Mathematica Scandinavica
0025-5521
1903-1807
Đan mạch
Cơ quản chủ quản: MATEMATISK INST , Mathematica Scandinavica
Lĩnh vực:
Mathematics (miscellaneous)
Các bài báo tiêu biểu
Locally compact quantum groups in the von Neumann algebraic setting 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.
Tập 92 Số 1 - Trang 68
Some remarks on the $C^*$-algebras associated with subshifts 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.
Tập 95 Số 1 - Trang 145
On the $KK$-theory of strongly self-absorbing $C^{*}$-algebras 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 $K_0(A\otimes \mathcal D)$.
Tập 104 Số 1 - Trang 95
Periodic points of equivariant maps We assume that $X$ is a compact connected polyhedron, $G$ is a finite group acting freely on $X$, and $f:X\to X$ a $G$-equivariant map. We find formulae for the least number of $n$-periodic points in the equivariant homotopy class of $f$, i.e., $\inf_h |(\mathrm{Fix}(h^n)|$ (where $h$ is $G$-homotopic to $f$). As an application we prove that the set of periodic points of an equivariant map is infinite provided the action on the rational homology of $X$ is trivial and the Lefschetz number $L(f^n)$ does not vanish for infinitely many indices $n$ commeasurable with the order of $G$. Moreover, at least linear growth, in $n$, of the number of points of period $n$ is shown.
Tập 107 Số 2 - Trang 224