Real and topological stable rank.

摘要

A locally compact normal Hausdorff space X of covering dimension m (m ≥ 2), and with the property that its one-point compactification has covering dimension 0, is constructed along the lines of a space constructed by E. van Douwen in 1992. The commutative C*-algebra A = C0(X), the set of all continuous complex valued functions that vanish at infinity, has the property that its topological stable rank is not equal to the topological stable rank of its multiplier algebra M(A). This provides an example in response to an open question raised by Rieffel in a 1983 paper.; In addition, the constructed space X is separable, locally countable but not second countable and has the property that every continuous complex valued function that vanishes at infinity has countable support. These properties of X are used to show that the C*-algebra A = C0(X) is the direct limit of AF-algebras and thus K1(A) = 0. The real rank of A equals zero, which follows from the dimension properties of X. However, the real rank of M(A) is not 0. With these three properties, A provides a counterexample to two conjectures posed by Brown and Perdersen in 1991.; Lastly we look at free modules over a C*-algebra. The topological stable rank of a free module is computed and shown to be the same as its Bass stable rank. The concept of connected stable rank is introduced and computed for a free module over a C*-algebra.