Non-stable \(K\)-theory and extremally rich \(C^\ast\)-algebras
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.
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