Divisibility properties for $$C^{\ast}$$-algebras.(English)Zbl 1339.46051

This paper deals with divisibility and comparison properties for Cuntz semigroups of $$C^{\ast}$$-algebras. Given a $$C^{\ast}$$-algebra $$A$$, the Cuntz semigroup $$\mathrm{Cu}(A)$$ is an ordered, abelian semigroup given by certain equivalence classes of positive elements in the stabilization of $$A$$, analogous to the construction of the Murray-von Neumann semigroup $$V(A)$$ as equivalence classes of projections.
The authors introduce and study three notions of divisibility for elements in $$\mathrm{Cu}(A)$$ (see Definition 3.1). Specifying to the class $$[1]$$ of the unit of a unital $$C^{\ast}$$-algebra $$A$$, we obtain the following divisibility numbers associated to $$A$$, for each $$m\in\mathbb{N}$$ (Definition 3.7):
(i)
$$\mathrm{Div}_m(A)$$ is the least $$n$$ such that exists $$x\in\mathrm{Cu}(A)$$ with $$mx\leq u\leq nx$$. (This means that $$[1]$$ is $$(m,n)$$-divisible.)
(ii)
$$\mathrm{Dec}_m(A)$$ is the least $$n$$ such that there exist $$x_1,x_2,\dots,x_m\in\mathrm{Cu}(A)$$ with $$x_1+x_2+\dots+x_m\leq u \leq nx_k$$ for all $$k$$. (This means that $$[1]$$ is $$(m,n)$$-decomposable.)
(iii)
$$w-\mathrm{Div}_m(A)$$ is the least $$n$$ such that there exists $$x_1,x_2,\dots,x_n\in\mathrm{Cu}(A)$$ with $$mx_k\leq u\leq x_1+x_2+\dots+ x_n$$ for all $$k$$. (This means that $$[1]$$ is weakly $$(m,n)$$-divisible.)
In each case, the value is set to $$\infty$$ if the condition is not satisfied for any $$n$$.
The paper studies the invariants $$\mathrm{Div}_m(A)$$, $$\mathrm{Dec}_m(A)$$ and $$w-\mathrm{Div}_m(A)$$ and their connection to structural properties of $$A$$. Let us just mention a few results (Corollary 5.4):
(i)
$$\mathrm{Div}_m(A)<\infty$$ if and only if there exists a full *-homomorphism from the cone over the $$m\times m$$-matrices to $$A$$.
(ii)
$$\mathrm{Dec}_m(A)<\infty$$ if and only if $$A$$ contains $$m$$ full, pairwise orthogonal positive elements.
(iii)
$$w-\mathrm{Div}_m(A)<\infty$$ if and only if $$A$$ has no irreducible representations of dimension $$<m$$. In particular, $$A$$ has a character if and only if $$w-\mathrm{Div}_2(A)=\infty$$ (Corollary 5.6).

A low value of $$\mathrm{Div}_m(A)$$ (or the other invariants) means that $$A$$ has good divisibility properties.
A crucial observation is that $$\mathrm{Div}_m(B)\leq\mathrm{Div}_m(A)$$ whenever there is a unital *-homomorphism from $$A$$ to $$B$$. Thus, establishing a lower bound on $$\mathrm{Div}_m(B)$$ can be used to prove that there exists no unital *-homomorphism from $$A$$ to $$B$$ when $$A$$ has good divisibility properties. This strategy was implicitly used in [M. Dadarlat et al., Math. Res. Lett. 16, No. 1, 23–26 (2009; Zbl 1180.46048)] to show that there exists a unital, simple, infinite-dimensional, nuclear $$C^*$$-algebra $$A$$ such that the Jiang-Su algebra $$\mathcal{Z}$$ does not embed as a unital subalgebra into $$A$$. One has $$\mathrm{Div}_3(\mathcal{Z})=4$$, but the example constructed in [loc. cit.] satisfies $$\mathrm{Div}_3(A)>4$$.
Every unital, simple, infinite-dimensional C*-algebra $$A$$ has ‘reasonable’ divisibility properties in the sense that the divisibility numbers $$\mathrm{Div}_m(A)$$, $$\mathrm{Dec}_m(A)$$ and $$w-\mathrm{Div}_m(A)$$ are finite for all $$m$$ (Example 3.11). However, there is no upper bound for how bad the divisibility properties can be. For every $$N$$, the authors construct a unital, simple, infinite-dimensional AH-algebra $$A$$ with $$N\leq w-\mathrm{Div}_2(A)\leq\mathrm{Div}_2(A)\leq 3N+4$$ (Theorem 7.9).
As an application, the authors obtain the surprising result that there exists a sequence ($$A_n$$) of unital, simple, infinite-dimensional $$C^*$$-algebras such that their product $$\prod_n A_n$$ and their ultraproduct over some free ultrapower have a character (Corollary 8.6).

MSC:

 46L05 General theory of $$C^*$$-algebras 46L06 Tensor products of $$C^*$$-algebras 46L35 Classifications of $$C^*$$-algebras

Zbl 1180.46048
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