By I. Kaplansky

ISBN-10: 0821816500

ISBN-13: 9780821816509

An algebraic prelude Continuity of automorphisms and derivations $C^*$-algebra axiomatics and uncomplicated effects Derivations of $C^*$-algebras Homogeneous $C^*$-algebras CCR-algebras $W^*$ and $AW^*$-algebras Miscellany Mappings maintaining invertible components Nonassociativity Bibliography

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Then (a) ihere exisis a chain of irreducible morphisms: M^OAM^I,- > Mn-i - ^ Mn and a morphism g : Mn —* N with (g • /„ • / n _i • • • f\) ®R/x^R ^ 0; and 52 Chapter 6 (b) there exists a chain of irreducible morphisms: and a morphism f : M —• Nn with (g\ • g2 • • • gn * / ) 0 R/x\2'R ^ 0. PROOF: We only prove (b), for (a) can be obtained by a dual argument. The proof proceeds by induction on n. For n = 0, there is nothing to prove. Assume n > 0. ^R ^ 0. Our assumption implies that / can never be an isomorphism.

More generally this shows that F(M) is a right End(M)-module. It is also easy to see that Hom Mod(( r)(F, G) is an tf-module for F, G € Mod(C). The following is known as Yoneda's lemma. 3) LEMMA. The functor c : C -> Mod(£) is fully faithful PROOF: It suffices to show the following is an isomorphism: where

2) THEOREM. (Yoshino [66], Dieterich [21]) Let T° be a connected component of T. Assume that R has only an isolated singularity and that T° is of bounded multiplicity type. Then T = T° and T is a finite graph. 3) COROLLARY. Let R be an isolated singularity. If T has a finite connected component T°, then T = T° and hence R is of finite representation type. This shows that if (£ is of finite representation type, then the AR quiver is a connected graph. 4)COROLLARY. Let R be an isolated singularity as above.

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