Dimension and stable rank in the K-theory of \(C^*\)-algebras.

*(English)*Zbl 0533.46046The so frequently called stability properties for (commutative, unital) \(C^*\)-algebras may be studied either by means of the Bass stable rank or, in view of Gel’fand’s duality, by the classical concept of dimension of their spectrum, the basic connecting relation in the complex case being \(Bsr({\mathcal C}(X))=\dim(X)/2+1,\) where (.) denotes ”integer part of” [cf. N. Vasershtejn, Funkts. Anal. Prilozh. 5, No.2, 17-27 (1971; Zbl 0239.16028)]. In the present article a notion of dimension for any Banach (or even topological) algebra A is appropriately formulated in terms of the density in \(A^ n\) of the set \(Lg_ n(A) (Rg_ n(A))\) of what are traditionally called in algebraic K-theory left (right) unimodular rows. However, as this notion dominates the Bass stable rank in general, and furtheron agrees with it in the original (and algebraic in character) case of \(A={\mathcal C}(X)\), the more convenient term ”topological stable rank” instead of ”dimension” is preferred.

Now, after examing the lowest-dimensional case which includes AF-\(C^*\)- algebras [see also A. G. Robertson, Math. Proc. Camb. Philos. Soc. 87, 413-418 (1980; Zbl 0429.46036); B. Blackadar and D. Handelman, J. Funct. Anal. 45, 297-340 (1982; Zbl 0513.46047)], the results are concerned with the behviour of the topological stable rank under passage to ideals and quotient algebras, inductive limits, matrix algebras over a given algebra, and notably with its nice one with respect to forming crossed product \(C^*\)-algebras for actions of the group of integers. Thus, the general theory is here provided for answering in the affirmative to the singnificant question as to whether the cancellation property holds for projective modules over irrational rotation \(C^*\)- algebras, in a subsequent paper [i.e., the author, ibid. 47, 285-302 (1983)].

Now, after examing the lowest-dimensional case which includes AF-\(C^*\)- algebras [see also A. G. Robertson, Math. Proc. Camb. Philos. Soc. 87, 413-418 (1980; Zbl 0429.46036); B. Blackadar and D. Handelman, J. Funct. Anal. 45, 297-340 (1982; Zbl 0513.46047)], the results are concerned with the behviour of the topological stable rank under passage to ideals and quotient algebras, inductive limits, matrix algebras over a given algebra, and notably with its nice one with respect to forming crossed product \(C^*\)-algebras for actions of the group of integers. Thus, the general theory is here provided for answering in the affirmative to the singnificant question as to whether the cancellation property holds for projective modules over irrational rotation \(C^*\)- algebras, in a subsequent paper [i.e., the author, ibid. 47, 285-302 (1983)].

Reviewer: G.F.Nassopoulos

##### MSC:

46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |

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

46L55 | Noncommutative dynamical systems |

18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |

55N15 | Topological \(K\)-theory |