New York Journal of Mathematics

New York J Math

The Top ological Snake Lemma

and Corona Algebras

C L Scho chet

Abstract We establish versions of the Snake Lemma from homological alge

bra in the context of top ological groups Banach spaces and op erator algebras



We apply this to ol to demonstrate that if f B B is a quasiunital C



map of separable C algebras so that it induces a map of Corona algebras

f QB QB andiff is mono then the induced map f is also mono

This pap er presents a crosscultural result we use ideas from homological alge

bra suitable top ologized in order to establish a functional analytic result

The Snake Lemma also known as the Sequence is a basic

result in Here is what it says Supp ose that one is given a

Ker Ker Ker

A

A A

B

B B

Cok

Cok Cok

Received August

Mathematics Subject Classication Primary G L Secondary A L

Key words and phrases Corona algebra multiplier algebra Snake Lemma homological alge



bra for C algebras

and fondly recalled as the one serious mathematical theorem ever to app ear in a ma jor motion

picture Its My Turn starring Jill Clayburgh

c

 State UniversityofNewYork

ISSN

C L Schochet



with exact rows in some ab elian category The Snake Lemma cf Macpage

asserts that there is a morphism

Ker Cok

which is natural with resp ect to diagrams and a long

Ker Ker Ker Cok Cok Cok

The maps not explicitly lab eled in and are induced by and in

the obvious way



The b oundary map is dened as follows

Ker

A

A

B

B B

Cok

Let a A b e an elementofKer Since is onto there is some a A

with aa Then

a a a

and so a Ker Im Thus there is some unique b B with b

a Finally dene

B Im Cok a b



The map is welldened and it is a morphism in the category

We supp ose that the following prop osition is wellknown The notation refers to

Prop osition Suppose that A is a A is an ideal A is the quotient ring

and similarly for B Further suppose that the maps and are ring homo

morphisms and that A is an ideal in B Then the map is a ring homomor

phism

For instance mo dules over some commutative ring Eventually the ring will b e the complex

numb ers

Here we assume for convenience that we are working with a category of mo dules over a

commutativeringsothatourobjectshave elements This is not necessary strictly sp eaking but

the alternative is to b e far more abstract than is needed for present purp oses

Clayburgh denes and proves that it is welldened in the op ening credits of the movie Her pro of is correct

Topological Snake Lemma

Pro of Wecheck directly using the denition of Supp ose that a a Ker



We wish to show that

a a a a

 

Cho ose elements a a A with a a and aa a Ker Then

i i



i

a a a a a a a



 

and so

a a a Ker Im



Let a A b e the unique elementwith

a a a a



Wehave

a a a

i i

i

and

a a a a



so that a and b oth a lie in Ker Im Thus there exist unique

i

elements b b B with

i

b a and b a

i

i

Of course

a b Cok

i i

and

a a b Cok



ust show that b b b Now so to complete this pro of wem



b b b b b b

 

so it suces to showthat b b b Im We compute



b b b a a a a a





and since is mono wehave

b b a Im b



as required This implies that the map is a ring map

Nowwe start to imp ose top ological conditions up on diagram

Prop osition Suppose that A is a topological group with subgroup A and quo

tient group A and similarly for B and suppose that the maps and are

continuous Give the various kernels the subgroup topology and the various coker

nels the quotient group topology Then al l of the maps in the term sequence

econtinuous ar

Pro of It is necessary only to showthat is continuous Let U Cok bean

op en set Wemust show that U is an op en set in Ker

Let B Cok b e the natural map It is continuous so the set U

is op en in B As B has the relative top ology in B this means that there is some

op en set V B with

U B V

C L Schochet

Then V is op en in A since is continuous and V is op en in A

since is an op en map Thus

V Ker

is an op en set in Ker To complete the argumentitwillthus suce to establish

that

U V Ker

that a U Then a U But This is a direct check Supp ose

a b for some b B given as p er the denition of andsob U

Then

b B V V

and b awith ax by the denition of soa V Then

a a V

as required

In the opp osite direction let a V Ker Then a a with

a V so a V Also a B sincea Ker Thus

a B V U

and so x a U

Recall that if A A is a continuous surjection of Banach spaces then it

has a continuous crosssection A A by the BartleGraves theorem BG

Theorem Mi Corollary on page We may use this section to explicitly

realize the map

Prop osition Suppose that A is a Banach space A is a closed Banach subspace

and A is the quotient Banach space and similarly for B and suppose that the

vertical maps arecontinuous Then we may realize the map

Ker Cok

am in terms of the BartleGraves section via the diagr

Ker

o

A

A

B

B

Cok

Pro of As any section continuous or not of may b e used in the denition of

we may as well use the section Then the comp osition Ker B

is obviously continuous Its image lies in the image of and since B has the

relative top ology in B wemay conclude that Ker B is also continuous

Comp osing with the continuous pro jection B Cok yields

Topological Snake Lemma

Note that as a consequence of the pro of we see that all BartleGraves sections

yield the same map

We continue to assume that is a diagram in the category of Banach spaces

and closed subspaces as in the previous prop osition

Prop osition K Thomsen If the map is a monomorphism then the map

is an isometry

Pro of This is a direct calculation Let a A and cho ose some a A with

aa Then

jj a jj inf jj a a jj

a A

a jj inf jj a

a A

but is mono hence an isometry

inf jja a jj

a A

jja jj

completing the pro of

We turn our attention to C algebras

Theorem Suppose in Diagram that A is a C algebra A is a closedideal

and A is the quotient algebra and similarly for B and suppose that the vertical

maps are C maps Then

the Snake sequence

Ker Ker Ker Cok Cok Cok

is an exact sequence of Banach spaces

The sequence

Ker Ker Ker

is an exact sequenceof C algebras and C maps

If is a monomorphism then is an isometry and the sequence reduces to

the sequence

Ker Cok Cok Cok

If A is a closed ideal in B then the map

Ker Cok

is also a map of C algebras

Pro of This simply applies the earlier results to the context of C algebras The

only point to check is that preserves the op eration and this we leave as an

exercise

If B is a C algebra then the multiplier algebra of B is denoted by MB and the

Corona algebra is denoted QB MBB

Recall H T that a homomorphism f B B is quasiunital

when there is a pro jection p MB such that the closed linear span of f B B has

C L Schochet

the form pB Thomsen shows that a homomorphism f B B extends to a

homomorphism Mf MB MB which is strictly continuous on the unit ball

if and only if f is quasiunital Of course if f do es extend then there is an induced

map f QB QB Thomsen also shows that if f is a monomorphism then so is



Mf

Prop osition Suppose that B and B are C algebras and f B B is a

quasiunital map Then the natural diagram

QB

B MB



f Mf

f

QB

B MB

leads to the exact sequence of Banach spaces

Kerf KerMf Cokf CokMf

Kerf Cokf

The map is continuous If Mf is mono then is an isometry and the sequence

degenerates to the exact sequence

Cokf CokMf

Kerf Cokf

and if f is the inclusion of an ideal then is a C map

Pro of This follows by sp ecializing the general results ab ove

Theorem Suppose that B and B areseparable C algebras and that f B B

is a quasiunital monomorphism Then the natural map

f QB QB

is a monomorphism

Pro of We apply Prop osition to obtain the sequence

Cokf

Kerf

Now Cokf is a quotient of the separable C algebra B as a metric

and hence is separable This plus the fact that is an isometry implies that

Kerf is separable On the other hand Kerf is an ideal in QB and we know

from L G Brown Br Corollary that QB has no nontrivial separable ideals

The conclusion is that Kerf and f is mono

Remark Klaus Thomsen has found a direct pro of of the ab ove result It will

b e included in S The original imp etus for this work came from wanting an explicit

realization of the map KK A B KK A B induced from a C map B B

This is indeed p ossible via the induced map f QB QB It is vital there to

knowthatif f is mono then so is f For details see S

Acknowledgements I wish to thank Terry Loring Huaxin Lin and esp ecially

Gert Pedersen and Klaus Thomsen for their help and encouragement

Here is the argument Let m MB be such that Mfm Then f mb

Mfmf b for all b B Since f is injective this means that mb for all b B

and hence m

Topological Snake Lemma

References

BG R G Bartle and L Graves Mappings between function spaces Trans Amer Math So c

 MR i Zbl

Br Lawrence G Brown Determination of A from M A and related matters C R Math

Rep Acad Sci Canada  MR j Zbl



H Nigel Higson Algebraic K theory of stable C algebras Advances in Math 

MR g Zbl

Mac S Mac Lane Homology Springerverlag New York MR Zbl

Mi E Michael Continuous Selections I Ann of Math  MR e

Zbl

S C L Scho chet The ne structure of the Kasparov groups I continuity of the KKpairing

preprint



homomorphisms between stable C algebras and T Klaus Thomsen Homotopy classes of

their multiplier algebras Duke Math J  MR m Zbl

Mathematics Department Wayne State University Detroit MI

claudemathwayneedu httpwwwmathwayneeduclaude

This pap er is available via httpnyjmalbanyedujhtml