On inductive limits of matrix algebras over higher dimensional spaces. I.

*(English)*Zbl 0901.46053Summary: Suppose that \(A\) is a \(C^*\)-algebra of real rank zero, and is an inductive limit of \(\displaystyle{\bigoplus_{i=1}^{k_n}} M_{[n,i]} (C(X_{n,i}))\), where the spaces \(X_{n,i}\) are finite CW complexes, and \([n,i]\) are positive integers. In this note, we will prove the following results.

(1) If \(M_Q\) is the UHF algebra with \(K_0(M_Q)= Q\), then \(A\otimes M_Q\) can be expressed as an inductive limit of finite direct sums of matrix algebras over \(C(S^1)\).

(2) If one supposes further that the cohomology groups \(\widetilde{H}^* (X_{n,i})\) are torsion free and that \(\sup_{n,i} \{\dim(X_{n,i}) \}<+\infty\) (one can replace this condition by the condition of slow dimension growth), then \(A\) itself can be expressed as an inductive limit of finite direct sums of matrix algebras over \(C(S^1)\).

Recall that a result of G. Elliott says that the class of \(C^*\)-algebras of real rank zero, which can be expressed as inductive limits of finite direct sums of matrix algebras over \(C(S^1)\), is completely classified by \(K\)-theory (graded ordered \(K\)-group with dimension range).

(1) If \(M_Q\) is the UHF algebra with \(K_0(M_Q)= Q\), then \(A\otimes M_Q\) can be expressed as an inductive limit of finite direct sums of matrix algebras over \(C(S^1)\).

(2) If one supposes further that the cohomology groups \(\widetilde{H}^* (X_{n,i})\) are torsion free and that \(\sup_{n,i} \{\dim(X_{n,i}) \}<+\infty\) (one can replace this condition by the condition of slow dimension growth), then \(A\) itself can be expressed as an inductive limit of finite direct sums of matrix algebras over \(C(S^1)\).

Recall that a result of G. Elliott says that the class of \(C^*\)-algebras of real rank zero, which can be expressed as inductive limits of finite direct sums of matrix algebras over \(C(S^1)\), is completely classified by \(K\)-theory (graded ordered \(K\)-group with dimension range).

##### MSC:

46L35 | Classifications of \(C^*\)-algebras |

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |

46L05 | General theory of \(C^*\)-algebras |

46M40 | Inductive and projective limits in functional analysis |