You are an expert at summarizing long articles. Proceed to summarize the following text: 
in applications of duality theory for crossed products one often has to know how common constructions such as induction and restriction of representations behave under duality . thus , after piecemeal results by several authors , echterhoff was led to prove that , for systems involving abelian groups , induction and restriction are dual to one another @xcite . he later used these results to great effect in his analysis of crossed products with continuous trace @xcite .    to state echterhoff s theorem precisely , we fix an action @xmath10 of a locally compact abelian group @xmath1 on a @xmath2-algebra @xmath3 , and a closed subgroup @xmath4 of @xmath1 . green s imprimitivity theorem gives an imprimitivity bimodule @xmath11 implementing a morita equivalence between @xmath12 and the imprimitivity algebra @xmath13 ; the latter algebra is naturally isomorphic to the crossed product @xmath14 by the dual action @xmath15 of the subgroup @xmath16 of @xmath17 . let @xmath18 denote the map on equivalence classes of representations given by induction of representations in the sense of green : if @xmath19 is a representation of @xmath12 on @xmath20 , then @xmath21 is by definition the natural left action on ( a completion of ) @xmath22 . similarly , let @xmath23 denote the map given by restricting a covariant representation @xmath24 of @xmath25 to the representation @xmath26 of @xmath27 . then echterhoff s theorem says that we have commutative diagrams @xmath28 and @xmath29 in which the horizontal arrows on the bottom are the bijections induced by the imprimitivity bimodule @xmath11 . as it stands , this theorem only makes sense for abelian groups , because otherwise there is no dual action . however , as echterhoff himself observed , this is an obvious case of a theorem about abelian groups which should extend to an arbitrary locally compact group @xmath1 and a closed normal subgroup @xmath4 , replacing dual actions by dual coactions . ( another such theorem is ( * ? ? ? * theorem  2.4 ) , which was extended to non - abelian groups in ( * ? ? ? * theorem  4.4 ) . ) this project has been carried through by echterhoff and the first and third authors @xcite , relating induction and restriction for @xmath30 to , respectively , the restriction and induction processes of mansfield @xcite   for crossed products by coactions .    here we study the dual situation in which we start with a coaction @xmath0 of a locally compact group @xmath1 on @xmath3 . our main theorems give ( in special cases ) commutative diagrams @xmath31 and @xmath32 similar to   and . this time , the horizontal arrows are the bijections given by mansfield s morita equivalence of @xmath33 with @xmath34 . these theorems are potentially very useful , since they reduce questions about mansfield induction and restriction to questions about the much more familiar and manageable theory of green @xcite . indeed , in the special case @xmath35 , diagrams  and   reduce to results used by gootman and lazar @xcite  to study induced ideals and the primitive ideal space of crossed products by coactions . we begin with a preliminary section in which we discuss our conventions about hilbert modules , basic facts about induction of representations , and a version of green s imprimitivity theorem for reduced crossed products . this is necessary because we want to avoid unnecessary assumptions of amenability ; for the same reason , we shall use the version of mansfield induction given in @xcite . we stress , however , that for amenable groups , the only part of @xcite  needed here is the extension of mansfield s theorem to full coactions and twisted systems . a special case of this extension is in @xcite , and the general case is hinted at in @xcite . our first main results in section  [ res - ind - sec ]   concern the `` @xmath36-@xmath37 '' diagram . the result is more general than that stated above in three respects . first , we consider two normal subgroups @xmath38 . second , we consider also the twisted crossed products of @xcite , which introduces another normal subgroup @xmath39 containing @xmath40 ; our strategy , both here and in section  [ ind - res - sec ] , is to prove the untwisted version , and then check that the constructions respect the twists . third , we prove that the diagram commutes in a very strong sense : all the maps in the diagram are implemented by hilbert modules , so their compositions are given by tensor products of modules , and we prove that the tensor products corresponding to the two alternative routes are naturally isomorphic . this not only gives a stronger theorem than the corresponding result of @xcite , but also a distinctly shorter proof . thus we can claim that this is another example of a theorem about abelian groups which becomes much cleaner when seen from the point of view of coactions . ( well , we would , would nt we . )    in section  [ ind - res - sec ]  we turn to the `` @xmath37-@xmath36 '' diagram . as in section  [ res - ind - sec ] , our main results are much more general than stated above . the overall pattern of this section is like that of section  [ res - ind - sec ] , but this time passing to twisted crossed products is harder . in echterhoff s paper , the second diagram amounts to the induction in stages of green @xcite . mansfield did not prove such a theorem for his induction process , so we are forced to do it here ( corollary  [ mansfield - stages - thm ] ) .    in section  [ me - infl - stab - sec ] we give some applications of our theorems . we prove that restriction and induction are compatible with morita equivalence in general , and in particular with the stabilization trick of @xcite , which allows us to replace twisted crossed products by ordinary ones . ( this result for abelian groups was a fundamental tool in @xcite . ) along the way we deduce , from the corresponding properties of @xmath37 and @xmath36 for crossed products by actions , that both @xmath36 and @xmath37 are respected by morita equivalence of coactions . we feel that this is a good illustration of how our results might be used to deduce information about coactions from known properties of actions . finally , in the last section , we use our main theorems to study the maps @xmath36 , @xmath37 , @xmath41 and @xmath42 on ideals of crossed products by coactions , obtaining generalizations to the case of nonamenable @xmath1 of many of the results in @xcite . here we see the power not only of @xmath36-@xmath37 ( and @xmath42-@xmath41 ) duality , but also that of the hilbert module techniques , since our maps on spaces of representations automatically give rise , via the rieffel correspondence , to maps on spaces of ideals . this research was carried out while the second author was visiting the university of newcastle in 1994 and 1995 , and while the first author was visiting arizona state university in 1995 . the various visitors are grateful to their respective hosts for their hospitality . [ prelim - sec ]    throughout , @xmath1 will be a locally compact group with modular function @xmath43 . we use left haar measure . the group @xmath2-algebra of @xmath1 is denoted @xmath44 ; a subscript @xmath45 , as in @xmath46 or @xmath47 , always indicates a reduced object . nondegenerate homomorphisms of @xmath2-algebras extend to homomorphisms of their multiplier algebras , and this will be done implicitly . we use the conventions of @xcite , @xcite , ( * ? ? ? * section  7 ) , and @xcite , although the latter uses maximal tensor products . our coactions use minimal tensor products , are injective , and are full , i.e. , use @xmath44 . let @xmath48 be a coaction . we let @xmath49 denote the crossed product , @xmath50 the dual action of @xmath1 on @xmath8 , and @xmath51 the normalization of @xmath0 . if @xmath4 is a closed normal subgroup of @xmath1 , we let @xmath52 denote the restricted coaction of @xmath53 on @xmath3 and @xmath7 the restricted crossed product . moreover we let @xmath54 denote the restriction to @xmath55 of a nondegenerate homomorphism @xmath56 of @xmath57 .    in @xcite the first two authors generalized mansfield s imprimitivity machine @xcite . when @xmath48 is a nondegenerate coaction and @xmath4 is a closed normal subgroup of @xmath1 , there are dense @xmath58-subalgebras @xmath59 and @xmath60 of @xmath8 and @xmath61 , respectively , such that @xmath59 is a ( full ) pre - hilbert @xmath60-module under right multiplication and inner product @xmath62 the integral converging strictly in @xmath63 . let @xmath64 denote the completion of the pre - hilbert module @xmath59 . then @xmath64 has a left @xmath9-module action determined by left multiplication on @xmath59 and @xmath65 when @xmath4 is amenable and @xmath0 is a reduced coaction , mansfield ( * ? ? ? * theorem  27 )  proves that @xmath66 is morita equivalent to @xmath7 . for nonamenable subgroups and full coactions , the corresponding result is :    ( * ? ? ? * corollary  3.4)[manimp ] let @xmath48 be a nondegenerate coaction and @xmath4 a closed normal subgroup of @xmath1 such that @xmath67 is faithful . then @xmath64 is an @xmath9  @xmath7 imprimitivity bimodule .    in view of the above theorem , if @xmath0 is a nondegenerate coaction and @xmath4 is a closed normal subgroup of @xmath1 , we say _ mansfield imprimitivity works _ for @xmath4 and @xmath0 whenever @xmath68 is faithful ( * ? ? ? * definition  3.5 ) . when mansfield imprimitivity works we let @xmath69 denote the extension to @xmath64 of the inner product @xmath70 on @xmath59 . mansfield s computations show that the left inner product @xmath71 for @xmath72 can be identified with the element @xmath73 of @xmath74 . when @xmath0 is nondegenerate , mansfield imprimitivity works if either @xmath4 is amenable or @xmath0 is normal ( * ? ? ? * lemma  3.2 ) . if mansfield imprimitivity works for @xmath4 and @xmath0 , then it also works for any closed subgroup contained in @xmath4 ( * ? ? ? * theorem  5.2 )  and any coaction morita equivalent to @xmath0 ( * ? ? ? * theorem  5.3 ) . if @xmath39 is a closed normal subgroup of @xmath1 containing @xmath4 and @xmath75 is a coaction , then mansfield imprimitivity works for @xmath4 and @xmath76 if and only if it works for @xmath4 and the inflated coaction @xmath77 ( * ? ? ? * theorem  5.4 ) . when we say @xmath78 is a twisted coaction , we mean @xmath39 is a closed normal subgroup of @xmath1 and @xmath79 is a twist for @xmath0 over @xmath80 @xcite . we let @xmath81 denote the twisting ideal of @xmath8 , @xmath82 the twisted crossed product , and @xmath83 the dual action of @xmath39 on @xmath84 . if further @xmath4 is a closed normal subgroup of @xmath1 contained in @xmath39 , then @xmath85 is also a twist for the restricted coaction @xmath86 over the quotient @xmath87 . let @xmath88 denote the twisting ideal of @xmath89 . then there is a restricted twisted crossed product @xmath90 the following result generalizes ( * ? ? ? * theorem  4.1 ) :    ( * ? ? ? * theorem  4.4)[mprimp ] let @xmath78 be a nondegenerate twisted coaction and @xmath4 a closed normal subgroup of @xmath1 contained in @xmath39 such that mansfield imprimitivity works for @xmath4 and @xmath0 . then the quotient @xmath91 is an @xmath92  @xmath93 imprimitivity bimodule . if @xmath78 is a nondegenerate twisted coaction , then mansfield imprimitivity works for @xmath39 and @xmath0 if and only if @xmath0 is normal ( * ? ? ? * lemma  3.6 ) . if @xmath78 is a nondegenerate normal twisted coaction and @xmath4 is a closed normal subgroup of @xmath1 contained in @xmath39 , then mansfield imprimitivity works for @xmath4 and @xmath0 since @xmath0 is normal , and also for @xmath4 and the morita equivalent stabilized coaction @xmath94 ( * ? ? ? * theorem  3.1 ) , ( * ? ? ? * theorem  5.5 ) . everything in this paper revolves around rieffel s induction process , so we should make our conventions explicit . for more detailed treatments of this material we refer the reader to @xcite , @xcite , @xcite . all our hilbert modules will be full , i.e. , the closed span of the inner product generates the @xmath2-algebra . if @xmath95 is a right hilbert @xmath96-module and @xmath3 acts nondegenerately on @xmath95 by adjointable @xmath96-module maps ( so there is a homomorphism @xmath97 such that @xmath98 ) , we say @xmath95 is a right - hilbert @xmath3  @xmath96 bimodule . ( this terminology first appears in @xcite . ) if @xmath95 is also a left hilbert @xmath3-module such that @xmath99 for @xmath100 , then of course @xmath95 is an @xmath3 @xmath96 imprimitivity bimodule . we denote the reverse bimodule by @xmath101 , with elements @xmath102 .    when @xmath95 is a right - hilbert @xmath3 @xmath96 bimodule , rieffel induction gives a functor @xmath103 and we leave out parts of the notation if confusion seems unlikely . actually , @xmath104 can be factored as @xmath105 followed by `` restriction '' from @xmath106 to @xmath107 . since we will need it a lot , we abstract this latter bit : if @xmath108 is a nondegenerate homomorphism , composition with @xmath19 gives a `` restriction '' map @xmath109 we view this as a rieffel induction process : @xmath96 becomes a right - hilbert @xmath3 @xmath96 bimodule via @xmath110 for @xmath111 , @xmath112 .    now suppose @xmath108 is a nondegenerate homomorphism and @xmath95 is a right - hilbert @xmath96  @xmath113 then @xmath95 becomes a right - hilbert @xmath3 @xmath113 bimodule via @xmath114 on the other hand , we can regard @xmath96 as a right - hilbert @xmath3 @xmath96 bimodule , and the map @xmath115 induces an isomorphism @xmath116 of right - hilbert @xmath3  @xmath113 bimodules . in particular , we get a commutative diagram @xmath117{{\operatorname{rep}}a } \end{diagram } \ ] ] where we use prescripts and postscripts to indicate the coefficient algebras when necessary . in general , we will omit parts of the notation , so that when we say @xmath118{{\operatorname{rep}}a } \end{diagram } \ ] ] is a commutative diagram , we mean @xmath95 is a right - hilbert @xmath3  @xmath96 bimodule , and similarly for @xmath119 and @xmath120 , and the equation @xmath121 holds in the strong sense that @xmath122 as right - hilbert @xmath3  @xmath113 bimodules .    recall that rieffel induction gives rise to maps between ideals , so if @xmath123 is an ideal of @xmath96 and if @xmath19 is any nondegenerate representation of @xmath96 with kernel @xmath123 , then @xmath124 is the kernel of @xmath125 . when we have a commutative diagram of hilbert modules as in , we of course get @xmath126 as maps from ideals of @xmath113 to ideals of @xmath3 .    we will often want to pass from a commutative diagram of hilbert modules to quotients . there is a subtle point that needs checking :    let @xmath95 and @xmath119 be right - hilbert @xmath3  @xmath96 and right - hilbert @xmath96 @xmath113 bimodules , respectively , and let @xmath39 be an ideal of @xmath113 . further , let @xmath127 and @xmath128 be the corresponding induced ideals of @xmath96 and @xmath3 , respectively , and let @xmath129 and @xmath130 be the quotient maps . then @xmath131 induces an isomorphism between the right - hilbert @xmath132 @xmath133 bimodules @xmath134 and @xmath135 . straightforward ; a slightly different version was given in ( * ? ? ? * lemma 1.10 ) .    [ quotient ] suppose the diagram @xmath136{{\operatorname{rep}}a } \end{diagram}\ ] ] commutes in the usual strong sense that @xmath137 . further suppose that @xmath39 is an ideal of @xmath113 , and set @xmath127 and @xmath128 . then the diagram @xmath138{{\operatorname{rep}}a / i } \end{diagram}\ ] ] also commutes in the usual strong sense .    since @xmath137 , certainly @xmath139 so the above lemma immediately gives the corollary . similarly , commutative diagrams with any number of vertices pass to quotients .      we will need an induction process for reduced crossed products by actions . we could deduce it from green s version by applying @xcite , but we give a direct argument since we need the explicit imprimitivity bimodule . let @xmath140 be an action and @xmath40 a closed subgroup of @xmath1 . recall that green s inducing process starts with the right - pre - hilbert @xmath141  @xmath142 bimodule @xmath141 , where the operations for @xmath143 , @xmath144 are given by @xmath145 the particular arrangement of modular functions comes from @xcite ; green uses unorthodox conventions . let @xmath146 be the completion of the pre - hilbert module @xmath141 , so @xmath146 is a right - hilbert @xmath147  @xmath148 bimodule . we need to know that the kernels of the regular representations match up , so that @xmath146 passes to a right - hilbert @xmath149 @xmath150 bimodule . [ reduced ] let @xmath140 be an action , @xmath40 a closed subgroup of @xmath1 , @xmath146 green s right - hilbert @xmath147 @xmath148 bimodule , and @xmath123 the kernel of the regular representation of @xmath148 . then the induced ideal @xmath151 is the kernel of the regular representation of @xmath147 . consequently , the quotient @xmath152 is a right - hilbert @xmath149  @xmath150 bimodule . this follows from induction in stages : the proof of ( * ? ? ? * proposition  8) shows that the diagram @xmath153{{\operatorname{rep}}b\times g } \end{diagram } \ ] ] commutes in the usual strong sense . if @xmath19 is any faithful representation of @xmath96 , then @xmath154 and @xmath155 are the regular representations of @xmath148 and @xmath147 , respectively . so the ideals @xmath156 and @xmath157 are the kernels of the regular representations . by commutativity of diagram , @xmath158 so indeed @xmath146 induces the kernel @xmath123 of @xmath159 to the kernel of @xmath160 . [ reduced - stages ] let @xmath140 be an action , @xmath40 a closed subgroup of @xmath1 , and @xmath161 the right - hilbert @xmath149  @xmath150 bimodule obtained in lemma  [ reduced ] . then the diagram @xmath162{{\operatorname{rep}}b\times_r g } \end{diagram } \ ] ] commutes in the usual strong sense . this follows from the above lemma and corollary  [ quotient ] . we emphasize that @xmath163 may be viewed as the completion of @xmath141 with respect to the norm induced by the @xmath150-valued pre - inner product . in particular , the actions of @xmath150 and @xmath149 on @xmath163 are determined by the covariant representations of @xmath164 and @xmath140 on @xmath141 . suppose we have a twisted coaction @xmath165 , and closed normal subgroups @xmath166 of @xmath1 contained in @xmath39 . in this section , we show that when mansfield imprimitivity works , the following diagram commutes in the usual strong sense : @xmath167 we will do this in two steps , first showing that the analogous untwisted diagram commutes , and then showing that the twisting ideals in the various crossed products match up properly , so that commutativity is preserved on taking quotients by these ideals .    for @xmath168 , a weak form of the following theorem was proven in ( * ? ? ? * theorem  4.1 ) . [ plain - res - ind - thm ] let @xmath48 be a nondegenerate coaction , and let @xmath166 be closed normal subgroups of @xmath1 such that mansfield imprimitivity works for @xmath40 ( which is automatic if @xmath40 is amenable ) . then the diagram @xmath169 commutes in the usual strong sense .    first note that by ( * ? ? ? * theorem  5.2 ) , mansfield imprimitivity also works for @xmath4 and @xmath0 , so @xmath64 is indeed an @xmath9  @xmath7 imprimitivity bimodule , and the above diagram makes sense . we shall actually prove that the hilbert module tensor product @xmath170 of mansfield bimodules is isomorphic to the reduction @xmath171 of green s bimodule , as a right - hilbert @xmath172 @xmath9 bimodule . this suffices , because then @xmath173 as a right - hilbert @xmath172 @xmath7 bimodule , and this is exactly what it means for the above diagram to commute in the strong sense . both bimodules in the tensor product   are completions of mansfield s dense subalgebra @xmath59 of @xmath8 for the appropriate inner products . our isomorphism will be the extension to @xmath174 of the map @xmath175 defined by @xmath176 note that , up to a modular function , @xmath177 is just mansfield s left @xmath178-valued inner product @xmath179 , and hence does indeed give an element of @xmath178 . in fact , if we define @xmath180 , then the map @xmath181 is a homeomorphism of @xmath178 ( with the inductive limit topology ) onto itself , which takes @xmath182 to @xmath183 . this latter set is dense in @xmath184 for the inductive limit topology ( ( * ? ? ? * lemma  25 ) ) ; it follows that the range of @xmath185 is also inductive limit dense in @xmath184 , and therefore in @xmath171 . it only remains to show that @xmath185 preserves the hilbert module structure . for the left action of @xmath172 , fix @xmath186 and @xmath187 . then : @xmath188 and @xmath189 for the right action of @xmath9 , fix @xmath186 , @xmath190 and @xmath191 ; then one similarly verifies that @xmath192 and @xmath193 for the right @xmath9-valued inner product , fix @xmath191 and compute : @xmath194    it now follows that @xmath185 is a right - hilbert @xmath172  @xmath9 bimodule isomorphism of @xmath195 onto @xmath171 . [ twisted - res - ind - cor ] let @xmath78 be a nondegenerate twisted coaction , and let @xmath166 be closed normal subgroups of @xmath1 contained in @xmath39 such that mansfield imprimitivity works for @xmath40 and @xmath0 ( which is automatic if @xmath40 is amenable ) . then the diagram @xmath196 commutes in the usual strong sense . we show the appropriate ideals in diagram match up and appeal to corollary  [ quotient ] ; because the diagram commutes , and the top and bottom maps are morita equivalences , we need only match up the ideals along three sides . let @xmath81 , @xmath197 , and @xmath198 be the twisting ideals of @xmath8 , @xmath7 , and @xmath199 , respectively . the reduced crossed product @xmath92 is by definition @xmath200 . if @xmath19 is a representation of @xmath8 with kernel @xmath81 ( for example , if @xmath201 ) , then @xmath19 induces a faithful representation @xmath202 of @xmath203 ; thus @xmath204 it is part of the content of ( * ? ? ? * theorem  4.4 )  that @xmath205 ( see ( * ? ? ? * equation  4.2 ) ) . hence the appropriate ideals match up along the top of diagram , and similarly along the bottom . since @xmath206 by corollary  [ reduced - stages ] , the ideals also match up along the right side of diagram . in this section we prove analogs of theorem  [ plain - res - ind - thm ]  and corollary  [ twisted - res - ind - cor ] , where now we use mansfield induction on the left sides of the diagrams and restriction on the right . as in the previous section , we first prove an untwisted version , and then show that the twisting ideals match up properly . [ plain - ind - res - thm ] let @xmath48 be a nondegenerate coaction , and let @xmath166 be closed normal subgroups of @xmath1 such that mansfield imprimitivity works for @xmath40 ( which is automatic if @xmath40 is amenable ) . then the diagram @xmath207 commutes in the usual strong sense . let us denote the hilbert module for mansfield induction from @xmath199 to @xmath7 by @xmath208 ; here we are identifying @xmath209 with @xmath210 , so @xmath208 is a completion of mansfield s dense subalgebra @xmath60 of @xmath7 . we shall prove that the hilbert module tensor product @xmath211 is isomorphic to @xmath212 as a right - hilbert @xmath9  @xmath199 bimodule . this suffices , because then @xmath213 as a right - hilbert @xmath9 @xmath199 bimodule , and this is exactly what it means for the above diagram to commute in the strong sense .    our map will be the extension to @xmath214 of the map @xmath215 defined by @xmath216 here the product @xmath217 makes sense in @xmath63 because @xmath218 belongs to @xmath219 . in other words , this product is given by the right action of @xmath60 on mansfield s bimodule @xmath59 , and hence @xmath220 does indeed belong to @xmath59 . we now show that @xmath221 preserves the right - hilbert bimodule structure . since @xmath8 on the left and @xmath199 on the right act by multiplication in @xmath63 , it is immediate that @xmath221 preserves these actions . to see that @xmath221 preserves the left @xmath4-action is a straightforward calculation , using the fact that each @xmath222 is fixed by @xmath223 for @xmath191 : @xmath224 to see that @xmath221 preserves the right @xmath199-valued inner products , note that @xmath225 on @xmath89 , and compute : @xmath226    it only remains to show that the range of @xmath221 is dense in @xmath212 . for this , note that @xmath227 , since if @xmath228 is @xmath229 , we may choose @xmath230 such that @xmath231 is identically @xmath232 on @xmath233 , and then @xmath234 . hence , @xmath235 is dense in @xmath212 . since @xmath212 is a nondegenerate right @xmath3-module , and @xmath236 is dense in @xmath3 , we therefore have @xmath237 it now follows that @xmath221 is a right - hilbert @xmath9 @xmath199 bimodule isomorphism of @xmath214 onto @xmath212 . [ mansfield - stages - thm ] ( mansfield induction in stages . ) let @xmath48 be a nondegenerate coaction , and let @xmath166 be closed normal subgroups of @xmath1 such that mansfield imprimitivity works for @xmath40 ( which is automatic if @xmath40 is amenable ) . then the diagram @xmath238 commutes in the usual strong sense . the corollary requires that the hilbert module tensor product @xmath211 be isomorphic to @xmath212 as right - hilbert @xmath8 @xmath199 bimodules ; we showed slightly more than this in the proof of theorem  [ plain - ind - res - thm ]  by including the left @xmath4-action . [ twisted - ind - res - cor ] let @xmath78 be a nondegenerate twisted coaction , and let @xmath166 be closed normal subgroups of @xmath1 contained in @xmath39 such that mansfield imprimitivity works for @xmath40 and @xmath0 ( which is automatic if @xmath40 is amenable ) . then the diagram @xmath239 commutes in the usual strong sense .    as with corollary  [ twisted - res - ind - cor ] , we need only show the appropriate ideals in the diagram match up , and appeal to corollary  [ quotient ] . as before , let @xmath81 , @xmath197 , and @xmath198 be the twisting ideals of @xmath8 , @xmath7 , and @xmath199 , respectively . the reduced crossed product @xmath92 is by definition @xmath200 . if @xmath19 is a representation of @xmath8 with kernel @xmath81 ( for example , if @xmath201 ) , then @xmath19 induces a faithful representation @xmath202 of @xmath203 ; thus @xmath240 it is part of the content of ( * ? ? ? * theorem  4.4 )  that @xmath205 ( see ( * ? ? ? * equation  4.2 ) ) . hence the appropriate ideals match up along the top of diagram , and similarly along the bottom . it only remains to see that the ideals match up along the right side of . we need @xmath241 now , our reduced version corollary  [ reduced - stages ]   of induction in stages gives @xmath242 so it will suffice to show two things :    1 . the ideal @xmath243 of @xmath244 is @xmath40-invariant ; 2 . if @xmath245 is an @xmath40-invariant ideal of @xmath9 , then @xmath246    these are shown abstractly in the next two lemmas , which although at most partly new , may have independent interest . let @xmath164 be an action , @xmath4 a closed normal subgroup of @xmath40 , and @xmath123 an @xmath40-invariant ideal of @xmath96 . then the ideal @xmath247 of @xmath248 is @xmath40-invariant for the decomposition action . the decomposition action of @xmath40 on @xmath249 leaves the kernel of the regular representation invariant , hence it indeed induces an action @xmath250 of @xmath40 on @xmath248 . explicitly , for @xmath191 , @xmath251 , and @xmath252 , @xmath253 where @xmath254 is the modular function of conjugation of @xmath40 on @xmath4 : @xmath255    let @xmath256 be the usual right - hilbert @xmath248 @xmath96 bimodule , so @xmath95 is a completion of @xmath257 . we will show @xmath95 is @xmath40-equivariant , which will imply that @xmath40-invariant ideals of @xmath96 induce to @xmath40-invariant ideals of @xmath248 . more precisely , we will construct a strongly continuous banach representation @xmath258 of @xmath40 on @xmath95 such that for @xmath251 , @xmath259 , @xmath260 , and @xmath261 we have @xmath262 a straightforward calculation then shows that if @xmath123 is @xmath40-invariant , so is @xmath263 . for @xmath251 , @xmath264 define @xmath265 by @xmath266 to show , take @xmath267 and compute : @xmath268 it is now clear that @xmath258 is a homomorphism of @xmath40 into the isometric automorphisms of the normed space @xmath257 , hence determines by continuity a homomorphism of @xmath40 into the isometric automorphisms of the banach space @xmath95 , and follows , again by continuity .    for , it suffices to take @xmath264 : for @xmath269 we have @xmath270    we next show @xmath258 is strongly continuous . by @xcite , it suffices to show that @xmath258 is strongly continuous for the inductive limit topology on @xmath257 . fix @xmath271 with compact support @xmath233 , and suppose @xmath272 in @xmath1 . since @xmath273 uniformly ( by a standard compactness argument ) , we need only find a compact set @xmath274 in @xmath4 and @xmath275 such that the support of @xmath276 is in @xmath274 for @xmath277 . to do this , choose a neighborhood @xmath278 of @xmath279 in @xmath1 with compact closure , and let @xmath275 be such that @xmath280 for @xmath277 . then the compact set @xmath281 will do .    finally , for , it suffices to take @xmath282 : for @xmath191 we have @xmath283    [ res - ind - invt - ideal - lem ] if @xmath164 is an action , @xmath4 is a closed normal subgroup of @xmath40 , and @xmath245 is an @xmath40-invariant ideal of @xmath248 , then @xmath284    we first show the analogous equality @xmath285 for full crossed products , assuming @xmath39 is an @xmath40-invariant ideal of @xmath249 . by ( * proposition 1 ) , the decomposition action of @xmath40 on @xmath249 is twisted over @xmath4 , and @xmath286 so follows from ( * ? ? ? * proposition 11 ) . now we have an @xmath40-equivariant commutative diagram @xmath287 where @xmath288 denotes the regular representation of @xmath249 . this gives an @xmath40-equivariant commutative diagram @xmath289 also , we have a diagram @xmath290 which commutes by our construction of the bottom arrow in lemma  [ reduced ] . since @xmath245 is an @xmath40-invariant ideal of @xmath248 , @xmath291 is an @xmath40-invariant ideal of @xmath249 , and we have @xmath292 by . since @xmath293 is surjective , @xmath294 is injective on ideals , and the lemma follows . in this section we show that our res - ind duality is compatible with certain standard constructions .    before discussing morita equivalence of coactions , we recall the concept of multiplier bimodules introduced in @xcite . a _ multiplier _ @xmath295 of @xmath296 consists of an @xmath3-linear map @xmath297 and a @xmath96-linear map @xmath298 such that @xmath299 for @xmath111 , @xmath260 . the _ multiplier bimodule _ @xmath300 consists of all multipliers of @xmath95 . an _ imprimitivity bimodule homomorphism _ @xmath301 consists of homomorphisms @xmath302 and @xmath303 and a bimodule map @xmath304 preserving the inner products : @xmath305 @xmath306 is called _ nondegenerate _ if @xmath307 and @xmath308 are nondegenerate . significantly , this implies an ostensibly stronger form of nondegeneracy :    [ iains - ib - hom - lem ] if @xmath309 is a nondegenerate imprimitivity bimodule homomorphism , then @xmath310    by symmetry , it suffices to show the second equality . since @xmath311 , and @xmath307 is nondegenerate , we have @xmath312 . now , since the range of the inner product @xmath313 spans @xmath3 , we have @xmath314 because we can factor @xmath311 , and @xmath315 , the pairing @xmath316 takes @xmath317 to @xmath318 , and we can deduce from our calculation that @xmath319 . the other inclusion is trivial , so this establishes the lemma . [ lem : nondegenerate ] let @xmath296 and @xmath320 be imprimitivity bimodules , and let @xmath321 be a nondegenerate homomorphism such that @xmath322 where @xmath323 is the linking algebra of @xmath95 , and similarly for @xmath119 . then @xmath306 restricts on the corners to give a nondegenerate imprimitivity bimodule homomorphism @xmath324 . since @xmath306 is nondegenerate , we have @xmath325 . since @xmath326 we deduce that there are unique linear maps @xmath327 such that @xmath328 the algebraic properties of @xmath306 show that @xmath329 is an imprimitivity bimodule homomorphism of @xmath296 to @xmath330 , and nondegeneracy of @xmath331 and @xmath332 is inherited from nondegeneracy of @xmath306 . we will need the following elementary fact about imprimitivity bimodule homomorphisms . [ surjection ] if @xmath333 is a nondegenerate imprimitivity bimodule homomorphism , then the diagram @xmath334 commutes in the usual strong sense . we must show @xmath335 as a right - hilbert @xmath3  @xmath318 bimodule . of course , @xmath336 is isomorphic to @xmath337 . define @xmath338 by @xmath339 the required properties of @xmath221 follow from the nondegeneracy of @xmath306 , and straightforward calculations showing that @xmath340 and @xmath341 for @xmath111 , @xmath259 , and @xmath342 .    following ( * ? ? ? * definition  3.3 ) ( see also @xcite , @xcite , @xcite ) , a _ coaction _ @xmath0 of @xmath1 on an imprimitivity bimodule @xmath296 is an imprimitivity bimodule homomorphism @xmath343 such that @xmath344 and @xmath345 are @xmath2-coactions , and satisfying @xmath346 as a consequence of the definition , we automatically have @xmath347 and @xmath348 for @xmath349 , @xmath350 . also , since by assumption @xmath351 and @xmath352 are nondegenerate @xmath2-homomorphisms , @xmath0 is automatically nondegenerate as an imprimitivity bimodule homomorphism . hence , by lemma  [ iains - ib - hom - lem ] , we have @xmath353 when such a @xmath0 exists we say @xmath344 and @xmath345 are _ morita equivalent _ , and we call @xmath354 a _ morita equivalence _ of @xmath351 and @xmath352 .    if @xmath4 is a closed normal subgroup of @xmath1 , then @xmath355 is a coaction of @xmath53 on @xmath95 , where @xmath356 is the canonical quotient map . a _ morita equivalence _ between twisted coactions @xmath357 and @xmath358 is an @xmath344 @xmath345 morita equivalence @xmath354 such that @xmath359 in this case , for any closed normal subgroup @xmath4 of @xmath1 contained in @xmath39 , there are @xmath84  @xmath360 and @xmath93  @xmath361 imprimitivity bimodules @xmath362 and @xmath363 , respectively . moreover , there is an action of @xmath39 on @xmath362 inducing a morita equivalence between the dual actions @xmath364 and @xmath365 , so there is an @xmath366  @xmath367 imprimitivity bimodule @xmath368 . the next result shows corollary  [ twisted - res - ind - cor ]  is compatible with morita equivalence :    [ me - res - ind - cube - thm ] if @xmath354 is a morita equivalence between nondegenerate twisted coactions @xmath357 and @xmath358 , and @xmath166 are closed normal subgroups of @xmath1 contained in @xmath39 such that mansfield imprimitivity works for @xmath40 and one of the coactions , then the cube @xmath369{{{\operatorname{rep}}}b\times_{g / k}g / n }          \arrow[2]{e }          \arrow{s,- }      \arrow{sw , t}{x\times_{g / k}g / n } \node[2]{{{\operatorname{rep}}}b\times_{g / k}g\times_r n }          \arrow[2]{s , r}{{{\operatorname{ind } } } }          \arrow{sw , t}{x\times_{g / k}g\times_r n}\\ \node{{{\operatorname{rep}}}a\times_{g / k}g / n }          \arrow[2]{e }          \arrow[2]{s , l}{{{\operatorname{res } } } } \node { }          \arrow{s , l}{{{\operatorname{res } } } } \node{{{\operatorname{rep}}}a\times_{g / k}g\times_r n }          \arrow[2]{s , r,3}{{{\operatorname{ind}}}}\\ \node[2]{{{\operatorname{rep}}}b\times_{g / k}g / h }          \arrow{e,- }          \arrow{sw , b}{x\times_{g / k}g / h } \node { }          \arrow{e } \node{{{\operatorname{rep}}}b\times_{g / k}g\times_r h }          \arrow{sw , b}{x\times_{g / k}g\times_r h}\\ \node{{{\operatorname{rep}}}a\times_{g / k}g / h }          \arrow[2]{e } \node[2]{{{\operatorname{rep}}}a\times_{g / k}g\times_r h } \end{diagram}\ ] ] commutes in the usual strong sense .    first note that by ( * ? ? ? * theorem  5.3 ) , if mansfield imprimitivity works for @xmath40 and one of the coactions , it works for the other , so the above cube makes sense .    because all the horizontal arrows are bijections , we need only show commutativity of three of the vertical faces , as well as the top and bottom . the front and back commute by corollary  [ twisted - res - ind - cor ] . the bottom face is just the top face with @xmath4 replaced by @xmath40 ; we show the top commutes . the untwisted version ( i.e. , with @xmath370 ) @xmath371 is ( * ? ? ? * proposition 4.5 ) . even though they use reduced coactions and require @xmath4 to be amenable , their arguments carry over to our setting since we assume that mansfield imprimitivity works , as pointed out in @xcite . so , it only remains to show the ideals in diagram  match up and appeal to corollary  [ quotient ] . the ideals match up along the top and bottom by ( * ? ? ? * theorem  4.4 ) , and along the left side by ( * ? ? ? * corollary 3.3 ) , and this is enough , since the top and bottom are morita equivalences .    for the left face , we use the following general lemma :    [ co - res - res - lem ] let @xmath354 be a morita equivalence between coactions @xmath344 and @xmath345 , and let @xmath4 be a closed normal subgroup of @xmath1 . then the diagram @xmath372 commutes . we aim to apply lemma  [ surjection ] , so we need a nondegenerate imprimitivity bimodule homomorphism @xmath373 with @xmath374 let @xmath375 be the linking algebra for @xmath95 ; then there is a coaction @xmath376 of @xmath1 on @xmath375 . by [ er , appendix ] , we have @xmath377 and @xmath378 by lemma  [ lem : nondegenerate ] , the nondegenerate homomorphism @xmath379 restricts on the corners to a nondegenerate imprimitivity bimodule homomorphism @xmath380 since the restrictions of @xmath381 to the diagonal corners @xmath7 and @xmath382 agree with @xmath383 and @xmath384 , respectively , we are done .    returning to the proof of theorem  [ me - res - ind - cube - thm ] , the untwisted version of the left - hand face of diagram  follows from lemma  [ co - res - res - lem ] . we need only show that the twisting ideals in each crossed product in diagram  match up , and apply corollary  [ quotient ] . that the ideals match up across the horizontal arrows is shown in ( * ? ? ? * corollary  3.3 ) . that the ideals match up along the vertical arrows follows from the proof of corollary  [ twisted - res - ind - cor ] . this completes the proof of theorem  [ me - res - ind - cube - thm ]    in the proof of theorem  [ me - res - ind - cube - thm ] , we were able to deduce commutativity of the right face of from the other five faces . this is a special case of a general `` @xmath37-@xmath37 '' diagram for actions which is related to ( * ? ? ? * theorem  3 )  and ( * ? ? ? * proposition  4.1.2 ) . we now show that corollary  [ twisted - ind - res - cor ]  is compatible with morita equivalence :    [ me - ind - res - cube - thm ] if @xmath354 is a morita equivalence between nondegenerate twisted coactions @xmath357 and @xmath358 , and @xmath166 are closed normal subgroups of @xmath1 contained in @xmath39 such that mansfield imprimitivity works for @xmath40 and one of the coactions , then the cube @xmath385{{{\operatorname{rep}}}b\times_{g / k}g / n }          \arrow[2]{e }          \arrow{sw , t}{x\times_{g / k}g / n } \node[2]{{{\operatorname{rep}}}b\times_{g / k}g\times_r n }          \arrow{sw , t}{x\times_{g / k}g\times_r n}\\ \node{{{\operatorname{rep}}}a\times_{g / k}g / n }          \arrow[2]{e } \node { }          \arrow{n , l}{{{\operatorname{ind } } } } \node{{{\operatorname{rep}}}a\times_{g / k}g\times_r n}\\ \node[2]{{{\operatorname{rep}}}b\times_{g / k}g / h }          \arrow{n,- }          \arrow{e,- }      \arrow{sw , b}{x\times_{g / k}g / h } \node { }          \arrow{e } \node{{{\operatorname{rep}}}b\times_{g / k}g\times_r h }          \arrow[2]{n , r}{{{\operatorname{res } } } }          \arrow{sw , b}{x\times_{g / k}g\times_r h}\\ \node{{{\operatorname{rep}}}a\times_{g / k}g / h }          \arrow[2]{n , l}{{{\operatorname{ind } } } }          \arrow[2]{e } \node[2]{{{\operatorname{rep}}}a\times_{g / k}g\times_r h }          \arrow[2]{n , r,3}{{{\operatorname{res } } } } \end{diagram}\ ] ] commutes in the usual strong sense . the front and back faces commute by corollary  [ twisted - ind - res - cor ] , and the top and bottom faces are the same as in , so it suffices to show commutativity of the right face . for this we use the following lemma :    [ rt - face - res - lem ] let @xmath386 be a morita equivalence between actions @xmath387 and @xmath388 , and let @xmath4 be a closed subgroup of @xmath40 . then the diagram @xmath389 commutes . we aim to apply lemma  [ surjection ] , so we need a nondegenerate imprimitivity bimodule homomorphism @xmath390 with @xmath391 let @xmath392 be the linking algebra for @xmath119 ; then there is an action @xmath393 of @xmath40 on @xmath392 . by [ er , appendix ] , we have @xmath394 and @xmath395 by lemma  [ lem : nondegenerate ] , the nondegenerate homomorphism @xmath396 restricts on the corners to a nondegenerate imprimitivity bimodule homomorphism @xmath397 since the restrictions of @xmath398 to the diagonal corners @xmath399 and @xmath400 agree with @xmath401 and @xmath402 , respectively , we are done .    returning to the proof of the right - hand face of diagram , we need only show that the kernels of the regular representations of each crossed product in diagram  match up , and apply corollary  [ quotient ] . that the kernels match up across the horizontal arrows is shown in @xcite . the kernels match up along the vertical arrows because the regular representation of @xmath4 is quasi - equivalent to the regular representation of @xmath40 restricted to @xmath4 . alternatively , we can argue as follows : we must show that @xmath403 by green induction in stages , @xmath404 since for @xmath4 normal , @xmath405 is @xmath40-invariant , equation  gives us @xmath406 this completes the proof of theorem  [ me - ind - res - cube - thm ] .    by ( * ? ? ? * proposition  2.3 ) , morita equivalence respects nondegeneracy of coactions . hence , in theorems  [ me - res - ind - cube - thm]and [ me - ind - res - cube - thm ] , one coaction is in fact nondegenerate if and only if the other one is . this time , we were able to deduce commutativity of the left face of from the other five faces . written in terms of tensor products , @xmath407 where we have used self - explanatory notation to distinguish the bimodules for @xmath3 and @xmath96 the special case @xmath168 is ( * ? ? ? * theorem 4.4 ) . a direct proof is presumably possible , although probably quite tedious . we turn to inflation of coactions : if @xmath39 is a closed normal subgroup of @xmath1 and @xmath75 is a coaction , composing with the natural embedding of @xmath408 in @xmath409 gives a coaction @xmath410 , called an _ inflated _ coaction . @xmath411 is trivially twisted over @xmath80 by @xmath412 , and ( * ? ? ? * example 2.14 )  gives a natural isomorphism of @xmath413 onto @xmath414 which takes @xmath415 to @xmath416 . the next two theorems show corollaries [ twisted - res - ind - cor ]  and [ twisted - ind - res - cor ]  are compatible with inflation :    [ inf - res - ind - cube - thm ] if @xmath166 are closed normal subgroups of @xmath1 contained in @xmath39 , and @xmath75 is a nondegenerate coaction such that mansfield imprimitivity works for @xmath40 and @xmath76 , then the cube @xmath417{{{\operatorname{rep}}}a\times k / n }          \arrow[2]{e }          \arrow{s,- } \node[2]{{{\operatorname{rep}}}a\times k\times_r n }          \arrow[2]{s , r}{{{\operatorname{ind}}}}\\ \node{{{\operatorname{rep}}}a\times_{g / k}g / n }          \arrow{ne , t}{\cong }          \arrow[2]{e }          \arrow[2]{s , l}{{{\operatorname{res } } } } \node { }          \arrow{s , l}{{{\operatorname{res } } } } \node{{{\operatorname{rep}}}a\times_{g / k}g\times_r n }          \arrow{ne , t}{\cong }          \arrow[2]{s , r,3}{{{\operatorname{ind}}}}\\ \node[2]{{{\operatorname{rep}}}a\times k / h }          \arrow{e,- } \node { }          \arrow{e } \node{{{\operatorname{rep}}}a\times k\times_r h}\\ \node{{{\operatorname{rep}}}a\times_{g / k}g / h }          \arrow{ne , b}{\cong }          \arrow[2]{e } \node[2]{{{\operatorname{rep}}}a\times_{g / k}g\times_r h }          \arrow{ne , b}{\cong } \end{diagram}\ ] ] commutes in the usual strong sense .    first , by ( * ? ? ? * theorem  5.4 ) , mansfield imprimitivity works for @xmath40 and the ( nondegenerate ) inflated coaction @xmath77 , so the above diagram makes sense . the front and back faces commute by theorem  [ plain - res - ind - thm ] . we show the top commutes ; the bottom is essentially the same . * proposition 4.8 and remark 4.9 )  give an imprimitivity bimodule homomorphism @xmath418 hence , lemma  [ surjection ]  gives a commutative diagram @xmath419 to pass from to the top of , we need to show the appropriate ideals @xmath197 in @xmath7 , @xmath420 in @xmath9 , and @xmath421 in both @xmath422 and @xmath423 match up . the ideals match up along the top ( by ( * ? ? ? * theorem  4.4 ) ) , bottom ( trivially ) , and left ( since @xmath424 ) , and this is enough .    finally , for the left face of , the diagram @xmath425 of nondegenerate homomorphisms commutes , since the diagram @xmath426 does . hence , the required diagram @xmath427 of right - hilbert bimodules commutes as well . [ inf - ind - res - cube - thm ] if @xmath166 are closed normal subgroups of @xmath1 contained in @xmath39 , and @xmath75 is a nondegenerate coaction such that mansfield imprimitivity works for @xmath40 and @xmath76 , then the cube @xmath428{{{\operatorname{rep}}}a\times k / n }          \arrow[2]{e } \node[2]{{{\operatorname{rep}}}a\times k\times_r n}\\ \node{{{\operatorname{rep}}}a\times_{g / k}g / n }          \arrow{ne , t}{\cong }          \arrow[2]{e } \node { }          \arrow{n , l}{{{\operatorname{ind } } } } \node{{{\operatorname{rep}}}a\times_{g / k}g\times_r n }          \arrow{ne , t}{\cong}\\ \node[2]{{{\operatorname{rep}}}a\times k / h }          \arrow{n,- }          \arrow{e,- } \node { }          \arrow{e } \node{{{\operatorname{rep}}}a\times k\times_r h }          \arrow[2]{n , r}{{{\operatorname{res}}}}\\ \node{{{\operatorname{rep}}}a\times_{g / k}g / h }          \arrow[2]{n , l}{{{\operatorname{ind } } } }          \arrow{ne , b}{\cong }          \arrow[2]{e } \node[2]{{{\operatorname{rep}}}a\times_{g / k}g\times_r h }          \arrow[2]{n , r,3}{{{\operatorname{res } } } }          \arrow{ne , b}{\cong } \end{diagram}\ ] ] commutes in the usual strong sense . the front and back faces commute by theorem  [ plain - ind - res - thm ] , and the top and bottom faces are the same as in . since the horizontal maps in the right face come from equivariant isomorphisms , the commutativity of this face follows from lemma  [ rt - face - res - lem ]  as in the proof of theorem  [ me - ind - res - cube - thm ] .    by ( * proposition  2.1 ) , in theorems  [ inf - res - ind - cube - thm]and [ inf - ind - res - cube - thm ] , @xmath411 is in fact nondegenerate if and only if @xmath76 is . moreover , when the coactions are nondegenerate , mansfield imprimitivity works for @xmath40 and @xmath411 if and only if it works for @xmath40 and @xmath76 . as before , we get commutativity of a sixth face of diagrams and from the other five faces . the left face of is new ; the special case @xmath168 follows from ( * ? ? ? * theorem 4.7 ) .    finally , we show that our res - ind duality is compatible with the stabilization trick of @xcite , as adapted to full coactions in @xcite . let @xmath78 be a nondegenerate twisted coaction such that mansfield imprimitivity works for @xmath0 and @xmath39 itself ; equivalently , such that @xmath0 is normal ( * ? ? ? * lemma  3.6 ) . then mansfield imprimitivity works for @xmath0 and any closed normal subgroup of @xmath1 contained in @xmath39 by ( * ? ? ? * theorem  5.2 )  ( or by ( * ? ? ? * lemma  3.2 ) , since @xmath0 is normal ) . * theorem 3.1)shows that the twisted coaction @xmath78 is morita equivalent to the ( nondegenerate ) inflated twisted coaction @xmath429 , so by ( * ? ? ? * theorems 5.3 and 5.4 ) , mansfield imprimitivity works for @xmath430 and any closed normal subgroup of @xmath1 contained in @xmath39 . thus we can chain together theorems [ me - res - ind - cube - thm ]  and [ inf - res - ind - cube - thm ] , and similarly theorems [ me - ind - res - cube - thm ]  and [ inf - ind - res - cube - thm ] , to obtain :    [ stab - cube - thm ] let @xmath78 be a nondegenerate normal twisted coaction , and let @xmath166 be closed normal subgroups of @xmath1 contained in @xmath39 . further let @xmath431 , which carries the double dual coaction of @xmath39 . then the cubes @xmath432{{{\operatorname{rep}}}b \times k / n }      \arrow[2]{e }      \arrow{s,- } \node[2]{{{\operatorname{rep}}}b\times k\times_r n }      \arrow[2]{s , r}{{{\operatorname{ind}}}}\\ \node{{{\operatorname{rep}}}a\times_{g / k}g / n }      \arrow{ne }      \arrow[2]{e }      \arrow[2]{s , l}{{{\operatorname{res } } } } \node { }      \arrow{s , l}{{{\operatorname{res } } } } \node{{{\operatorname{rep}}}a\times_{g / k}g\times_r n }      \arrow{ne }      \arrow[2]{s , r,3}{{{\operatorname{ind}}}}\\ \node[2]{{{\operatorname{rep}}}b\times k / h }      \arrow{e,- } \node { }      \arrow{e } \node{{{\operatorname{rep}}}b\times k\times_r h}\\ \node{{{\operatorname{rep}}}a\times_{g / k}g / h }      \arrow{ne }      \arrow[2]{e } \node[2]{{{\operatorname{rep}}}a\times_{g / k}g\times_r h }      \arrow{ne } \end{diagram}\ ] ] and @xmath433{{{\operatorname{rep}}}b\times k / n }          \arrow[2]{e } \node[2]{{{\operatorname{rep}}}b\times k\times_r n}\\ \node{{{\operatorname{rep}}}a\times_{g / k}g / n }          \arrow{ne }          \arrow[2]{e } \node { }          \arrow{n , l}{{{\operatorname{ind } } } } \node{{{\operatorname{rep}}}a\times_{g / k}g\times_r n }          \arrow{ne}\\ \node[2]{{{\operatorname{rep}}}b\times k / h }      \arrow{n,- }          \arrow{e,- } \node { }          \arrow{e } \node{{{\operatorname{rep}}}b\times k\times_r h }      \arrow[2]{n , r}{{{\operatorname{res}}}}\\ \node{{{\operatorname{rep}}}a\times_{g / k}g / h }      \arrow[2]{n , l}{{{\operatorname{ind } } } }          \arrow{ne }          \arrow[2]{e } \node[2]{{{\operatorname{rep}}}a\times_{g / k}g\times_r h }      \arrow[2]{n , r,3}{{{\operatorname{res } } } }          \arrow{ne } \end{diagram}\ ] ] both commute in the usual strong sense . again , the left face of  is new ; when @xmath168 it reduces to ( * ? ? ? * theorem 4.7 ) . in this section , as a sample application of our @xmath36-@xmath37 duality , we generalize some results of gootman and lazar @xciteconcerning restriction and induction of ideals in crossed products by coactions of amenable groups , to nonamenable groups . nilsen @xcite  has recently proved similar results , using different , representation - theoretic techniques . our methods , based on our res - ind duality results , appear to be more efficient than those of gootman and lazar . with some additional effort , we could further generalize to the setting of intermediate twisted crossed products by coactions of nonamenable groups , but we feel that to do so at this point would only muddy the waters . first , recall from ( * ? ? ? * proposition  9 )  that a nondegenerate homomorphism @xmath434 gives rise to maps @xmath435 and @xmath436 between spaces of ideals . by definition , @xmath437 , and @xmath438 is the ideal generated by @xmath439 . if in addition @xmath95 is a @xmath96  @xmath113 imprimitivity bimodule , then by definition we have the following commutative diagram : @xmath440 green defines a map @xmath441 by requiring the diagram @xmath442 to commute . ( actually , we have abstracted green s definition a bit in order to bring the properties of the maps into high relief . ) so , @xmath42 is to @xmath37 as @xmath41 is to @xmath36 , and in fact @xmath41 is a special case of @xmath42 , just as @xmath36 is a special case of @xmath37 .        * @xmath443 ; @xmath444 ; * @xmath445 ; @xmath446 ; * @xmath37 , @xmath36 , @xmath42 and @xmath41 are order - preserving * @xmath37 and @xmath36 preserve arbitrary intersections ; * @xmath42 and @xmath41 preserve arbitrary sups ; * @xmath447 ; @xmath448 ; * @xmath449 ; @xmath450 ; * @xmath451 is the smallest ideal @xmath452 of @xmath96 such that @xmath453 ; * @xmath454 is the largest ideal @xmath455 of @xmath3 such that @xmath456 ; * @xmath457 is the smallest ideal @xmath458 of @xmath113 such that @xmath459 ; * @xmath460 is the largest ideal @xmath455 of @xmath3 such that @xmath461    now consider an action @xmath140 . green applies the above machinery to the canonical map @xmath462 to get maps @xmath463 and @xmath464 he then applies the same machinery to the canonical map @xmath465 and the @xmath466  @xmath96 imprimitivity bimodule @xmath95 from his imprimitivity theorem to get maps @xmath467 and @xmath468 ( green uses @xmath469 as the imprimitivity algebra , rather than the isomorphic co - crossed product @xmath466 . )        * @xmath454 and @xmath474 are @xmath1-invariant ; * @xmath475 is the largest @xmath1-invariant ideal of @xmath96 contained in @xmath245 ; * @xmath476 is the smallest @xmath1-invariant ideal of @xmath96 containing @xmath245 ; * @xmath245 is @xmath1-invariant if and only if @xmath477 . we will have to adapt green s machinery to reduced crossed products . let @xmath140 be an action , and let @xmath478 be the regular representation . the commutative diagram @xmath479{m(b\times_{r}g ) } \end{diagram}\ ] ] gives in turn commutative diagrams @xmath480{{\mathcal i}(b\times_{r}g ) } \end{diagram}\ ] ] and @xmath481{{\mathcal i}(b\times_{r}g ) , }          \arrow{nw , b}{{{\operatorname{res}}}_{\rho\circ i_b } }          \arrow{n , r}{\rho^ * } \end{diagram}\ ] ] the horizontal @xmath41 and @xmath36 are what green uses ; we will need the diagonal ones , together with their associated maps @xmath482 and @xmath483 . similarly , the commutative diagram @xmath484 gives in turn commutative diagrams @xmath485{s , l}{\cong }          \arrow{se , b}{\cong }      \node[2]{{\mathcal i}(b\times g ) }          \arrow[2]{w , t}{{\operatorname{ex } } }          \arrow{sw , b}{{\operatorname{sub } } }          \arrow[2]{s , r}{\rho_*}\\      \node[2]{{\mathcal i}(b ) }          \arrow{sw , t}{\cong}\\      \node{{\mathcal i}(b\times_{r}g\times g ) }      \node[2]{{\mathcal i}(b\times_{r}g ) }          \arrow[2]{w , b}{{\operatorname{ex } } }          \arrow{nw , t}{{\operatorname{sub } } } \end{diagram}\ ] ] and @xmath486{s , l}{\cong }          \arrow{se , b}{\cong }          \arrow[2]{e , t}{{{\operatorname{res } } } }      \node[2]{{\mathcal i}(b\times g)}\\      \node[2]{{\mathcal i}(b ) }          \arrow{sw , t}{\cong }          \arrow{ne , b}{{{\operatorname{ind } } } }          \arrow{se , t}{{{\operatorname{ind}}}}\\      \node{{\mathcal i}(b\times_{r}g\times g ) }          \arrow[2]{e , b}{{{\operatorname{res } } } }      \node[2]{{\mathcal i}(b\times_{r}g ) . }          \arrow[2]{n , r}{\rho^ * } \end{diagram}\ ] ] diagrams  and , together with the @xmath1-equivariance and injectivity of @xmath487 , imply that all of green s results in ( * ? ? ? * lemma  10(ii ) and proposition  11)carry over to reduced crossed products . in particular , those results summarized in proposition  [ green - sum - prop ]  carry over ; we will cite these as simply @xcite  without further comment . we now consider a nondegenerate normal coaction @xmath48 . we would like to apply the general abstract nonsense of the beginning of this section to obtain maps @xmath42 and @xmath41 among the spaces @xmath488 and @xmath489 to go along with the maps @xmath36 and @xmath37 . we use the canonical map @xmath490 to get a map @xmath491 ( in fact , we ve already used this map in diagram  in the case @xmath492 . ) similarly , we use the canonical map @xmath493 and the @xmath494  @xmath3 imprimitivity bimodule @xmath119 from mansfield s imprimitivity theorem to get a map @xmath495    in this situation , theorem  [ plain - res - ind - thm ]  in the case @xmath168 and @xmath496 gives us the commutativity of the following diagram : @xmath497 by the same token , theorem  [ plain - ind - res - thm ]  reduces to the commutativity of @xmath498 which is just the definition of @xmath37 . [ sub - ex - thm ] let @xmath78 be a nondegenerate twisted coaction , and let @xmath166 be closed normal subgroups of @xmath1 contained in @xmath39 such that mansfield imprimitivity works for @xmath40 and @xmath0 . then the following diagrams commute : @xmath499 @xmath500    the theorem follows immediately from corollaries [ twisted - res - ind - cor ]  and [ twisted - ind - res - cor ] , together with elementary properties of @xmath36 , @xmath41 , @xmath42 and @xmath37 . for example , by @xcite , corollary  [ twisted - ind - res - cor ]  gives a commutative diagram @xmath501 now fix @xmath502 . by proposition  [ gan - prop ] , @xmath457 is the smallest ideal @xmath245 of @xmath503 such that @xmath504 ; hence @xmath505 is the smallest ideal @xmath39 of @xmath506 such that @xmath507 . again using proposition  [ gan - prop ] , this implies that @xmath508 so diagram  commutes .      for the rest of this section , we will only need the special case @xmath168 and @xmath509 of theorem  [ sub - ex - thm ] . for reference , this gives the commutativity of the following triangles : @xmath510 @xmath511 ( again , diagram  has simply reduced to the definition of @xmath42 . ) we are now able to generalize some of the coaction results of @xcite . let @xmath48 be a nondegenerate normal coaction . adapting ( * ? ? ? * definition  2.4 )  to full coactions , we say that an ideal @xmath123 of @xmath3 is _ @xmath1-invariant _ if @xmath512 where @xmath513 is the quotient map and @xmath514 is the left regular representation . we do not know if this implies that @xmath0 restricts to a coaction on @xmath123 ( unless @xmath1 is amenable ) , but we do get a coaction @xmath515 of @xmath1 on @xmath132 . since we will not need this fact here , we omit the proof .            green shows ( * ? ? ? * proposition  13 )  that for an action @xmath140 with @xmath1 amenable , @xmath519 on @xmath473 . this is definitely not true for non - amenable @xmath1 : @xmath520 , while @xmath521 if and only if @xmath522 . for all we know , even if we pass to reduced crossed products , @xmath41 and @xmath37 can be different on @xmath473 . even for trivial actions , the question reduces to the unsolved problem of whether there exists a locally compact group @xmath1 for which @xmath46 is not exact . we do not know how to prove the companion result for @xmath42 and @xmath36 ( generalizing ( * ? ? ? * proposition  3.14(iv ) ) ) ; it certainly does nt follow from @xcite  and duality as in the proof of lemma  [ ex = ind - lem ] . * @xmath543 is the smallest @xmath1-invariant ideal of @xmath3 containing @xmath123 ; * @xmath544 is the largest @xmath1-invariant ideal of @xmath8 contained in @xmath245 ; * @xmath545 is the smallest @xmath1-invariant ideal of @xmath8 containing @xmath245 . \(i ) let @xmath39 be the intersection of all invariant ideals of @xmath3 containing @xmath123 ; since @xmath516 is closed under intersections , @xmath39 is invariant and hence is the smallest such ideal of @xmath3 . now @xmath543 is invariant , and @xmath443 because @xmath526 is increasing ; so we get @xmath546 taking @xmath41 and using proposition  [ gan - prop](vi ) , we have @xmath547 so equality holds throughout . in particular , @xmath548 since @xmath41 is injective on invariant ideals , we get @xmath549
consider a coaction @xmath0 of a locally compact group @xmath1 on a @xmath2-algebra @xmath3 , and a closed normal subgroup @xmath4 of @xmath1 . 
 we prove , following results of echterhoff for abelian @xmath1 , that mansfield s imprimitivity between @xmath5 and @xmath6 implements equivalences between mansfield induction of representations from @xmath7 to @xmath8 and restriction of representations from @xmath9 to @xmath8 , and between restriction of representations from @xmath8 to @xmath7 and green induction of representations from @xmath8 to @xmath9 . 
 this allows us to deduce properties of mansfield induction from the known theory of ordinary crossed products .