Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · funct-an/9506003

A remark on trace properties of K-cycles

Fabio Cipriani·Daniele Guido·Sergio Scarlatti
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we discuss trace properties of $d^+$-summable $K$-cycles considered by A.Connes in [\rfr(Conn4)]. More precisely we give a proof of a trace theorem on the algebra $\A$ of a $K$--cycle stated in [\rfr(Conn4)], namely we show that a natural functional on $\A$ is a trace functional. Then we discuss whether this functional gives a trace on the whole universal graded differential algebra $\Q(\A)$. On the one hand we prove that the regularity conditions on $K$-cycles considered in [\rfr(Conn4)] imply the trace property on $\Q(\A)$. On the other hand, by constructing an explicit counterexample, we remark that the sole $K$-cycle assumption is not sufficient for such a property to hold.

Explore related subjects

funct-anmath.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Fabio Cipriani, Daniele Guido, Sergio Scarlatti. 1995-06-14. A remark on trace properties of K-cycles. https://arxiv.org/abs/funct-an/9506003

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2024

Tensor Products of Fell Bundles over Groups

We extend the theory of tensor products of C*-algebras to the larger category of Fell bundles over locally compact groups. We prove that, like in the case of C*-algebras, there exist maximal and minimal tensor products. Given two Fell bundles, we compare the tensor products of their cross-sectional algebras with the cross-sectional algebras of their tensor products. As applications we prove that, under certain conditions, the cross-sectional C*-algebra of a Fell bundle is nuclear or exact whenever so is its fiber over the unit element of the group.

funct-an↗
arXiv2020

Isomorphism classes for quantum Heisenberg manifolds

We embed the quantum Heisenberg manifold in a crossed product algebra. This enables us to show that, in the irrational case, all tracial states on $\dc$ induce the same homomorphism on the K_0-group. We conclude that two irrational quantum Heisenberg manifolds $\dc$ and $D^c_{μ' ν'}$ are isomorphic if and only if the parameters $(μ,ν)$ and $(μ',ν')$ belong to the same orbit under the usual action of $GL_2(\ZZ)$ on the torus.

funct-an↗
arXiv2010

Hilbert C*-modules over monotone complete C*-algebras

The aim of the present paper is to describe self-duality and C*- reflexivity of Hilbert {\bf A}-modules $\cal M$ over monotone complete C*-algebras {\bf A} by the completeness of the unit ball of $\cal M$ with respect to two types of convergence being defined, and by a structural criterion. The derived results generalize earlier results of {\sc H.~Widom} [Duke Math.~J.~23, 309-324, MR 17 \# 1228] and {\sc W.~L.~Paschke} [Trans. Amer.~Math.~Soc.~182, 443-468, MR 50 \# 8087, Canadian J.~Math.~26, 1272-1280, MR 57 \# 10433]. For Hilbert C*-modules over commutative AW*-algebras the equivalence of the self-duality property and of the Kaplansky-Hilbert property is reproved, (cf. {\sc M.~Ozawa} [J.~Math.~Soc.~Japan 36, 589-609, MR 85m:46068] ). Especially, one derives that for a C*-algebra {\bf A} the {\bf A}-valued inner pro\-duct of every Hilbert {\bf A}-module $\cal M$ can be continued to an {\bf A}-valued inner product on it's {\bf A}-dual Banach {\bf A}-module $\cal M$' turning $\cal M$' to a self-dual Hilbert {\bf A}-module if and only if {\bf A} is monotone complete (or, equivalently, additively complete) generalizing a result of {\sc M.~Hamana} [Internat.~J.~Math.~3(1992), 185-204]. A classification of countably generated self-dual Hilbert {\bf A}-modules over monotone complete C*-algebras {\bf A} is established. The set of all bounded module operators ${\bf End}_A (\cal M)$ on self-dual Hilbert {\bf A}-modules $\cal M$ over monotone complete C*-algebras {\bf A} is proved again to be a monotone complete

funct-an↗