective C*-algebras. We give new characterisations of connectivity for exact and for nuclear separable C*-algebras and show that an extension of connective separable nuclear C*-algebras is connective. We establish connectivity or lack of connectivity for C*-algebras associated to certain classes of groups: virtually abelian groups, linear connected nilpotent Lie groups and linear connected semisimple Lie groups