Mathematica Scandinavica

In this paper we complete in several aspects the picture of locally compact quantum groups. First of all we give a definition of a locally compact quantum group in the von Neumann algebraic setting and show how to deduce from it a $C^*$-algebraic quantum group. Further we prove several results about locally compact quantum groups which are important for applications, but were not yet settled in our paper [8]. We prove a serious strengthening of the left invariance of the Haar weight, and we give several formulas connecting the locally compact quantum group with its dual. Loosely speaking we show how the antipode of the locally compact quantum group determines the modular group and modular conjugation of the dual locally compact quantum group.

We point out incorrect lemmas in some papers regarding the $C^*$-algebras associated with subshifts written by the second named author. To recover the incorrect lemmas and the affected main results, we will describe an alternative construction of $C^*$-algebras associated with subshifts. The resulting $C^*$-algebras are generally different from the originally constructed $C^*$-algebras associated with subshifts and they fit the mentioned papers including the incorrect results. The simplicity conditions and the K-theory formulae for the originally constructed $C^*$-algebras are described. We also introduce a condition called $(*)$ for subshifts such that under this condition the new $C^*$-algebras and the original $C^*$-algebras are canonically isomorphic to each other. We finally present a subshift for which the two kinds of algebras have different K-theory groups.

Let $\mathcal D$ and $A$ be unital and separable $C^{*}$-algebras; let $\mathcal D$ be strongly self-absorbing. It is known that any two unital ${}^*$-homomorphisms from $\mathcal D$ to $A \otimes \mathcal D$ are approximately unitarily equivalent. We show that, if $\mathcal D$ is also $K_{1}$-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of $\mathcal D$ is asymptotically inner. Moreover, the space of automorphisms of $\mathcal D$ is compactly-contractible (in the point-norm topology) in the sense that for any compact Hausdorff space $X$, the set of homotopy classes $[X,(\mathrm{Aut}(\mathcal D)]$ reduces to a point. The respective statement holds for the space of unital endomorphisms of $\mathcal D$. As an application, we give a description of the Kasparov group $KK(\mathcal D, A\otimes \mathcal D)$ in terms of $^*$-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group $KK(\mathcal D, A\otimes \mathcal D)$ is isomorphic to $K_0(A\otimes \mathcal D)$.