Maps on Triangular Matrix Algebras

Transcription

1 Maps o ragular Matrx lgebras HMED RMZI SOUROUR Departmet of Mathematcs ad Statstcs Uversty of Vctora Vctora, BC V8W 3P4 CND sourour@mathuvcca bstract We surveys results about somorphsms, Jorda somorphsms, Le somorphsms, ra oe preservers ad other maps o tragular matrx algebras ey-words Isomorphsm, Jorda maps, Le maps, Commutatvty preservers, Ra oe, ragular algebras Itroducto By M ( R), respectvely ( R), we deote the algebra of all matrces, respectvely all upper tragular matrces, over a rg R (ll rgs cosdered here are assumed to have a detty ) For every fte sequece of postve tegers, 2,,, satsfyg 2 =, we assocate a algebra (,, 2, )( R) cosstg of all matrces over R of the form = M where s a matrx We call such a algebra a bloc upper tragular matrx algebra Whe = for every, ths algebra s ust I ths expostory artcle, we are cocered wth characterzg specal maps o tragular matrx algebras, partcular algebra somorphsms, as well as Jorda ad Le somorphsms lso commutatvty preservg maps ad ra oe preservg maps 2 lgebra Isomorphsms ad Jorda Isomorphsms Recall that a automorphsm of a algebra s called er f t s of the form X a X, for a vertble elemet lthough ot every algebra automorphsm of the full matrx algebra M ( R) over a commutatve rg R s er (see [7]), ezla [9] proved that every algebra automorphsm of ( R) s er related result s [8] he case of bloc upper tragular algebras s qute dfferet Ideed M ( R) s tself a tragular matrx algebra ad, as stated before, there are outer M R However, f R automorphsms of ( ) s such that every automorphsm of ( R) M s er, (eg, f R s a uque factorzato doma), the ([4]) every automorphsm of,, 2, R s also er ( )( )

2 By a Jorda somorphsm betwee assocatve algebras, we mea a bectve mappg ϕ satsfyg ( B B) = ϕ( ) ϕ( B) ϕ( B) ϕ( ) ϕ Every somorphsm ad at-somorphsm s evdetly a Jorda somorphsm s may famlar algebras, every Jorda automorphsms of ( R), for certa rgs R, s ether a automorphsm or a atautomorphsm hs s proved [2] whe R s the rg of complex umbers ad [2] whe R s a 2-torso free commutatve rg havg o dempotets other tha 0 ad hs s also the case for bloc tragular matrx algebras (,, 2, ) over a feld [3] 3 Le Isomorphsms Le somorphsm from a assocatve algebra to a assocatve algebra B s a bectve mappg ϕ : B such that ( ab ba) = ϕ( a) ϕ( b) ϕ( b) ϕ( a) ϕ We ote that f α s a (assocatve) somorphsm or the egatve of a at-somorphsm from to B ad γ s a lear map from to the cetre of B, such that γ ( ab ba) = 0 for every a ad b, the α γ s a Le somorphsm, provded t s ectve he cotet of the followg heorem s that the coverse holds for tragular matrx algebras over felds We deote by tr(), the trace of the matrx, ad by F, a arbtrary feld he followg result f proved [] heorem Let = (, 2,r )( F ) ad B = ( m, m2,ms )( F ) be bloc upper tragular algebras M ( F ) ad M m ( F ) respectvely If ϕ s a Le somorphsm from oto B, the m =, r = s ad there exsts a vertble matrx B B ad a lear fuctoal τ o satsfyg τ ( I) such that ether (a) ϕ ( ) B B τ( )I or =, ad =, (b) ϕ ( ) = B B τ( )I m, ad =, m r where a s the "traspose" relatve to the at-dagoal (e, the dagoal that cota the postos (, ) he mappg τ s gve by τ ( ) = tr( D), where D s a dagoal matrx such that tr( D ) ad the dagoal etres every oe of the blocs that determe are detcal, for a rg R havg o otrval dempotets, are characterzed [6] ad for more geeral rgs [3] he results are too techcal to state here Le automorphsms of ( R) lear map ϕ from a algebra to a algebra B s sad to be commutatvty preservg f ϕ ( a) commutes wth ϕ ( b) for every par of commutg elemets a, b O the face of t, ths s a sweepg geeralzato of the cocept of Le somorphsm Nevertheless, may famlar algebras t turs out that these maps dffer oly slghtly from Le somorphsms Evdetly every Le somorphsm ϕ betwee algebras ad B preserves commutatvty both drectos, as does the map c ϕ τ for a o-zero scalar c ad a lear map τ from to the cetre of B hat the coverse s true whe = B = ( F ), for a feld F, s a restatemet of results that appear [0] ad [2], whch we state presetly

3 heorem 2 ([0], [2]) Let F be a feld, ad let ϕ be a lear mappgs from ( F ), 3, to tself he followg codtos are equvalet (a) ϕ preserves commutatvty both drectos (b) here exsts a o-zero scalar c F, a lear fuctoal f o ad a vertble matrx S such that ϕ taes oe of the followg forms ϕ( ) = cs S f ( )I or ϕ( ) = cs S f ( )I (c) here exsts a Le somorphsm α of F, a o-scalar c F ad a lear ( ) mappg g from ( F ) to ts ceter such that ϕ = c α g he above result s false for = 2 4 Ra Oe Preservg Maps map ϕ from a space S of matrces to a space S 2 of matrces s sad to preserve matrces of ra oe f ϕ ( ) s of ra oe wheever has ra oe It s sad preserve ra oe matrces both drectos whe ϕ ( ) s of ra oe f ad oly f has ra oe Characterzg lear maps o spaces of matrces or operators that preserve ra 0e operators has bee a actve area of research for qute a whle Of all the so called "lear preserver problems", ths s arguably the most basc Ideed several other questos about preservers may be reduced to, or solved wth the help of, raoe preservers We ow descrbe certa maps that are "buldg blocs" of the most geeral addtve maps that preserve ra oe By a left multplcato o a algebra we mea a mappg L defed by ( x) ax L a =, for every x, where a s a elemet of Rght multplcatos are defed aalogously a Ra ssume that c a c~ s a automorphsm of the uderlyg feld F, ad C = [ c ] M m ( F ) We deote the matrx [ c~ ] by C ~ ~ Evdetly the map C a C ~ preserves every ra We say that C a C s the map duced o the space of matrces by the feld-automorphsm c a c~ other two types of ra oe preservers are the followg Let each of f, f 2, f be a addtve mappg from F to F such that f s f = f f, Defe bectve, ad let ( ), 2 a mappg fˆ o a tragular algebra = ( ), wth =, by ([ a ]) = [ ] fˆ, b where b = f ( ), b a f ( ) a a b =, for 2 f =, ad a hs s a surectve addtve mappg o ad t preserves ra oe matrces, but oly whe = For f ad f, f 2, f as above, defe a mappg f ( o a tragular algebra = ( ), wth =, a smlar fasho except that the "acto" s o the last colum stead of the frst row, more ( precsely f ( C ) = ( fˆ ( C )) ga ths s a addtve mappg o preservg ra oe matrces, but oly whe =

4 We ow preset a result from [] Specal cases of ths are obtaed [5] ad [2] heorem 3 ([]) Let ( ) bloc upper tragular algebra ( F ) such that ( F ) Let : = be a M, 2 ϕ be a surectve addtve mappg that preserves ra oe matrces he ϕ s a composto of some or all of the followg maps: () Left multplcato by a vertble matrx (2) Rght multplcato by a vertble matrx ~ (3) he map C a C, duced by a feld automorphsm a a a~ of F (4) he map fˆ defed 3 above, but oly whe = (5) he map f ( defed 32 above, but oly whe = (6) he traspose wth repsect to the atdagoal a hs s preset oly whe =, e, = for every Corollary If ϕ s as heorem 3, the: (a) ϕ s ectve; (b) ϕ preserves every ra, e, ra ϕ ( ) = ra, for every Remar Whe ϕ s lear, the obvously maps of type (), (v), ad (v) heorem 3 caot be preset ad so ϕ s a composto of a left multplcato, a rght multplcato, ad possbly the map a hs s actually true for maps o spaces of matrces much more geeral tha tragular algebras, see [] Refereces: [] J Bell ad R Sourour, ddtve ra-oe preservg mappgs o tragular matrx algebra, Lear lgebra ppl 32 (2000), 3-33 [2] Bedar, M Brešar ad M Chebotar, Jorda somorphsms of tragular matrx algebras over a coected commutatve rg, Lear lgebra ppl 32 (2000), [3] C Cao, utomorphsms of certa Le algebras of upper tragular matrces over a commutatve rg, J lgebra 89 (997), pp [4] WS Cheug ad R Sourour, utomorphsms of tragular algebras, preprt [5] W L Cho ad MH Lm, Lear preservers o tragular matrces, Lear lgebra ppl 269 (998), [6] D Doovæ, utomorphsms of the Le algebra of upper tragular matrces over a coected commutatve rg, J lgebra 70 (994), 0-0 [7] Isaacs, utomorphsms of matrx algebras over commutatve rgs, Lear lgebra ppl 3 (980), pp25-23 [8] Jødrup, utomorphsms ad dervatos of upper tragular matrx rgs, Lear lgebra ppl 22 (995), pp [9] P ezla, ote o algebra automorphsms of tragular matrces over commutatve rgs, Lear lgebra ppl 35 (990), pp8-84

5 [0] L Marcoux ad R Sourour, Commutatvty preservg lear maps ad Le automorphsms of tragular matrx algebras, Lear lgebra ppl 288 (999), [] L Marcoux ad R Sourour, Le somorphsms of Nest lgebras, J Fuct al 64 (999), [2] L Molar ad P Semrl, Some lear preserver problems o upper tragular matrces, L Multlear lg 45 (998), [3] R Sourour, Jorda somorphsms of rgs of bloc tragular matrx algebras, preprt