We show that quotients of local Banach algebras are local Banach algebras. This solves a problem posed by B. Blackadar in his book on K-Theory [BLA 1]. Furthermore our techniques allow to prove additional density Theorems for the K-Theory of normed algebras stable under inverses and logarithms.

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We propose an algorithm for computing nondegenerate hysteresis points arising in bifurcation problems with two parameters such as G:R²×R^N→R^N. A combination of methods in [5] and [7] requires finding zeros of an extended system of 4N+3 variables. Shintani and Kanda [6] proposes another algorithm, in which computation of extended system of 3N+3 variables is sufficient. On the contrary, in our algorithm, we should find zeros of another extended system of 3N+2 variables.

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By the technique of Longo's sector, the following two results are obtained. (1) A proof is given to Ocneanu's announcement about the non-existence of paragroups for Coxeter graphs E₇ and D_odd. (2) Subfactors of the type III₁. AFD factor with index 3 are classified.

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