Strict comparison and \(\mathcal{Z}\)-absorption of nuclear \(C^*\)-algebras. (English) Zbl 1277.46028

Matui, Hiroki; Sato, Yasuhiko

Acta Math. 209, No. 1, 179-196 (2012).

A \(C^*\)-algebra is said to be \(\mathcal Z\)-stable (or \(\mathcal Z\)-absorbing) if \(A\otimes\mathcal Z\cong A\), where \(\mathcal Z\) is the Jiang-Su algebra [X.-H. Jiang and H.-B. Su, Am. J. Math. 121, No. 2, 359–413 (1999; Zbl 0923.46069)]. \(\mathcal Z\)-stable \(C^*\)-algebras are well behaved, and are the main subject of the program of the \(K\)-theoretical classification of nuclear \(C^*\)-algebras.
To characterize \(\mathcal Z\)-stable \(C^*\)-algebras, A. S. Toms and W. Winter [Can. J. Math. 60, No. 3, 703–720 (2008; Zbl 1157.46034)] conjectured that if \(A\) is a simple unital separable nuclear \(C^*\)-algebra, then the following are equivalent:

(1)

\(A\) is \(\mathcal Z\)-stable;

(2)

\(A\) has strict comparison on positive elements, i.e., if \(d_\tau(a)<d_\tau(b)\) for any trace \(\tau\) of \(A\), where \(a, b\) are positive elements in \(\mathrm{M}_n(A)\) for some \(n\in\mathbb N\), then there is a sequence \(r_k\in\mathrm{M}_n(A)\), \(k=1, 2, \dots \), such that \(\lim_{k\to\infty}r_k^*br_k=a\);

(3)

\(A\) has finite nuclear dimension.

(The implication \((1)\Rightarrow (2)\) is known by a result of M. Rørdam [Int. J. Math. 15, No. 10, 1065–1084 (2004; Zbl 1077.46054)], and \((3)\Rightarrow (1)\) was shown by W. Winter [Invent. Math. 187, No. 2, 259–342 (2012; Zbl 1280.46041)].)
In this paper, the authors study the implication \((2)\Rightarrow (1)\) and prove a remarkable result that if \(A\) has finitely many extremal traces, then the strict comparison of \(A\) implies the \(\mathcal Z\)-stability of \(A\). Therefore, (1) and (2) are equivalent in this case.
Besides the \(\mathcal Z\)-stability and the strict comparison, the authors also consider another two (more technical) properties for the \(C^*\)-algebra \(A\):

(4)

any completely positive map \(A\to A\) can be excised in small central sequences, and

(5)

property (SI),

and show that each of them is equivalent to (1) or (2) in the case of finitely many extremal traces. Moreover, these two properties also serve as the main technical devices in the paper.
Note that recently this result was generalized independently by E. Kirchberg and M. Rørdam [“Central sequence \(C^*\)-algebras and tensorial absorption of the Jiang-Su algebra”, J. Reine Angew. Math. (to appear), doi:10.1515/crelle-2012-0118, arXiv:1209.5311], [Y. Sato, “Trace spaces of simple nuclear \(C^*\)-algebras with finite-dimensional extreme boundary”, arXiv:1209.3000], and [A. Toms, S. White and W. Winter, “\(\mathcal Z\)-stability and finite dimensional tracial boundaries”, arXiv:1209.3292] to the case that the extreme boundary of the trace simplex is closed and of finite topological dimension.

Reviewer: Zhuang Niu (Laramie)

MSC:

46L35

Classifications of \(C^*\)-algebras

Keywords:

Jiang-Su algebra; \(\mathcal Z\)-absorption; strict comparison; completely positive map; nuclear \(C^*\)-algebras

Citations:

Zbl 0923.46069; Zbl 1077.46054; Zbl 1157.46034; Zbl 1280.46041

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