132 C. L. Schochet with exact rows in some abelian category. 2 asserts that there is a morphism :Ker( )! Cok( ) The Snake Lemma (cf. [Mac, page5]) whi

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1 New York Journal of Mathematics New York J. Math. 5 (1999) 131{137. The Topological Snake Lemma and Corona Algebras C. L. Schochet Abstract. We establish versions of the Snake Lemma from homological algebra in the context of topological groups, anach spaces, and operator algebras. We apply this tool to demonstrate that if f :! is a quasi-unital C - map of separable C -algebras, so that it induces a map of Corona algebras f : Q!Q,andiff is mono, then the induced map f is also mono. This paper presents a cross-cultural result: we use ideas from homological algebra, suitable topologized, in order to establish a functional analytic result. The Snake Lemma (also known as the Kernel-Cokernel Sequence) 1 is a basic result in homological algebra. Here is what it says. Suppose that one is given a commutative diagram (1) Ker( ) Ker( ) A A A Cok( ) Cok( ) Received August 3, Mathematics Subject Classication. Primary 18G35, 46L5 Secondary 22A5, 46L85. Key words and phrases. Corona algebra, multiplier algebra, Snake Lemma, homological algebra for C -algebras. 1 and fondly recalled as the one serious mathematical theorem ever to appear in a major motion picture: \It's My Turn," starring Jill Clayburgh (198). 131 c1999 State University of New York ISSN

2 132 C. L. Schochet with exact rows in some abelian category. 2 asserts that there is a morphism :Ker( )! Cok( ) The Snake Lemma (cf. [Mac, page5]) which is natural with respect to diagrams and a long exact sequence (2) Ker( ) Ker( ) Cok( ) Cok( ) : The maps not explicitly labeled in (1) and (2) are induced by,, and in the obvious way. The boundary map is dened as follows. 3 Ker( ) A A Cok( ) Let a 2 A be an element ofker( ). Since is onto, there is some a 2 A with (a) =a. Then (a) = (a) = (a )= and so (a) 2 Ker( ) = Im( ). Thus there is some unique b 2 with (b )= (a). Finally, dene (a )=[b ] 2 =Im( )=Cok( ): The map is well-dened and it is a morphism in the category. 4 We suppose that the following proposition is well-known. The notation refers to (1). Proposition 3. Suppose that A is a ring, A is an ideal, A is the quotient ring, and similarly for. Further, suppose that the maps,, and are ring homomorphisms, and that (A ) is an ideal in. Then the map is a ring homomorphism. 2 For instance, modules over some commutative ring. Eventually the ring will be the complex numbers. 3 Here we assume for convenience that we are working with a category of modules over a commutative ringsothatourobjectshave elements. This is not necessary, strictly speaking, but the alternative is to be far more abstract than is needed for present purposes. 4 Clayburgh denes and proves that it is well-dened in the opening credits of the movie. Her proof is correct.

3 Topological Snake Lemma 133 Proof. We check directly using the denition of. Suppose that a 1 a 2 2 Ker( ). We wish to show that (a 1 a 2) =(a 1)(a 2): Choose elements a i a2 A with (a i )=a i and (a) =a 1 a 2 2 Ker( ). Then (a ; a 1 a 2 )=a 1 a 2 ; a 1 a 2 = and so Let a 2 A be the unique element with a ; a 1 a 2 2 Ker( ) = Im( ): (a )=a ; a 1 a 2 : We have (a i )= (a i )=(a i )= and (a) = (a) =(a 1 a 2) = so that (a) and both (a i ) lie in Ker( ) = Im( ). Thus there exist unique elements b i b 2 with (b i)=(a i ) and (b )=(a): Of course (a i )=[b i] 2 Cok( ) and (a 1 a 2) =[b ] 2 Cok( ) so to complete this proof we must show that [b 1][b 2]=[b ]. Now [b ] ; [b 1][b 2]=[b ; b 1 b 2] so it suces to show thatb ; b 1 b 2 2 Im( ). We compute: and since is mono we have (b ; b 1 b 2)=(a ; a 1 a 2 )= (a )= (a ) b ; b 1 b 2 = (a ) 2 Im( ) as required. This implies that the map is a ring map. Now we start to impose topological conditions upon diagram (1). Proposition 4. Suppose that A is a topological group with subgroup A and quotient group A, and similarly for, and suppose that the maps,, and are continuous. Give the various kernels the subgroup topology and the various cokernels the quotient group topology. Then all of the maps in the 6-term sequence (2) are continuous. Proof. It is necessary only to show that is continuous. Let U Cok( )bean open set. We must show that ;1 (U) is an open set in Ker( ). Let :! Cok( ) be the natural map. It is continuous, so the set ;1 (U) is open in. As has the relative topology in, this means that there is some open set V with ;1 (U) = \ V:

4 o 134 C. L. Schochet Then ;1 (V ) is open in A, since is continuous, and ;1 (V ) is open in A, since is an open map. Thus ;1 (V ) \ Ker( ) is an open set in Ker( ). To complete the argument itwillthus suce to establish that () ;1 (U) = ;1 (V ) \ Ker( ): This is a direct check. Suppose that a 2 ;1 (U). Then (a ) 2 U. ut (a )=[b ] for some b 2 given as per the denition of, andsob 2 ;1 (U). Then b 2 \ V V and (b )=(a) with (a) =x by the denition of, soa 2 ;1 (V ). Then a = (a) 2 ;1 (V ) as required. In the opposite direction, let a 2 ;1 (V ) \ Ker( ). Then a = (a) with a 2 ;1 (V ), so (a) 2 V. Also, (a) 2,sincea 2 Ker( ). Thus and so (x) =[(a)] 2 U. (a) 2 \ V = ;1 (U) Recall that if : A! A is a continuous surjection of anach spaces then it has a continuous cross-section : A! A by the artle-graves theorem ([G, Theorem 4], [Mi, Corollary on page 364]). We may use this section to explicitly realize the map. Proposition 5. Suppose that A is a anach space, A is a closed anach subspace, and A is the quotient anach space, and similarly for, and suppose that the vertical maps are continuous. Then we may realize the map :Ker( ) ;! Cok( ) in terms of the artle-graves section via the diagram Ker( ) A A Cok( ) Proof. As any section (continuous or not) of may be used in the denition of, we may as well use the section. Then the composition : Ker( )! is obviously continuous. Its image lies in the image of, and since has the relative topology in we may conclude that :Ker( )! is also continuous. Composing with the continuous projection! Cok( ) yields.

5 Topological Snake Lemma 135 Note that as a consequence of the proof we see that all artle-graves sections yield the same map. We continue to assume that (1) is a diagram in the category of anach spaces and closed subspaces as in the previous proposition. Proposition 6 (K. Thomsen). If the map is a monomorphism, then the map is an isometry. Proof. This is a direct calculation. Let a 2 A and choose some a 2 A with (a) =a. Then jj(a )jj = inf jj(a) ; (a )jj a 2A but is mono, hence an isometry completing the proof. We turn our attention to C -algebras. = inf a 2A jj(a) ; (a )jj = inf a 2A jja ; (a )jj = jja jj Theorem 7. Suppose in Diagram (1) that A is a C -algebra, A is a closed ideal, and A is the quotient algebra, and similarly for, and suppose that the vertical maps are C -maps. Then 1. the Snake sequence Ker( ) Ker( ) Cok( ) Cok( ) is an exact sequence of anach spaces. 2. The sequence Ker( ) Ker( ) is an exact sequence ofc -algebras and C -maps. 3. If is a monomorphism then is an isometry and the sequence reduces to the sequence Ker( ) Cok( ) Cok( ) 4. If (A ) is a closed ideal in then the map is also a map of C -algebras. :Ker( )! Cok( ) Proof. This simply applies the earlier results to the context of C -algebras. The only point to check is that preserves the -operation, and this we leave as an exercise. If is a C -algebra then the multiplier algebra of is denoted by M and the Corona algebra is denoted Q = M=. Recall [H, 1.1.6], [T, 2.6] that a -homomorphism f :! is quasi-unital when there is a projection p 2M such that the closed linear span of f() has

6 136 C. L. Schochet the form p. Thomsen shows that a -homomorphism f :! extends to a -homomorphism Mf : M!M which is strictly continuous on the unit ball if and only if f is quasi-unital. Of course if f does extend then there is an induced map f : Q!Q. Thomsen also shows that if f is a monomorphism then so is Mf. 5 Proposition 8. Suppose that and are C -algebras and f :! is a quasi-unital map. Then the natural diagram M Q f Mf f M Q leads to the exact sequence of anach spaces Ker(f) Ker(Mf) Ker( f) Cok(f) Cok(Mf) Cok( f) : The map is continuous. If Mf is mono then is an isometry and the sequence degenerates to the exact sequence Ker( f) Cok(f) Cok(Mf) Cok( f) and if f is the inclusion of an ideal then is a C -map. Proof. This follows by specializing the general results above. Theorem 9. Suppose that and are separable C -algebras and that f :! is a quasi-unital monomorphism. Then the natural map f : Q!Q is a monomorphism. Proof. We apply Proposition 8 to obtain the sequence Ker( f) Cok(f) ::: Now Cok(f) is a quotient of the separable C -algebra (as a metric vector space) and hence is separable. This, plus the fact that is an isometry, implies that Ker( f) is separable. On the other hand, Ker( f) is an ideal in Q and we know from L. G. rown [r, Corollary 6] that Q has no non-trivial separable ideals. The conclusion is that Ker( f)=and f is mono. Remark 1. Klaus Thomsen has found a direct proof of the above result. It will be included in [S]. The original impetus for this work came from wanting an explicit realization of the map KK 1 (A )! KK 1 (A ) induced from a C -map!. This is indeed possible, via the induced map f : Q!Q. It is vital there to know thatiff is mono then so is f. For details see [S]. Acknowledgements. I wish to thank Terry Loring, Huaxin Lin, and especially Gert Pedersen and Klaus Thomsen for their help and encouragement. 5 Here is the argument. Let m 2 M be such that Mf(m) =. Then f(mb) = Mf(m)f(b) = for all b 2. Since f is injective this means that mb = for all b 2 and hence m =.

7 References Topological Snake Lemma 137 [G] R. G. artle and L. Graves, Mappings between function spaces, Trans. Amer. Math. Soc. 72 (1952), 4{413, MR 13,951i, Zbl [r] Lawrence G. rown, Determination of A from M(A) and related matters, C. R. Math. Rep. Acad. Sci. Canada 1 (1988), 273{278, MR 89j:4661, Zbl [H] Nigel Higson, Algebraic K-theory of stable C -algebras, Advances in Math. 67 (1988), 1{14, MR 89g:4611, Zbl [Mac] S. Mac Lane, Homology, Springer-verlag New York, 1963, MR 28 #122, Zbl [Mi] E. Michael, Continuous Selections, I. Ann. of Math. 63 (1956), 361{382, MR 17,99e, Zbl [S] C. L. Schochet, The ne structure of the Kasparov groups I: continuity of the KK-pairing, preprint. [T] Klaus Thomsen, Homotopy classes of -homomorphisms between stable C -algebras and their multiplier algebras, Duke Math. J. 61 (199), 67{14, MR 91m:46115, Zbl Mathematics Department, Wayne State University, Detroit, MI 4822 claude@math.wayne.edu This paper is available via