Non-stable \(K\)-theory and extremally rich \(C^\ast\)-algebras. (English) Zbl 1303.46062
A unital \(C^\ast\)-algebra \(A\) has stable rank one if the set \(A^{-1}\) of invertible elements is dense in \(A\), is isometrically rich if the set \(A^{-1}_l \cup A^{-1}_r\) of one-sided invertible elements is dense, and is extremally rich if the set \(A^{-1}_q\) of quasi-invertible elements is dense. A \(C^\ast\)-algebra \(A\) has weak cancellation if any pair of projections \(p, q\) in \(A\) that generate the same closed ideal \(I\) of \(A\) and have the same image in \(K_0(I)\) must be Murray-von Neumann equivalent in \(A\). If the matrix algebra \(\mathbb M_n(A)\) with coefficients from \(A\) has weak cancellation for every \(n\), or equivalently, if \(A \otimes \mathcal K\) has weak cancellation, one says that \(A\) has stable weak cancellation. “A main question in this paper is whether every extremally rich \(C^\ast\)-algebra has weak cancellation. Some answers to this question are in Section 7.3 and Corollary 3.6, which state that every isometrically rich \(C^\ast\)-algebra has weak cancellation.”
One says that \(A\) has \(K_1\)-surjectivity if the map from the “classifying space” \(\mathcal U( A)/\mathcal U_0(A)\) to \(K_1(A)\) is surjective, \(K_1\)-injectivity if this map is injective, and \(K_1\)-bijectivity if it is bijective. Theorem 4.4 states that every extremally rich \(C^\ast\)-algebra with weak cancellation has \(K_1\)-surjectivity. In Corollary 5.3, the authors show that if \(A\) is extremally rich with weak cancellation, then \(K_e(A) = \varinjlim_n (\mathcal E(\mathbb M_n ( A))/\mathrm{homotopy})\) and one says that \(A\) has \(K_e\)-surjectivity, \(K_e\)-injectivity, or \(K_e\)-bijectivity if the map from \(\mathcal E(A)/\mathrm{homotopy}\) to \(K_e(A)\) is respectively surjective, injective, or bijective. These properties actually imply the corresponding \(K_1\)-properties. In Theorem 4.7, the authors show that every extremally rich \(C^*\)-algebra with weak cancellation has \(K_e\)-surjectivity.
One says that a (non-unital) \(C^\ast\)-algebra \(B\) has good index theory if some kind of “Stinespring lifting” holds, i.e., whenever \(B\) is embedded as an ideal in a unital \(C^\ast\)-algebra \(A\), then any unitary \(u\) in \(A/B\), which vanishes by the index map \(\partial_1 : K_1(B) \to K_0(B)\), lifts to an element of \(A\). Theorem 5.1 states that every extremally rich \(C^\ast\)-algebra with weak cancellation has good index theory.
The main result of Section 6 is that every extremally rich \(C^\ast\)-algebra with weak cancellation has \(K_1\)-injectivity. The authors also show in Section 6 that every extremally rich \(C^\ast\)-algebra with weak cancellation has weak \(K_0\)-surjectivity and \(K_e\)-injectivity.
One says that \(A\) has \(K_1\)-surjectivity if the map from the “classifying space” \(\mathcal U( A)/\mathcal U_0(A)\) to \(K_1(A)\) is surjective, \(K_1\)-injectivity if this map is injective, and \(K_1\)-bijectivity if it is bijective. Theorem 4.4 states that every extremally rich \(C^\ast\)-algebra with weak cancellation has \(K_1\)-surjectivity. In Corollary 5.3, the authors show that if \(A\) is extremally rich with weak cancellation, then \(K_e(A) = \varinjlim_n (\mathcal E(\mathbb M_n ( A))/\mathrm{homotopy})\) and one says that \(A\) has \(K_e\)-surjectivity, \(K_e\)-injectivity, or \(K_e\)-bijectivity if the map from \(\mathcal E(A)/\mathrm{homotopy}\) to \(K_e(A)\) is respectively surjective, injective, or bijective. These properties actually imply the corresponding \(K_1\)-properties. In Theorem 4.7, the authors show that every extremally rich \(C^*\)-algebra with weak cancellation has \(K_e\)-surjectivity.
One says that a (non-unital) \(C^\ast\)-algebra \(B\) has good index theory if some kind of “Stinespring lifting” holds, i.e., whenever \(B\) is embedded as an ideal in a unital \(C^\ast\)-algebra \(A\), then any unitary \(u\) in \(A/B\), which vanishes by the index map \(\partial_1 : K_1(B) \to K_0(B)\), lifts to an element of \(A\). Theorem 5.1 states that every extremally rich \(C^\ast\)-algebra with weak cancellation has good index theory.
The main result of Section 6 is that every extremally rich \(C^\ast\)-algebra with weak cancellation has \(K_1\)-injectivity. The authors also show in Section 6 that every extremally rich \(C^\ast\)-algebra with weak cancellation has weak \(K_0\)-surjectivity and \(K_e\)-injectivity.
Reviewer: Do Ngoc Diep (Hanoi)
MSC:
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 19K14 | \(K_0\) as an ordered group, traces |
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