On the Cones of Lower Semicontinuous Traces and 2-quasitraces of a C*-algebra
نویسنده
چکیده
The basic properties of the cones of lower semicontinuous traces and 2-quasitraces are studied. These properties include: compactness and Hausdorffness of the given cone, continuity of the corresponding functor, and a suitable notion of dual space. These results are applied to the study of the Cuntz semigroup of some classes of C*-algebras. It is shown that if a C*-algebra absorbs the Jiang-Su algebra, then the subsemigroup of its Cuntz semigroup consisting of the purely non-compact elements, is isomorphic to the dual space of the cone of lower semicontinuous 2-quasitraces. This yields a computation of the Cuntz semigroup for the following two classes of C*-algebras: C*-algebras that absorb the Jiang-Su algebra and have no non-zero simple subquotients, and simple C*-algebras that absorb the Jiang-Su algebra.
منابع مشابه
The Cone of Lower Semicontinuous Traces on a C*-algebra
The cone of lower semicontinuous traces is studied with a view to its use as an invariant. Its properties include compactness, Hausdorffness, and continuity with respect to inductive limits. A suitable notion of dual cone is given. The cone of lower semicontinuous 2-quasitraces on a (non-exact) C*-algebra is considered as well. These results are applied to the study of the Cuntz semigroup. It i...
متن کاملOn the Comparison of Positive Elements of a C*-algebra by Lower Semicontinuous Traces
It is shown in this paper that two positive elements of a C*algebra agree on all lower semicontinuous traces if and only if they are equivalent in the sense of Cuntz and Pedersen. A similar result is also obtained in the more general case where the two elements are comparable by their values on the lower semicontinuous traces. This result is used to give a characterization of the functions on t...
متن کاملQuasitraces Need Not Be Traces
We prove the title, establishing the existence of a quasitrace on a (unital, simple) C-algebra (of real rank zero and stable rank one), which is not a tracial state.
متن کاملOn some open problems in cone metric space over Banach algebra
In this paper we prove an analogue of Banach and Kannan fixed point theorems by generalizing the Lipschitz constat $k$, in generalized Lipschitz mapping on cone metric space over Banach algebra, which are answers for the open problems proposed by Sastry et al, [K. P. R. Sastry, G. A. Naidu, T. Bakeshie, Fixed point theorems in cone metric spaces with Banach algebra cones, Int. J. of Math. Sci. ...
متن کاملOn Polar Cones and Differentiability in Reflexive Banach Spaces
Let $X$ be a Banach space, $Csubset X$ be a closed convex set included in a well-based cone $K$, and also let $sigma_C$ be the support function which is defined on $C$. In this note, we first study the existence of a bounded base for the cone $K$, then using the obtained results, we find some geometric conditions for the set $C$, so that ${mathop{rm int}}(mathrm{dom} sigma_C) neqem...
متن کامل