Certain \(C^*\)-algebras with real rank zero and their corona and multiplier algebras. II
The author investivates the structures of projections and \(K\)-theory of purely infinite simple \(C^*\)-algebras, their multiplier and corona algebra. For a \(\sigma\)-unital purely infinite \(C^*\)-algebra \(\mathcal A\) the author shows that the homotopy classes \(D_ h(M({\mathcal A}\otimes {\mathcal K}))\) of nontrivial projections in the multiplier algebra \(M({\mathcal A}\otimes {\mathcal K})\) can be identified with the disjoint union \(K_ 0({\mathcal A})\dot{\cup} K_ 0({\mathcal A})\dot{\cup}\{\infty\}\), where \(\{\infty\}\) denotes the path component of projections in \(M({\mathcal A}\otimes {\mathcal K})\) whose images in the generalized Calkin algebra \({\mathcal C}:= C({\mathcal A}\otimes {\mathcal K})= M({\mathcal A}\otimes {\mathcal K})/{\mathcal A}\otimes {\mathcal K}\) are nontrivial projections. Furthermore, there is a bijection between \(D_ h({\mathcal C})\) and \(K_ 1({\mathcal A})\). As consequences of these results the author proves for the Cuntz algebras \({\mathcal O}_ n\) \((2\leq n< \infty)\) that the number of homotopy classes of nontrivial projection in \(\mathcal A\), in \(M({\mathcal A}\otimes {\mathcal K})\) and in \(\mathcal C\) are \(n-1\), \(2n-1\) and 1 respectively. If \(\mathcal A\) is a type III factor then the number of path components of nontrivial projections in \(M({\mathcal A}\otimes {\mathcal K})\) and in \(M({\mathcal C})\) are 3 and 1 respectively. In the second section the author shows that the generalized Fredholm index map defines a bijection between the set of path components of unitaries in the generalized Calkin algebra \({\mathcal C}\) and \(D_ h({\mathcal A})\). In the third section the author considers matroid algebras. Let \(\mathcal A\) be a separable infinite matroid algebra; then it is well-known [\textit{G. A. Elliott}, Ann. Math., II. Ser. 100, 407-422 (1974; Zbl 0296.46062)] that there exists a unique proper closed ideal \(J\) of the multiplier algebra \(M({\mathcal A})\) which contains \(\mathcal A\) strictly. The author determines the path components of nontrivial projections in the corona algebra of a separable nonunital finite matroid algebra \(\mathcal A\), i.e. \(D_ h(C({\mathcal A}))\cong K_ 0(M({\mathcal A}))/ K_ 0({\mathcal A})\). If \(\mathcal A\) is a separable infinite matroid algebra then the author proves the following relations: \[ D_ h(M({\mathcal A}))\cong D_ h(J)\dot{\cup} D_ h(J)\dot{\cup} \{\infty\},\qquad D_ h(C({\mathcal A}))\cong D_ h(J/{\mathcal A})\dot{\cup} D_ h(J/{\mathcal A})\dot{\cup}\overline{\{\infty\}}, \] where \(\overline{\{\infty\}}\) is the image of \(\{\infty\}\) in the corona algebra \(C({\mathcal A})\), furthermore the relations \[ D_ h(J/{\mathcal A})\cong K_ 0(J)/ K_ 0({\mathcal A}),\qquad D_ h(M({\mathcal A})/J)= \{0\} \] are shown. In section 4 the author considers \(^*\)-isomorphism classes \(\widetilde H({\mathcal A})\) of the set \(H({\mathcal A})\) of all nonzero \(\sigma\)-unital hereditary \(C^*\)-subalgebras of a purely infinite simple \(C^*\)- algebra \(A\) concluding that there is a surjective mapping \(\phi: K_ 0({\mathcal A})\to \widetilde H_ 0({\mathcal A})\) defined by \(\phi[p]:= [p{\mathcal A} p]\) for each nontrivial projection \(p\in {\mathcal A}\), where \(\widetilde H_ 0({\mathcal A})\) denotes the set \(\widetilde H({\mathcal A})\) without the \(^*\)-isomorphism class of \(\sigma\)-unital stable hereditary \(C^*\)- subalgebras. Furthermore, \(\widetilde H_ 0({\mathcal A})\) is a group, which is group isomorphic to \(\widetilde H_ 0({\mathcal A}\otimes {\mathcal K})\), but unfortunately \(\phi\) is not a group homomorphism in general. In the final section the author uses his recently derived results [Pac. J. Math. 145, 181-200 (1990; Zbl 0718.46056)] to show that if \(\mathcal A\) is a unital \(C^*\)-algebra with real rank zero, then every unitary element in \(\mathcal A\) can be factored into a product of two unitary elements, one which is homotopic to the identity in \(\mathcal A\) and the other has a matrix form whose entries are all partial isometries in \(\mathcal A\), where the matrix form is written with respect to a decomposition \(1= \sum_{i= 1}^ n p_ n\). If \(\mathcal A\) is a \(\sigma\)-unital, nonunital \(C^*\)- algebra with real rank zero, then the same factorization holds for the unitary group of the multiplier and the corona algebras.
\(K\)-theory of purely infinite simple \(C^*\)- algebras
separable nonunital finite matroid algebra
scientific article; zbMATH DE number 149763
scientific article