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

Robert, Leonel; Rørdam, Mikael

Proc. Lond. Math. Soc. (3) 106, No. 6, 1330-1370 (2013).

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).

Reviewer: Hannes Thiel (Münster)

MSC:

46L05	General theory of \(C^*\)-algebras
46L06	Tensor products of \(C^*\)-algebras
46L35	Classifications of \(C^*\)-algebras

Keywords:

\(C^{\ast}\)-algebras; Cuntz semigroup; almost divisible; almost unperforated; Kirchberg’s covering number; tensor products; ultrapowers

Citations:

Zbl 1180.46048