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Prestabilization for \(K_ 1\) of Banach algebras. (English) Zbl 0629.20021

Let A be a ring and B an ideal of A satisfying the n stable range condition. It follows from earlier results of Bass and the first author that the natural map \(GL_ n(B)\to K_ 1(A,B)\) is onto. The purpose of this paper is to obtain some explicit descriptions of its kernel in various interesting cases including, for example, Banach algebras and others which extend to \(K_ 1(A,B)\) some previous results on \(K_ 1(A)\) due to P. Menal - J. Moncasi and M. Godefroid.
Reviewer: P.Menal

MSC:

20G35 Linear algebraic groups over adèles and other rings and schemes
20H25 Other matrix groups over rings
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
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References:

[1] Bass, H., \(K\)-theory and stable algebra, Publ. Math. IHES, 22, 5-60 (1964) · Zbl 0248.18025
[2] Bass, H., Algebraic K-theory, ((1968), Benjamin), 762
[3] Bass, H.; Milnor, J.; Serre, J.-., Solution of the congruence subgroup problem for \(SL_n\) (n ⩾ 3) and \(SP_{2n}\) (n ⩾ 2), Publ. Math. IHES, 33, 59-137 (1967) · Zbl 0174.05203
[4] Goodearl, K. R., Von Neumann Regular Rings, ((1979), Pitman), 369
[5] Herman, R.; Vaserstein, L., The stable range of \(C\)∗-algebas, Invent. Math., 77, 3, 553-555 (1984)
[6] van der Kallen, W., Injective stability for \(K_2\), Lecture Notes in Math., 551, 71-154 (1976) · Zbl 0349.18009
[7] van der Kallen, W., Stability for \(K_2\) of Dedekind rings of arithmetic type, Lecture Notes in Math., 854, 217-248 (1981)
[8] van der Kallen, W., A group structure on certain orbit sets of unimodular rows, J. Algebra, 82, 363-397 (1983) · Zbl 0518.20035
[9] W. van der Kallen, Vasterstein’s prestabilization theorem over commutative rings, to appear.; W. van der Kallen, Vasterstein’s prestabilization theorem over commutative rings, to appear. · Zbl 0614.16019
[10] Menal, P.; Moncasi, J., \(K_1\) of von Neumann regular rings, J. Pure Appl. Algebra, 33, 3, 295-312 (1984) · Zbl 0541.16021
[11] Milnor, J., Introduction to Algebraic \(K\)-Theory, (Ann. Math. Stud., 72 (1971), Princeton U. P) · Zbl 0237.18005
[12] Rieffel, M. A., Dimension and stable rank in the \(K\)-theory of \(C\)∗-algebaas, (Proc. London Math. Soc. (3), 46 (1983)), 301-333, (2) · Zbl 0533.46046
[13] Vasertein, L. N., Math. Ussr—Sb., 8, 383-400 (1969), transl. · Zbl 0238.20057
[14] Vaserstein, L. N., Math Notes, 5, 141-148 (1969), transl. · Zbl 0195.32202
[15] Vaserstein, L. N., Functional Anal. Appl., 5, 102-110 (1971), transl. · Zbl 0239.16028
[16] Vaserstein, L. N., Math. Ussr—Sb., 22, 271-303 (1974), transl. · Zbl 0305.18007
[17] Vaserstein, L. N., Russian Math. Survéys, 31, 4, 89-156 (1976), transl. · Zbl 0359.18015
[18] Vaserstein, L. N.; Suslin, A. A., Math. USSR—Izv., 10, 937-1001 (1976), transl. · Zbl 0379.13009
[19] Vaserstein, L. N., Bass’s first stable range condition, J. Pure Appl. Algebra, 34, 2-3, 319-330 (1984) · Zbl 0547.16017
[20] L. N. Vaserstein, Computation of \(K_1 Comm. Algebra \); L. N. Vaserstein, Computation of \(K_1 Comm. Algebra \)
[21] Godefroid, M., Déterminants sur certains anneaux non commutatifs, C. R. Acad. Sci. Paris Ser. I Math., 301, 10, 467-470 (1985) · Zbl 0584.16012
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