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The stable range of $C\sp*$-algebras. (English) Zbl 0559.46025
For any $C\sp*$-algebra A with 1 it is proved that its stable range (introduced by H. Bass for any ring A) equals its topological stable range (introduced by M. A. Rieffel for topological rings A). This extends an old result of Vaserstein on commutative $C\sp*$-algebras. The example of the disc algebra shows that the result cannot be extended to commutative Banach algebras.

##### MSC:
 46L05 General theory of $C^*$-algebras 46M20 Methods of algebraic topology in functional analysis 18F25 Algebraic $K$-theory and $L$-theory
##### Keywords:
topological stable range; disc algebra
Full Text:
##### References:
 [1] Bass, H.:K-theory and stable algebra. Publ. Math. IHES22, 5-60 (1964) · Zbl 0248.18025 [2] Blackadar, B.: A stable cancellation theorem for simpleC *-algebras*. Proc. London Math. Soc. (3)47, 303-307 (1983) · Zbl 0541.46056 · doi:10.1112/plms/s3-47.2.303 [3] Handelman, D.G.: Stable range inAW *-algebras. Proc. Amer. Math. Soc.76, 241-249 (1979) · Zbl 0427.46040 [4] Kadison, R.V., Ringrose, J.R.: Fundamentals of Operator Algebras I. New York: Academic Press 1982 [5] Menal, Pere Moncasi, J.: On regular rings with stable range 2. Journal of Pure and Applied Algebra24, 25-40 (1982) · Zbl 0484.16006 · doi:10.1016/0022-4049(82)90056-1 [6] Rieffel, M.A.: Dimension and stable rank in theK-theory ofC *-algebras. Proc. London Math. Soc. (3)46, 301-333 (1983) · Zbl 0533.46046 · doi:10.1112/plms/s3-46.2.301 [7] Rieffel, M.A.: The cancellation theorem for projective modules over irrational rotationC *-algebras. Proc. London Math. Soc. (3)47, 285-382 (1983) · Zbl 0541.46055 · doi:10.1112/plms/s3-47.2.285 [8] Robertson, A.G.: Stable range inC *-algebra Math. Proc. Comb. Phil. Soc.87, 413-418 (1980) · Zbl 0429.46036 · doi:10.1017/S030500410005684X [9] Suslin, A.A., Vaserstein, L.N.: Serre’s problem on projective modules over polynomial rings and algebraicK-theory. Izv. Akad. Nauk.40 (5), 87-149 (1976); In Russian, translated in Math. USSR Izvestija10, (No. 5) (1976) · Zbl 0338.13015 [10] Vaserstein, L.N.: Stable range of rings and dimension of topological spaces. Funk. An. Pril.5 (2) 17-27 (1971), (in Russian, translated in Funct. An. Appl.) · Zbl 0239.16028 [11] Vaserstein, L.N.:K-theory and the congruence subgroup problem. Math. Zametki5 (2) 233-244 (1969); In Russian, translated in Math. Notes · Zbl 0279.20037 [12] Vaserstein, L.N.: On stabilization for Milnor functorK 2, Uspekhi Mat. Nauk.30 (1) 224 (1975) (in Russian)