Abstract. Let A be an exact category, that is, an extension-closed full subcategory of an abelian category. First, we give new characterizations of an almost split sequence in A, which yields some necessary and sufficient conditions for A to have an almost split sequence with prescribed end terms. Then, we study when an almost split sequence in A induces an almost split sequence in an exact subcategory C of A. In case A has almost split sequences and C is Ext-finite and Krull-Schmidt, we obtain a necessary and sufficient condition for C to have almost split sequences. Finally, we show some applications of these results.

a b s t r a c tThis paper studies the existence of Auslander-Reiten sequences in subcategories of mod(Λ), where Λ is a finite dimensional algebra over a field. The two main theorems give necessary and sufficient conditions for the existence of Auslander-Reiten sequences in subcategories. Theorem. Let M be a subcategory of mod(Λ) closed under extensions and direct summands, and let M be an indecomposable module in M such that Ext 1 (M,M) ̸ = 0 for someM in M.Then the following are equivalent.(i) DTrM has an M-precover in the stable category mod(Λ),We also have the dual result of the above theorem. Together they strengthen the results in [3,4], and in Kleiner (1997) [7].

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Abstract. Let A be a unital separable simple exact C*-algebra. Suppose that either 1. A is purely infinite, or 2. A ⊗ K has strict comparison of positive elements and stable rank one, and A has unique tracial state.Then for all X ∈ M (A ⊗ K ) , X is a commutator if and only if X does not have the form α1 M (A ⊗K ) + x , for some α ∈ C − {0} and for some x belonging to a proper ideal ofMathematics subject classification (2010): 46L35.