TOPOLOGICALLY IRREDUCIBLE REPRESENTATIONS AND RADICALS IN BANACH ALGEBRAS

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1 AND RADICALS IN BANACH ALGEBRAS P G DIXON [Receved 6 Jue 995 Revsed 7 August 995 ad 5 February 996] Itroducto The Jacobso radcal of a assocatve algebra s the tersecto of the kerels of the strctly rreducble represetatos It s atural, whe studyg ormed algebras, to cosder cotuous topologcally rreducble represetatos o Baach spaces, that s, cotuous homomorphsms of the algebra oto algebras of bouded operators o Baach spaces for whch o o-trval closed subspace s varat (where o-trval meas havg o-zero dmeso ad codmeso) Aga, oe looks at the tersecto of the kerels of all these represetatos of a gve algebra We shall show (Theorem 8) that ths s, a reasoable sese, a topologcal radcal For Baach algebras, topologcal rreducblty s more geeral tha strct rreducblty, so our log-term asprato s to use topologcally rreducble represetatos to study Jacobso radcal Baach algebras However, whlst t s easy to fd cotuous topologcally rreducble represetatos whch are ot strctly rreducble, t s ot mmedately clear that the tersecto of the kerels of these ca be strctly smaller tha the Jacobso radcal Oe way to costruct a topologcally rreducble represetato of a ormed algebra A s to fd a cotuous homomorphsm φ: A B to a Baach algebra B such that φ( A) s dese B ad B has strctly rreducble represetatos The every strctly rreducble represetato of B duces a cotuous topologcally rreducble represetato of A Ths costructo s due to Meyer [ 2], who calls such represetatos stadard We shall use t x 9 to produce a o-commutatve Baach algebra whch the radcal descrbed above s strctly smaller tha the Jacobso radcal Durg the preparato of ths paper, the author asked Charles Read whether hs work o the Ivarat Subspace Problem could be exteded to produce a quas-lpotet operator o a Baach space wth o closed varat subspace Read was able to do ths [ 7] ad hs example gves a secod Baach algebra whch the radcal assocated wth topologcally rreducble represetatos s strctly smaller tha the Jacobso radcal Both examples are mportat our theory Read s example has the mert of beg commutatve; ours, whch s substatally easer, dstgushes the Jacobso radcal from the radcal assocated wth the stroger codto of topologcal trastvty A ormed represetato π of a algebra A o a ormed space X s sad to be topologcally trastve f, wheever { x,, x}, { y,, y} are fte subsets of X wth { x,, x} learly depedet ad ε > 0, there s a elemet a A wth π( ax ) y < ε ( ) It follows from Jacobso s Desty Theorem Proc Lodo Math Soc (3) 74 (997) Mathematcs Subject Classfcato: 46H5, 46H25, 6Nxx

2 75 that all stadard represetatos of ormed algebras have ths property It s atural to ask [ 4, p 460; 2, p 32] whether every topologcally rreducble represetato s topologcally trastve We shall observe (Corollary 55) that Eflo s soluto of the Ivarat Subspace Problem for Baach spaces gves a couter-example, because topologcally trastve represetatos of commutatve algebras must be oe-dmesoal Geeralzg ths, we show (Corollary 50) that all topologcally trastve represetatos of PI-algebras are fte-dmesoal Read s example shows that the radcals assocated wth topologcally rreducble ad wth topologcally trastve represetatos are dstct Our dscusso of the radcals assocated wth these varous types of represetato requres a abstract theory of topologcal radcals topologcal algebras We devote x 6 to settg up such a theory The ma problem s to choose the correct deftos: the theory seems to have uusually sestve depedece o tal codtos, to borrow a phrase from Chaos Theory May reasoable varats o our chose axoms seem ot to provde the desred results (though we have ot searched for couter-examples to establsh ths, sce our prcpal cocer s wth specfc radcals rather tha the axomatcs) Wth ths theory place, we ca produce topologcal radcals from maps whch satsfy most but ot all of the axoms (UTRs ad OTRs) Ths eables us to relate the ew radcals to each other ad to a topologcal radcal derved from the Baer radcal Our theory of topologcal radcals has a varat whch apples to all ormed algebras, ot just to Baach algebras Ths s useful order to have a axom about the radcal of a cotuous homomorphc mage Ufortuately, the Jacobso radcal s ot a topologcal radcal ths verso: t s ot ecessarly closed! However, as we show x0, the tersecto of the kerels of the cotuous strctly rreducble represetatos o Baach spaces provdes a good alteratve whch cocdes wth the Jacobso radcal Baach algebras I x we ote the cosequeces of ot requrg the represetato space to be complete ad the paper cocludes wth a lst of ope questos I should lke to thak Dr Joh Reso for potg out errors a earler draft of ths paper 2 Deftos ad abbrevatos All algebras cosdered wll be lear assocatve algebras over the complex feld They wll ot ecessarly be commutatve or utal A represetato of a algebra A s a homomorphsm π of A to the algebra of all operators o a vector space X We shall call π a ormed represetato of the algebra A f X s a ormed space ad π s a homomorphsm of A to the algebra L( X) of all bouded operators o X We ca look at varous refemets of ths cocept: we may make A a ormed or Baach algebra, we may the requre the represetato to be cotuous (wth respect to the gve orm o A ad the operator orm o L( X) ) We shall geerally do ths; otherwse, we should be gorg the topology o A Also, we may requre X to be a Baach space Ths too s a sesble opto, though we shall cosder, x, the cosequeces of usg complete spaces A represetato πof a algebra A o a vector space X s sad to be strctly rreducble f there s o subspace Y X wth { 0} Y X ad π( a)( Y ) Y for

3 76 P G DIXON all a A A ormed represetato π : A L( X) of a algebra A o a ormed space X s sad to be topologcally rreducble (TI) f there s o closed subspace Y X wth { 0} Y X ad π( a)( Y) Y for all a A For ay algebra A, we deote by J( A) the Jacobso radcal of A, whch s the tersecto of the kerels of all the strctly rreducble represetatos of A For a ormed algebra A, we defe the TI radcal T ( A) to be the tersecto of the kerels of all the cotuous TI represetatos of A o Baach spaces Equvaletly, we ca defe a left Baach A-module to be topologcally smple f t has o closed submodule The T( A) s the tersecto of the ahlators of the topologcally smple left Baach A-modules A represetato π of a algebra A o a vector space X s sad to be trastve (or strctly dese) f, wheever { x,, x}, { y,, y} are fte subsets of X wth { x,, x} learly depedet, there s a elemet a A wth π( ax ) = y ( ) I fact, f ths holds for = 2, t holds for all ad the topologcal verso of Jacobso s Desty Theorem [ 4, 423; 8, (247)] says that, for Baach algebras, every strctly rreducble represetato s trastve There s a obvous topologcal aalogue: for each postve teger, a ormed represetato π of a algebra A o a ormed space X s sad to be topologcally -trastve ( -TT) f, wheever { x,, x}, { y,, y} are subsets of X wth { x,, x} learly depedet ad ε > 0, there s a elemet a A wth π( a) x y < ε ( ) A represetato s sad to be topologcally trastve (TT) f t s topologcally -trastve for all postve tegers Ths s topologcally completely rreducble Palmer s termology ad s equvalet to sayg that π( A) s dese L( X) the strog operator topology (the topology gve by the semorms T Tx ( x X) ) It s ot kow whether or ot -TT for some 2 mples TT We wrte T( A), T ( A) for the tersecto of the kerels of all the cotuous -TT, TT (respectvely) represetatos of A o Baach spaces A partcular type of cotuous TT represetato arses as follows Let ρ be a strctly rreducble represetato of a Baach algebra B o a lear space X The there s a uque Baach space orm o X makg the represetato ormed ad cotuous [ 4, 426(a), 425] Let A be a ormed algebra ad φ: A B a cotuous homomorphsm such that φ( A) s dese B The π = ρφ s a TT represetato of A o X Followg Meyer [ 2], we call such TT represetatos stadard The tersecto of the kerels of all the stadard TT represetatos of a gve ormed algebra A wll be deoted S( A) For a ormed algebra A, the clusos T( A) T ( A) T ( A) T ( A) J( A) ( m ) m are clear What s ot mmedately clear s whether ay of these clusos ca be strct 3 Elemetary propertes of TI represetatos Some of our later examples wll produce, ter ala, TI represetatos whch are ot strctly rreducble, but t s worth otg ow that satsfyg these requremets aloe s qute easy

4 77 EXAMPLE 3 Let A = l( S2) be the semgroup algebra of the free semgroup 2 o two geerators X, Y Let Tbe the ulateral shft o H = l : T(ξ,ξ 2,ξ 3,) = ( 0,ξ,ξ 2,), T (ξ,ξ 2,ξ 3,) = (ξ 2,ξ 3,ξ 4,) 2 Let π be the cotuous represetato of A o l defed by π(δ X ) = T, π(δ Y ) = T It s easy to see that π( A) s a *-subalgebra of L( H) wth scalar commutat, so, by vo Neuma s Double Commutat Theorem, ts strog closure s L( H), that s, π s TT; but π( A)(( 00,,, )) = l, so π s ot strctly rreducble REMARK 32 For *-represetatos of C*-algebras, Kadso s Trastvty Theorem says that TI mples strctly rreducble ([ ]; see also [ 3, 522; 20, 27]) I the example above, π( A) s ot closed L( H) We shall be seekg to relate the TI radcal to radcals defable wthout referece to represetatos I oe drecto ths s easy, provded the algebra s complete: every strctly rreducble represetato of a Baach algebra A has the same kerel as some cotuous strctly rreducble represetato of A o a Baach space [ 4, 429; 8, (247)] Hece the TI radcal of a Baach algebra s cotaed the Jacobso radcal, whch has may characterzatos ot drectly volvg represetatos (largest quas-regular deal, largest deal of topologcally lpotet elemets, tersecto of the maxmal modular left deals) It s ot mmedately clear that ths cluso ca be strct Example 3 above, the algebra A s semsmple so there are may other represetatos whch are strctly rreducble but we shall gve a example later where ths s so I the other drecto, the oly results we kow stem from the followg proposto PROPOSITION 33 [ 4, 425(a), 449(a)] The kerel of a TI represetato of a algebra A o a ormed space s a prme deal of A Hece, the tersecto of the kerels of the TI represetatos of a algebra A cotas the Baer radcal of A The Baer radcal or prme radcal β( A) of a algebra A s the tersecto of all the prme deals of A; equvaletly, t s the smallest deal I of A such that A/ Ihas o o-zero lpotet deals [ 4, 446] It s the smallest of three radcals, the others beg the Levtzk radcal ad the l radcal, that cocde for Baach algebras [ 4] However, Corollary 94 below shows that, for complete ormed algebras, the TI radcal does ot ecessarly cota the other two radcals Whe we cosder cotuous TI represetatos of a ormed algebra, we have the further formato that the kerels of the represetatos are closed Cosequetly, the TI radcal of a ormed algebra A cotas the closure of the Baer radcal β( A) However, A/β( A) mght fal to be semprme, whch case the premage A of ts Baer radcal s also cluded the TI radcal, as s ts closure, ad so o Ths leads us to costruct ( Corollary 68) a ew radcal, the closed-baer radcal β, to gve a good lower boud for the TI radcal

5 78 P G DIXON 4 Classcal problems The dffculty of workg wth TI represetatos s well llustrated by ther relato to some famous problems of fuctoal aalyss PROPOSITION 4 The followg are equvalet ( ad true ): () there s a sgly-geerated ( as a Baach algebra) Baach algebra wth a cotuous fathful TI represetato o a fte-dmesoal Baach space; (2) there s a sgly-geerated Baach algebra wth a cotuous o-zero TI represetato o a fte-dmesoal Baach space; (3) there s a operator o a fte-dmesoal Baach space wth o o-trval closed varat subspace Proof The truth of (3) s Eflo s soluto of the Ivarat Subspace Problem for Baach Spaces [ 7] (see also [ 5;, Chapter XIV]) The proof of the equvalece of (), (2) ad (3) s straghtforward Note the sharp cotrast wth the stuato for strctly rreducble represetatos: t follows from Schur s Lemma that strctly rreducble represetatos of commutatve Baach algebras must be oe-dmesoal [ 4, 429] PROPOSITION 42 The followg are equvalet ( ad true ): () there s a sgly-geerated ( as a Baach algebra) radcal Baach algebra wth a cotuous fathful TI represetato o a fte-dmesoal Baach space; (2) there s a sgly-geerated ( as a Baach algebra) radcal Baach algebra wth a cotuous o-zero TI represetato o a fte-dmesoal Baach space; (3) there s a quas-lpotet operator o a fte-dmesoal Baach space wth o o-trval closed varat subspace Proof The truth of (3) s a recet result of Read [ 7] We prove the equvalece of (), (2) ad (3) I (2) (3), f π : A L( X) s a cotuous, TI represetato wth π( a) 0 for some a A, the the desred operator π( a) s quas-lpotet I (3) (), f T s the gve quas-lpotet operator o X, the the closed subalgebra of L( X) that t geerates s radcal REMARK 43 Sce fte-dmesoal subspaces are automatcally closed, all fte-dmesoal TI represetatos of algebras are strctly rreducble Hece, f a radcal algebra has TI represetatos, they must be fte-dmesoal Fally, we ote that the famous problem of the exstece of a topologcally smple commutatve radcal Baach algebra s equvalet to askg for a commutatve radcal Baach algebra for whch the left regular represetato s TI

6 79 5 Topologcally trastve represetatos The obvous frst questo about TT represetatos s whether there are TI represetatos whch are ot TT Oe way whch such represetatos mght occur s as the left regular represetatos of radcal Baach algebras wth o o-trval closed left deals, f such exst 5 If A s a Baach algebra of dmeso greater tha left regular represetato of A o A s ot 2-TT Proof We beg by provg ths uder the assumpto ( ) there are elemets x, y Asuch that xad xyare learly depedet Suppose the represetato s 2-TT The, for every ε> 0 there s a elemet a A such that axy x <ε ad ax <ε The, for every ε> 0, x axy x + ax y <ε( + y ) Thus x = 0, cotradctg ( ) Now assume that ( ) s false The, for every a, x, x2 A wth { x, x2} learly depedet, the elemets ax ad ax2 le the same -dmesoal subspace (spaed by a) Thus we ca ot make choces of a whch brg ax, ax2 deftely close to two gve learly depedet vectors y, y2; so the left regular represetato s ot 2-TT Aother approach to costructg TI, o-tt represetatos leads to the Ivarat Subspace Problem, ad therefore succeeds The followg theorem s probably the best topologcal aalogue of Schur s Lemma o strctly rreducble represetatos (Remember that L( X) here deotes the algebra of all bouded operators o X) 53 2 If (π, X) s a -TT represetato of a ( ot ecessarly ormed ) algebra A o a ormed space X, the { T L( X) : Tπ( a) = π( a) T ( a A) }= I I partcular, f A s commutatve the dm X =, the the REMARK 52 The problem of whether there exsts a radcal Baach algebra wth o o-trval closed left deals les betwee two usolved problems: the exstece of a topologcally smple radcal Baach algebra ad the exstece of a topologcally smple commutatve radcal Baach algebra Proof Suppose Tπ( a) = π( a) T ad T s ot a multple of the detty Let ξ X be such that ξ ad Tξ are learly depedet Let η,ζ X be arbtrary If π were 2-TT, we could fd a sequece ( b) A wth π( b)ξ η ad π( b)( Tξ) ζ However, π( b)( Tξ) = T(π( b)ξ) Tη Therefore ζ = Tη, cotradctg the arbtraress of ζ

7 80 COROLLARY 54 P G DIXON For a commutatve Baach algebra, T ( A) = J( A ) 2 Applyg Theorem 53 to the fte-dmesoal TI represetato derved from the soluto to the Ivarat Subspace Problem (Proposto 4()), we obta the followg corollary COROLLARY 55 There s a commutatve Baach algebra wth a cotuous TI represetato whch s ot 2-TT Read s ew example (Proposto 42) yelds a sgfcatly stroger statemet COROLLARY 56 There s a commutatve Baach algebra wth T ( A) T ( A ) 2 REMARK 57 Beauzamy [, Chapter XIV] ad Read s papers [ 6, 7] o the varat subspace problem gve examples where the Baach space s l, so, these corollares, the pathology may be cofed to the algebra (whose structure s uclear) ad the represetato, rather tha the Baach space We cojecture that there are examples wth straghtforward algebras ad Baach spaces, the pathology beg cofed just to the represetatos It s terestg to explore geeralzatos of Theorem 53 to algebras satsfyg polyomal dettes Bearg md the Amtsur Levtzk Theorem, that M( ) satsfes the stadard polyomal dettes Sk for k 2, we make the followg cojecture CONJECTURE 58 Let If A s a algebra satsfyg the stadard S2 A polyomal detty, the every -TT represetato of o a ormed space X has dm X < (ad s therefore a strctly rreducble represetato wth dm X < ) The obvous approach goes as follows Suppose π : A L( X) s a -TT represetato wth dm X Let { e,, e} be a learly depedet set X Now the algebra M( ) of matrces does ot satsfy S2 Let T,, T2 be lear mappgs o the spa of { e,, e} such that S2 ( T,, T2 ) 0 Sce π s -TT, we ca fd a,, a2 A wth π( a) ej approxmatg Te j ( 2, j ) Ufortuately, we have o cotrol over the orms a ad so the elemets π( a )π( a ) e 2 j mght ot approxmate to T T e j 2 59 Let If A s a algebra satsfyg the stadard polyomal detty S +, the every 2 -TT represetato of A o a ormed space X has dm X < ( ad s therefore a strctly rreducble represetato wth dm X [ ( + )]) 2 Proof The followg otato wll be useful: f B ={ b,, bk} A ad π s our represetato of A, the SB ( ) =± S(π( b),,π( b)), k k

8 8 where we shall gore the sg (Ths s a coveet shorthad; a more detaled proof merely requres a straghtforward, but obfuscatg, replacemet of ths otato by oe depedet o partcular ordergs of the subsets B for whch SB ( ) s used) We wrte S( ) = I, the detty operator We shall prove, by ducto o, that f π : A L( X) s a 2 -TT represetato of a algebra A o a fte-dmesoal ormed space X ad x X\{ 0 }, the there exst a,, a+ A such that { S( B) x: B { a,, a+ }} s learly depedet I partcular, ths mples that so A does ot satsfy the detty S+ The ducto starts trvally at = 0 Suppose the result has bee proved for, where Gve a 2 -TT represetato π : A L( X) ad x X \{ 0 }, we use the ducto hypothess to fd a, a2,, a A such that the set E0 ={ S( B) x: B { a,, a}} s learly depedet As a frst approxmato to a+, we fd a elemet b A such that π( b ) x spa E, that s, the set 0 π( S ( a,, a )) =± S( { a,, a }) 0, s learly depedet Let W=, W2, W3,, W2 be a eumerato of all the subsets of { a,, a}, ordered so that W Wj ( j) (where W deotes the cardalty of W) We costruct successve approxmatos b to the desred a such that E ={ S( W { b }) x: k} E s learly depedet We wll the set a+ = b2 We have already descrbed the costructo of b Suppose bk has bee costructed as above Each SW ( { bk} ) x( < k) may be wrtte SW ( { b} ) x= ± SU ( )π( b) SVx ( ), where the summato s over all parttos U V = W ad s therefore a cotuous fucto of the vectors π( bk) S( Wj) x X ( j ) The ducto hypothess o b meas that the set k E ={ S( B) x: B { a,, a }, B ={ b }} k k F ={ S( W { b }) x: k } { S( B) x: B { a,, a }} s learly depedet whe bk = bk (makg Fk = Ek ) If S( Wk { bk } ) x s ot the lear spa of Ek, the we may set bk = bk Suppose otherwse We shall set bk = bk + dk for some perturbato dk whch s small the sese that the vectors π( dk) S( Wj) x are small for all j < k Our basc dea s that by the stablty of lear depedece uder small perturbatos (see, for example, [ 0, Corollary 207]), the fact that Fk s learly depedet remas true as bk s perturbed away from bk, provded that dk s suffcetly small the above sese Choose a vector y outsde the spa of E The the set s learly depedet By the ducto hypothess o a, the pots S( Wj) x ( j 2 ) are learly depedet These 2 pots may be separated by π; k k k k k k k + k k E { S( W { b }) x+ y} 0

9 82 P G DIXON we ca therefore fd small dk so that π( dk) S( Wk) x approxmates y Now SW ( { b} ) x= SW ( { b }) x+ π( d) SW ( ) x+ ± SU ( )π( d) SVx ( ) k k k k k k k where the sum s take over all parttos U V = Wk wth U The last term s a cotuous fucto of the vectors π( dk) S( Wj) x ( j < k) Therefore, f we choose suffcetly small dk wth π( dk) S( Wk) x suffcetly close to y, the we ca esure that the perturbato from E { S( W { b }) x+ y} to E = F { S( W { b }) x} k k k k k k k s small eough to preserve lear depedece Ths completes the ducto bk a+ = b2 step for the whole proof step the costructo of the Puttg the completes the ducto COROLLARY 50 Every TT represetato of a PI-algebra s fte-dmesoal ad hece strctly rreducble Proof If A s a PI-algebra, the A/β( A) s PI ad so, by a corollary of Kaplasky s Theorem [ 9, Theorem 628], satsfes a stadard detty Every TT represetato π : A L( X) has kerπ β( A) ad so duces a represetato π : A/β( A) L( X) wth π ( A/β( A)) = π( A) I partcular, π s TT By Theorem 59, π s strctly rreducble, so π s strctly rreducble 6 A geeral theory of radcals I ths secto, we develop a lttle of a geeral theory of radcals ormed algebras The calculatos are geerally straghtforward oce the correct deftos are place; but the theory s qute sestve to these Our attempts based o oly slghtly dfferet deftos, such as the obvous aalogue of Dvsky s algebrac defto or a defto cludg o-closed deals (4) below, have foudered o seemgly sgfcat techcaltes Nevertheless, several varatos do work I the followg defto we preset smultaeously our algebrac ad topologcal otos of a radcal, ad some of our theorems wll exst both cotexts I x 0 we shall dscuss the varat of the topologcal verso whch radcals are defed for complete algebras DEFINITION 6 By a radcal (respectvely, a topologcal radcal ) we mea a map R assocatg wth each algebra (Baach algebra) A a (closed) deal RA ( ) Ð Asuch that the followg hold: () RRA ( ( )) = RA ( ); (2) RA ( / RA ( )) ={ 0 }, where { 0} deotes the zero coset A/ RA ( ); (3) f A, Bare (Baach) algebras ad φ: A Bs a (cotuous) epmorphsm, the φ( RA ( )) RB ( ) ( epmorphsm here meas just surjectve homomorphsm ); (4) f I s a (closed) deal of A, the (a) RI () s a (closed) deal of Aad (b) RI () RA ( ) I

10 83 We say that R s a heredtary ( topologcal ) radcal f t satsfes (2), (3), (4) ad (5) f I s a (closed) deal of A, the R() I R( A) I (Note that (5) mples ()) We say that a algebra A s R-semsmple f R( A) ={ 0} ad R-radcal f RA ( ) = A We have preferred to cast our theory terms of maps, rather tha radcal property used by other authors (for example, Dvsky [ 3, p 3]), to facltate geeralzatos (Defto 62) Traslato betwee the two forms s easy: the property correspodg to a radcal R s A beg equal to R( A) ; the map correspodg to a gve property assocates wth a (Baach) algebra A the largest (closed) deal of A wth the gve property Dvsky s defto ad ts obvous topologcal aalogue, stated terms of a map R, are our deftos wth (3) ad (4) replaced by: (3) f A, Bare (Baach) algebras, A = R( A) ad φ: A Bs a (cotuous) epmorphsm, the B = R( B) ; (4) f I s a (closed) deal of A wth R() I = I, the I R( A) I the algebrac case, ths s easly equvalet to our defto, but t s ot clear whether ths s so the topologcal case (Clearly ( 3) ( 3) ad ( 4) ( 4 ), both cases Also the mplcato (( 2 ) & ( 3 ) & ( 4) ) ( 4) ca be proved the topologcal case by the method of Dvsky s Theorem 47 f the radcal satsfes a weak o-trvalty codto: that all Baach algebras wth zero multplcato be radcal) We clam that our defto s at least as aesthetcally satsfyg as Dvsky s ad s easer to work wth the topologcal case ad we leave the detaled vestgato of the relato betwee the two for others to study Codto (5) has the followg equvalet ( the presece of (4)) formulato (see [ 3, p 23, Lemma 68]): (5) every (closed) deal I of A wth I R( A) has I = R( I) We shall order radcal ad other such maps by cluso: we wrte R S to mea RA ( ) SA ( ) for all algebras A(all Baach algebras A, the topologcal case) DEFINITION 62 We shall say that a map A R( A) whch assocates wth each (Baach) algebra a (closed) deal s a uder radcal ( UR ) (respectvely, uder topologcal radcal ( UTR )) f t satsfes (), (3) ad (4) We shall say that t s a over radcal ( OR ) (respectvely, over topologcal radcal ( OTR )) f t satsfes (2), (3) ad (4) The reaso for the termology s that we shall show how a radcal ca be costructed above a gve UR (Theorem 66) ad below a gve OR (Theorem 60) (We avod the Lat prefxes sub ad super lest commo usage of the latter should suggest somethg stroger tha radcal) Oe way whch UTRs arse s tryg to covert algebrac radcals to topologcal radcals by takg closures 63 Let R be a UR For Baach algebras A, defe RA ( ) = RA ( ) The R s a UTR

11 84 P G DIXON The proof s straghtforward Ufortuately, the map R eed ot satsfy (2), eve f Rdoes The atural example of ths s the followg EXAMPLE64 Let be the Baach algebra [0 ] of all bouded, A ( C,, ) cotuous, complex-valued fuctos o the ut terval, wth covoluto multplcato ad let β be the Baer radcal map The, by usg the Ttchmarsh Covoluto Theorem, β( A) s the deal of all fuctos vashg o a eghbourhood of zero Hece β( A) s the deal of fuctos vashg at zero The quotet A/β( A) s the oe-dmesoal algebra wth zero multplcato, so codto (2) fals If RA ( / RA ( )) { 0 }, the we have to look at the verse mage Aof RA ( / RA ( )) uder the quotet map A A/ RA ( ) There s o reaso why the quotet of A by ths deal should be R -semsmple, so we aga look at the verse mage of the radcal We may expect to have to cotue ths process trasftely to get a topologcal radcal DEFINITION 65 Let be a map assocatg wth each (Baach) algebra a R A (closed) deal RA ( ) We defe a trasfte sequece of such maps ( R) by: () R0( A) ={ 0 }; () R+ ( A) = q ( R( A/ R( A))), where q: A A/ R( A) s the quotet map (so, for example, R = R); () for lmt ordals λ, R ( A) = R ( A) the algebrac case, ad + λ R ( A) = R ( A) λ the topologcal case The trasfte sequece of sets ( R( A)) s mootoc o-decreasg, so t must stablse at the th stage, where s at most the cardalty of A We the wrte R ( A) = R ( A) = R ( A) For example, f R = β, the R ( A) = R2( A) for the algebra A of Example 64 above 66 If R s a UR ( respectvely, a UTR ), the so s R, for every ordal, ad R s a radcal ( topologcal radcal ) Proof We prove the topologcal case, whch s the oe of most terest to us ow The algebrac case s smlar ad easer The fact that R satsfes codto (2) s easy: because the sequece has stablsed, R ( A) = q ( R( A/ R ( A))), that s, RA ( / R( A)) ={ 0 } It follows that R( A/ R( A)) ={ 0} for all, ad so R ( A/ R ( A)) ={ 0 } <λ <λ

12 85 The rest of the proof cossts of showg, by trasfte ducto o, that R satsfes codtos (), (3) ad (4) The result s trvally true for = 0, whch starts the ducto Most of the work les the ducto step to successor ordals Suppose R s a topologcal radcal We wrte ( ) to mea codto ( ) o R () We have R ( R ( A)) q ( R( R ( A)/ R ( A))) ( ( ( / ( )))) ( ( / ( ))) (by ( ) ) + + = + = q R R A R A = q R A R A = R+ ( A) (3) Suppose φ: A B s a cotuous epmorphsm betwee Baach algebras By ( 3 ), we have φ( R( A)) R( B) Hece φ duces a cotuous homomorphsm ψ : A/ R( A) B/ R( B) By ( 3 ), ψ( RA ( / R( A))) RBR ( / ( B)) Wrtg qa: A A/ R( A) ad qb: B B/ R( B), we have R+ ( B) = qb ( R( B/ R( B))) qb (ψ( R( A/ R( A)))) = qb (ψ( { x+ R( A)) : x R+ ( A)) } ={ φ( x) + y: x R+ ( A), y R( B) } Therefore φ( R+ ( A)) R+ ( B) (4) Suppose I s a closed deal of the Baach algebra A The ( 4) mples that R() I s a closed deal of A ad R() I R( A) I Now I/ R() I s a closed deal of A/ R( I), so R( I/ R( I)) s a closed deal of A/ R( I) by ( 4 ) (a) By defto, R+ ( I) = qi ( R( I/ R( I))), where qi: A A/ R( I) s the quotet map Therefore R+ ( I) s a closed deal of qi ( A/ R( I)) = A By ( 4 ) (b), RI ( / R( I)) RA ( / R( I)) I/ R( I) Let us wrte qa for the quotet map A A/ R( A) The the fact that R( I) R( A) produces a atural map p: A/ R( I) A/ R( A) Applyg ( 3) to p gves R( A/ R ( A)) p( R( A/ R ( I))) q q ( R( A/ R ( I))), whece = A I + = A I + R ( A) q ( R( A/ R ( A))) q ( R( A/ R ( I))) R ( I) (It s provg ( 4) + that a theory whch allows o-closed deals (4) has problems If I s ot closed, the we caot guaratee that R() I s closed A, wthout whch A/ R( I) s ot a ormed algebra a quotet orm) The ducto step at lmt ordals s easer ad s therefore omtted The proof of Theorem 66 s ow completed by observg that each of propertes (), (3) ad (4) for follows from the correspodg property for R R by choosg suffcetly large so that R (Ɣ) = R(Ɣ) for the two algebras Ɣ

13 86 P G DIXON volved, that s, for Ɣ = A ad R ( A) (), Ɣ = A ad B (3), ad Ɣ = I ad A (4) COROLLARY 67 The topologcal radcal R ( A) of a Baach algebra A s the smallest closed deal I of A such that A/ I s R-semsmple The trasfte sequece of sets ( R ( A)) s mootoc o-creasg, so t must stablse at the th stage, where s at most the cardalty of A We the wrte + R ( A) = R ( A) = R ( A) 60 It follows that R ( R( A)) = R( A) for all, ad so R( R( A)) = R( A) The rest of the proof cossts of showg by trasfte ducto o, that R satsfes codtos (2), (3) ad (4) The result s trvally true for = 0, whch starts the ducto Suppose R s a OTR Aga, we wrte ( ) to mea codto ( ) o R + (2) Sce R ( A) R ( A), there s a atural cotuous homomorphsm + A/ R ( A) A/ R ( A) R ( A/ R ( A)) = R( R ( A/ R ( A))) R( R ( A)/ R ( A)) = { 0}, COROLLARY 68 The Baer radcal β gves rse to the topologcal radcal β, ad β( A ), for a Baach algebra A, s the smallest closed deal I of A such that A/ I cotas o o-zero lpotet deals (We recall that the Baer radcal β( A) ca be characterzed as the smallest deal I of A such that A/ I cotas o o-zero lpotet deals) DEFINITION 69 Let R be a map assocatg wth each (Baach) algebra A, a (closed) deal RA ( ) We defe a trasfte sequece ( R) by: () 0 R ( A) = A; () + R ( A) = R( R ( A)) (so, for example, R = R); () for lmt ordals λ, λ R ( A) = R ( A) If R s a OR ( respectvely, a OTR ), the so s R, for every ordal, ad R s a radcal ( topologcal radcal ) Proof Aga, we prove the topologcal case, the algebrac case beg smlar ad easer Frst we observe that (4) mples that all of the R( A) are closed deals of A ad hece so s R( A) The fact that R satsfes codto () s easy: because the sequece has stablsed, R ( A) = R( R ( A)) + Applyg ( 3) to ths map shows that R ( A/ R ( A)) maps to R ( A/ R ( A)), + + whch s the zero coset, by ( 2 ) Therefore R ( A/ R ( A)) R ( A)/ R ( A) ; + + fact, R ( A/ R ( A)) s a closed deal of R ( A)/ R ( A) Applyg ( 4) to ths deal, we have where the last step uses ( 2 ) <λ

14 87 (3) Suppose φ: A B s a cotuous epmorphsm betwee Baach algebras By ( 3 ), we have φ( R ( A)) R ( B) The argumet the algebrac case cotues wth + φ( R ( A)) = φ( RR ( ( A))) R(φ( R( A))) RR ( ( B)), but ths fals the topologcal case because, for the last step, we eed φ( R ( A)) to be a closed deal of R ( B) to apply ( 4 ) Istead, we argue that φ( R ( A)) s a closed deal of R ( B), so, by ( 4 ), ( ) + R φ( R ( A)) R( R ( B)) = R ( B) Now let ( ) = I φ φ( R ( A)) R ( A) + The R ( A) s a closed deal of I, so R ( A) = R( R ( A)) R( I), by ( 4 ) Thus, by applyg ( 3) to the mappg φ: I φ( R( A)), we obta ( ) + + φ( R ( A)) φ( R( I)) R φ( R ( A)) R ( B) (4) Suppose I s a closed deal of the Baach algebra A The, frst, ( 4 ) (a) mples that R () I s a closed deal of A ad so, applyg () 4 (a) to ths deal, + we see that R () I = R( R ()) I s a closed deal of A Secodly, () 4 (b) mples that R () I R ( A), so R () I s a closed deal of R ( A), ad () 4 (b) appled to ths deal gves + + R ( I) = R( R ( I)) R( R ( A)) = R ( A) Aga, we omt the ducto step at lmt ordals, whch s straghtforward The proof s completed by observg that each of propertes (), (3) ad (4) for R follows from the correspodg property for R by choosg suffcetly large so that R (Ɣ) = R (Ɣ) for the two algebras Ɣ volved 6 Let R be a UR ( UTR ) ad S a OR ( OTR ), wth R S The R S I partcular, f S s a radcal ( topologcal radcal ), the R S; f R s a radcal ( topologcal radcal ), the R S If R s a radcal ad S s a topologcal radcal, wth R S, the R S Proof The proof cossts of three steps (a) We frst show, by trasfte ducto o, that R S Ths s trval for = 0 ad the step to lmt ordals s easy For the successor step, suppose R S The + R ( A)/ R ( A) = R( A/ R ( A)) S( A/ R ( A)) + Sce R( A) S( A), we have a atural (cotuous) epmorphsm of A/ R( A) oto A/ S( A) Ths maps S( A/ R ( A)) to S( A/ S( A)) ={ 0 } Therefore so ad so R ( A) S( A) S( A/ R ( A)) S( A)/ R ( A), R ( A)/ R ( A) S( A)/ R ( A), +

15 88 P G DIXON β (b) Next, we show that R S for all β Ths s easer, the successor step β β beg that f R S the R( A) s a (closed) deal S ( A), so β β+ RA ( ) = RRA ( ( )) SRA ( ( )) SS ( ( A)) = S ( A) (c) Sce every R s UR (UTR), we ca apply step (b) to R place of R β to get R( A) S ( A) for all ordals, βad all (Baach) algebras A Hece R ( A) S ( A) COROLLARY 62 If R s a UR ( UTR ), the R s the smallest ( topologcal ) radcal greater tha or equal to R: that s, R R ad f S s a ( topologcal ) radcal wth S R, the R S Lkewse, f S s a OR ( OTR ), the S s the greatest ( topologcal ) radcal less tha or equal to S 7 Heredtary radcals It s atural to ask whether, Theorem 63, the map R satsfyg axom (5) would mply R satsfyg (5) The aswer s egatve, as the followg example shows EXAMPLE 7 Let A be the commutatve Baach algebra geerated by + { X: =, 2, 3, } subject to the relatos X = 0 ( =, 2, 3,) ad XX j = 0 ( j) That s, f A0 s the algebra defed, algebracally, by these geerators ad relatos, wth the orm gve by λ = λ j, j jx j j the A s the completo of ( A0, ) A typcal elemet of A s just a fte j sum x = j λjx = wth x = j λ j < Let y = 2 X ad let B be the set of elemets of A of the form j 2 j λjx The I = y + B s a closed deal of A Now a elemet j j λ jx of A s lpotet f ad oly f sup { / j: λ j 0} < ad, sce A s commutatve, β( A) ad β( I) are just the sets of lpotet elemets of A ad I, respectvely Thus β( A) = A ad β() I = B; so β( A) I β() I REMARK 72 I ths example, (5) seems to fal a rather trval way Ideed, for every heredtary radcal R, f I s a closed deal a Baach algebra A, 2 we have ( R( A) I) R ( I) Oe cosequece of ths s that f R β the RA ( ) I R 2( I), where R2 s costructed from Ras Defto 65 It s temptg to cojecture that ths wll form the start of a trasfte ducto leadg to R beg heredtary, but our attempts to carry out ths pla have bee thwarted by that pereal problem of Baach algebra theory: the fact that the sum of two closed deals s ot ecessarly closed We do ot eve kow whether or ot β s heredtary 8 The TT radcals 8 For =, 2,,, the map A T( A) whch assocates wth every Baach algebra A ts topologcally -trastve radcal s a heredtary topologcal radcal

16 89 Proof Codto () follows from (5) below ad (2) ad (3) are straghtforward (4)(a) We must show that f I s a closed deal of A, the T() I s a deal of A Suppose a A ad b T() I ; the π() b = 0 for every cotuous -TT represetato π of I ad we eed to show that π( ab) = 0 ad π( ba) = 0 for all such represetatos Cosder π( ab) where π : I L( X) s -TT If x X ad c I the π( c)π( ab) x = π( cab) x = π( ca)π( b) x = 0 Ths, for all c I, mples π( ab) x = 0, because π s TI Hece π( ab) = 0 Lkewse, π( ba)π( cx ) = π( b)π( acx ) = 0 ad { π( cx ) : c I, x X} s dese X, so π( ba) = 0 (4)(b) Every cotuous represetato π of A restrcts to a cotuous represetato of a closed deal I It suffces to show that, for <, f π s -TT the π I s -TT or zero It wll the follow that T( I) T( A) for all We beg by showg that f π I s o-zero, the t s TI To see ths, suppose x, y Xwth x 0 ad ε > 0; let bbe ay elemet of Isuch that π( b) x 0 (If o such b exsts, the π( I)(π( A) x) π( IA) x π( I) x = { 0 }, so π( I) X = 0, cotrary to assumpto) We may the fd a A such that π( a)(π( b) x) y < ε, so we have π( ab) x y < ε ad ab I We ow show that π I s -TT Let x,, x, y,, y X wth x,, x learly depedet ad ε > 0 Choose η > 0 such that every set { z,, z} wth z x <η ( ) s learly depedet Sce πrestrcts to a TI represetato o I, by the above, we ca fd, for each a elemet b I such that π( b ) x x < η/ 2 We the fd c A ( ) such that The So Let The π( c ) x δ x < η/ 2 π( b ) (, j ) j j π( bc) x δ π( b) x < η/ 2 (, j ) j j π( bc) x δ x < η/ (, j ) j j b = b c I = π( bx ) x < η ( j ) j j Therefore the set { π( bx ),,π( bx ) } s learly depedet Let a Abe such that π( a)π( b) x y < ε ad ab I s the desred elemet such that π( ab) seds x j ear to y j for all j (5) The fact that, for every closed deal I of a Baach algebra, T() I T( A) I follows from Lemma 82 below LEMMA 82 If I s a closed deal of a Baach algebra A ad, the for every cotuous -TT represetato π of I o a Baach space ( X, ) j j

17 90 P G DIXON there s a cotuous -TT represetato ρ of A o a Baach space ( W, ) such that kerπ = kerρ I Proof Let π : I L( X) be a cotuous TI represetato of the deal I o a Baach space X We shall descrbe the costructo of a cotuous represetato ρ : A L( W) ad the show that, for every k <, f π s k-tt, the so s ρ Let { } Y = π( I) X = π( b ) x : b I, x X ( ), = 23,,, wth orm = = = Let Z be the completo of ( Y, ) The the cluso map Y X s cotuous of orm at most π ad so exteds to a cotuous map θ: Z X Let ( W,) = ( Z,)/ ker θad deote by θ : W X the jectve map duced by θ We wsh to defe π : A L( Y) by ( ) π ( a) π( b ) x = π( ab ) x ( b I, x X ( ), = 23,,, ), { } y = f b x : y = π( b ) x as above = = but we frst eed to show that ths s well defed If m π( b ) x = π( b ) x, = j= = = j= j j the, for all c I, ( ) π( c) π( ab ) x = π( cab ) x = π( ca) π( b ) x = m = π( ca) π( b ) x j= m = π( cab ) j= j x j j j ( m ) = π( c) π( ab ) x j j Because π s TI, t follows that m π( ab ) x = π( ab ) x = j= j j

18 9 To prove that π( a) L( Y) for all a A ad that π s cotuous, we ote that, for every presetato y = π( b ) x of a gve elemet y Y we have π() a y = π( ab ) x ab x a b x It follows that = = = = We ca ow exted each π() a L( Y) to π() 2 a L( Z) by cotuty, thus defg a cotuous represetato π 2: A L( Z) For b I, ad y = π( b ) x Y we have that s, = π() a y a y π ( b)( y) = π( bb ) x = π( b)π( b ) x = π( b) y; = = 2 By cotuty (that s, π() 2 b L( Z), π() b L( X) ad θ: Z X beg cotuous) ad the fact that Y s dese Z, t follows that 2 π ( b)( y) = π( b)(θ( y)) π ( b)( z) = π( b)(θ( z)) ( z Z, b I) Let a A, z Z; the for all c I we have π( c)θ(π ( a) z) = π ( c)π ( a) z = π ( ca) z = π( ca)θ( z) Therefore, f θ( z) = 0 the π( c)θ(π 2( a) z) = 0 for all c I ad so, because π s TI o X, we have θ(π 2( a) z) = 0 Thus, for each a A, there s a well-defed mappg ρ( a) : W W such that ρ( a)( z+ kerθ) = π ( a) z+ ker θ Clearly ρ s a cotuous represetato of A o the Baach space W Equato () mples that ρ( b)(w) = π( b)(θ(w)) (w W, b I) (2) Now suppose that π s k-tt wth k < We show that ρ s k-tt Sce Y/ kerθ s dese W, t suffces to show that for every learly ( ) ( k) ( ) ( k) depedet w,,w W, every y,, y Y/ kerθ ad ε > 0 there s () j () j a a A wth ρ( a)w y < ε ( j k) We shall, fact, show that ths ca be doe wth a a I: thus (by (2)) we shall show that () j () j π( ax ) y < ε ( j k), () j () j () () k where x = θ(w ) Sce θ s jectve, the set { x,, x } s learly depedet We ca wrte y = b x j k, wth the b I, x X, by the smple expedet of makg the sets : 0 dsjot Because the 2 () j () j () j = π( ) ( ) () j Nj = { x } ()

19 92 P G DIXON orgal represetato π s k-tt, we ca fd c I ( ) so that () j () j π( c) x x < ε (, j k) b Let a = = bc The π( ax ) y = π( bc) x π( b) x () j () j () j () j = = () j () j = π( b)(π( c) x x ) = () j () j b c x x = <ε π( ) (by the defto of ) Fally, equato (2), together wth the fact that θ( W) = θ( Z) s dese X, shows that f b I, the ρ( b) = 0 f ad oly f π( b) = 0 Thus kerπ = ker ρ I 9 Stadard TI represetatos Let us recall that a stadard TI represetato of a ormed algebra A s a represetato π of A o a Baach space X whch s of the form π= φρ where φ s a cotuous homomorphsm of A oto a dese subalgebra of a Baach algebra B ad ρ s a strctly rreducble represetato of B o X Notce that there s o pot broadeg ths defto by droppg the completeess requremets o ether X or B, separately We have already remarked that f X were complete, or just a geeral lear space, the t would automatcally have a sutable Baach space structure If B were complete, we should requre that ρ be cotuous, order to make π cotuous Ths beg so, we eed oly complete B ad exted ρ to the completo to rega our orgal scearo Sce ρ may be factored through the prmtve algebra B/ ker ρ, we ca characterze S( A) as the tersecto of the kerels of the cotuous homomorphsms of Aoto dese subalgebras of prmtve (or semsmple) Baach algebras 9 OTR Hece S The map A S( A ), defed for Baach algebras A, s a s a topologcal radcal Proof Notato: throughout ths proof, B wll be a arbtrary semsmple Baach algebra, ad φmappg to B wll be a cotuous homomorphsm wth dese rage We shall use the charactersato of S( A) as the tersecto of the kerels of such mappgs φ: A B (2) Every φ: A B duces a cotuous homomorphsm φ : A/ S( A) B wth the same rage The tersecto of the kerels of these duced homomorphsms s zero, so S( A/ S( A)) ={ 0 } (3) If A, A2 are Baach algebras ad ψ: A A2 s a cotuous epmorphsm, the every φ: A2 B gves rse to a cotuous homomorphsm φψ: A B wth dese rage The fact that ψ ( S( A )) S( A ) follows mmedately 2

20 93 (4)(a) Let I be a closed deal of A ad let a A, b S() I ; we show that ab S() I, the case of ba beg smlar Gve φ: I B as above, we have φ( b) = 0 For arbtrary c I we have Sce φ( I) s dese B, t follows that Bφ( ab) ={ 0} ad hece, sce B s semsmple, that φ( ab) = 0 (4)(b) Let I be a closed deal of A Let φ: A B as usual Now I Ð A mples φ( I) Ð B (because φ( A) s dese B) Therefore φ( I) Ð B Therefore φ( I) s semsmple (the Jacobso radcal s heredtary) Thus φ: I φ( I) s a cotuous homomorphsm to a semsmple Baach algebra wth dese rage The fact that SI () SA ( ) Ifollows mmedately 92 The topologcal radcals S ad T ( ) are related to the closed-baer radcal β ad the Jacobso radcal J by the equaltes β T T T T S J ( m ) φ( c)φ( ab) = φ( cab) = φ( ca)φ( b) = 0 We ca summarze the relatoshps betwee our varous topologcal radcals as follows m Proof The equalty β T s Proposto 33 ad the other equaltes are trval The result follows by Theorem 6 The ma result of ths secto s that the last of these equaltes s proper EXAMPLE 93 We costruct a example of a radcal Baach algebra A whch has a jectve, dese embeddg φ to a semsmple Baach algebra B Hece, S( A) ={ 0 } The algebra B s the algebra regrettably called A [ 5] Let A0 be the algebra o symbols X, X2, subject to the followg relatos: every moomal X X cotag more tha occurreces of X r, where = max {,, r}, must vash It follows that X X = 0 f r ( + )!, where = max { r,, r} Ths algebra s gve a orm by λm = λ, = 0 = 0 where the λ are scalars ad the M moomals The Baach algebra B s the completo of ( A0, ) We shall costruct the radcal Baach algebra A as the completo of A0 a larger orm, so that there s a atural cotuous embeddg of A to B, whose rage cotas A0 ad s therefore dese For each moomal M = X X, we defe ( M) = max{,, } ad M = (( + ) ) k M M k We dstgush three cases (a) If ( M ( ) + )! k( k), the M Mk = 0 (b) If ( M ( ) + )! k( k+ 2 k), the M M = 0 r r ( +! )! Cosder a product of 2 moomals 2k, where k+ 2k

21 94 P G DIXON (c) If ether (a) or (b) hold, the there are at least two values of ( M ( ) + )! > kad t follows from the defto of M that k 2k 2k M M k M M for whch (3) I all three cases, (3) holds We exted the orm to geeral elemets of A 0 by defg λ M = λ M, = 0 = 0 where the λ are scalars ad the M moomals We the defe A to be the completo of A0 ths orm, whch s clearly detfed wth the set of fte sums = 0λM such that λ M < = 0 Hece the atural embeddg of A0 to B exteds to a atural embeddg of A to B It follows from (3) that k x x2k k x x2k k 2 / k for all x,, x2k A Sce ( k ) 0 as k, we see that A s topologcally lpotet, ad so a fortor radcal COROLLARY 94 Notce that the TI represetatos of A restrct to TI represetatos of the (complete) ormed algebra A0 Ths s terestg, sce A0 s locally lpotet ad hece l, but ot semprme Ths example prevets us from extedg Proposto 33 to the Levtzk ad l radcals There s a locally lpotet ormed algebra wth a separatg famly of cotuous TI represetatos It would be terestg to kow how geeral ths costructo ca be made, so as to produce a wde varety of S-semsmple, Jacobso-radcal algebras The followg theorem s a frst step that drecto 95 Let ( B, ) be a Baach algebra wth a creasg famly of lpotet subalgebras M ( = 23,,,) such that M = { 0} The there are a radcal Baach algebra ( A, ) ad a cotuous jectve homomorphsm φ: A B such that = M φ( A) B Proof Let M = = M For x M defe { k k } = x : f ν( ) m : x = m, m M, = = where the creasg sequece of postve real umbers ν( ) wll be defed later The fucto s clearly a orm o M If ν( ) for all the s submultplcatve ad x x for all x M Let A be the completo of ( M, )

22 95 ( ) ( 2N) Now cosder x,, x M wth ( ) ( ) x = m ( 2N), ( ) ( ) ( 2N) for some m M The m m = { 2 } Mr where r max,, N, so 2N ( ) ( 2N) ( ) ( 2N) x x max ν( ) m m ( ) ( 2N),, 2N ( ) ( 2N) ( ) ( 2N) 2N 2N ν( )ν( 2N) ( ) ( 2N) m m 2N ν( N) 2N ν( )ν( 2N) ( ) ( 2N) m m, 2N ν( N) 2N 2N where (4) holds because at least two of the j exceed N; for otherwse, the sequece m m would cota at least N cosecutve terms belogg to N M ad therefore m m = 0, sce M = { 0 } Thus N ( ) ( 2N) ( ) ( 2N) x x ν( N) x x for all x,, x M ad hece, by cotuty, A Takg ν( ) = makes A topologcally lpotet, ad hece radcal The equalty x x ( x M) shows that the atural embeddg φ : ( M, ) ( B, ) s cotuous ad therefore exteds to a cotuous homomorphsm φ: A B However, t s ot clear that φ: A B s ecessarly jectve If ot, we obta a jectve map by smply replacg A by A/ ker φ N (4) 0 Radcals complete algebras Our theory of topologcal radcals ca be developed equally well the cotext of complete ormed algebras: smply replace Baach algebra by ormed algebra throughout x 6 Secto 8 may be treated lkewse: the maps T are x topologcal radcals for ormed algebras I 9, the same recpe apples, except that the algebra B must rema Baach ad the equalty S J of Corollary 92 o loger apples (see Example 06 below) Ths alteratve theory has the advatage that ts axom (3) gves formato about the behavour of the radcal uder all cotuous homomorphsms, ot just those wth complete rage However, t has a major dsadvatage: t excludes the Jacobso radcal because the Jacobso radcal of a complete ormed algebra s ot ecessarly closed EXAMPLE0 Let B be the subalgebra of ( C[0, ], ) cosstg of the polyomals The B s algebracally somorphc to the subalgebra of [ X] cosstg of polyomals wthout costat term Therefore J( B) ={ 0 }, deed, B has a separatg famly of (dscotuous) trastve -dmesoal represetatos Now let A be the subalgebra of ( C[0, ], ) cosstg of all fuctos whch are polyomal o a eghbourhood of 0 There s a obvous homomorphsm of A oto B ad so the Jacobso radcal J( A) s cotaed the verse mage of { 0 }; that s, the set of fuctos vashg a eghbourhood of 0 The reverse

23 96 P G DIXON cluso s obvous: fact, f f A vashes o a eghbourhood of 0, the f s lpotet A Thus J( A) ={ f A: f 0 o a eghbourhood of 0}, whch s ot closed: J( A) s the deal of fuctos vashg at zero Ths example s very smlar to Example 64 As there, the quotet A/ J ( A) s the oe-dmesoal algebra wth zero multplcato, so J does ot satsfy axom (2), ad we have to go to J to obta a topologcal radcal However, ths s oly oe of the possble ways to get a topologcal radcal for ormed algebras whch reduces to the Jacobso radcal for Baach algebras I order to study these, we eed some basc formato about represetatos of complete algebras Ths s a lttle-studed topc It s well-kow that strctly rreducble represetatos of Baach algebras are trastve (whch meas that we do ot have to cosder radcals based o dfferet degrees of trastvty), but the followg varato whch the completeess hypothess s o the space rather tha the algebra seems ot to be avalable the lterature 02 Every strctly rreducble ormed represetato of a algebra o a Baach space s trastve Proof Let π : A L( X) be a strctly rreducble represetato of A o the Baach space X Let us wrte U( X) for the algebra of all lear edomorphsms of X The Schur s Lemma tells us that { S U( X) : Sπ( a) = π( a) S ( a A) } s a dvso algebra We are terested the set π( A) ={ S L( X) : Sπ( a) = π( a) S ( a A) } The every S π( A) s bjectve, ad therefore, by Baach s Isomorphsm Theorem, has a verse π( A) Thus π( A) s a ormed dvso algebra By the Gelfad Mazur Theorem, π( A) cossts of the scalar multples of the detty The remader of the proof s stadard (see [ 4, Theorem 423]) REMARK 03 If π : A L( X) s a cotuous strctly rreducble represetato of a ormed algebra A o a Baach space X, the we may regard π: A π( A) as a cotuous homomorphsm of A to the Baach algebra π( A) ad the detty map π( A) L( X) as a strctly rreducble represetato of π( A) Thus π s stadard I lookg for topologcal radcals of ormed algebras correspodg to the Jacobso radcal, let us restrct our atteto to those based o cotuous represetatos o Baach spaces The atural cocept to defe s the followg DEFINITION 04 For a ormed algebra, let deote the tersecto of A I( A) the kerels of the cotuous strctly rreducble represetatos of A o Baach spaces If A s Baach, J( A) = I( A) We shall show that I s a (heredtary) topologcal radcal, dfferet from J

24 97 05 The mappg I s a heredtary topologcal radcal Proof The proof proceeds as Theorem 8, the aalogue of Lemma 82 gog as follows If I s a closed deal of a ormed algebra A ad π s a cotuous strctly rreducble represetato of I o a Baach space X, we show that there s a cotuous strctly rreducble represetato πˆ of A o X such that ker π = ker πˆ I Let A be the completo of A ad let I be the closure of I A The I s a closed deal A Sce π s cotuous, t exteds to a cotuous strctly rreducble represetato π : I L( X) We use the usual algebrac method to exted ths to a represetato πˆ mappg A to the algebra of edomorphsms of the vector space X: we choose ay o-zero x0 X; every y X may be wrtte the form y = π( b) x0 for some b I; for a A we defe π( ˆ a) by π( ˆ a) y= π( ab) x0 It s easy to check that πˆ s a well-defed homomorphsm The restrcto πˆ A to A s strctly rreducble, sce t s a exteso of the strctly rreducble represetato π It remas to show that π( ˆ a) L( X) ad that πˆ s cotuous The mappg b π( b) x 0: I X s cotuous, surjectve, ad therefore ope Thus, there s a costat K > 0 such that for every y X there s a b I wth b K y such that y = π( b) x The Thus π( ˆ a) L( X) ad πˆ s cotuous of orm at most K π x m 0 ˆ π( a) y π ab x K π a y x ( a A, y X), 0 0 ˆ π( a) K π a x ( a A) We have J( A) I( A) for all ormed algebras A Therefore, by Theorem 6, J I The followg example shows that we ca have J I EXAMPLE06 Let B be the subalgebra of ( C[0, ], ) cosstg of the polyomals dscussed Example 0 The J( B) ={ 0 }, ad so J ( A) ={ 0 } O the other had, ay cotuous TI represetato of B would exted by cotuty to a TI represetato of ( C[0, ], ), whch s mpossble, sce β( C[0, ] ) = C[0, ] Therefore T ( B) = B, so I( B) = B 07 Ths example shows that to make Theorem 92 work for complete ormed algebras we have to replace J by I: related by β T T T T S I ( m ), Proof Every cotuous, strctly rreducble represetato π : A L( X ) exteds to a rreducble represetato π : A L( X), ad s therefore a stadard represetato of A Thus S I ad t follows from Theorem 6 that S I The other equaltes are obvous 0 The topologcal radcals of geeral ormed algebras are β J I 0