IRREDUCIBLE COVARIANT REPRESENTATIONS ASSOCIATED TO AN R-DISCRETE GROUPOID

UPB Sc Bull Sere A Vol 69 No 7 ISSN 3-77 IRREDUCIBLE COVARIANT REPRESENTATIONS ASSOCIATED TO AN R-DISCRETE GROUPOID Roxaa VIDICAN Ue perech covarate poztv defte ( T ) relatv la u grupod r-dcret G e poate aoca prtr-o teoremă de tp Steprg o reprezetare covartă ( U ) Vom tabl o codţe eceară ş ufcetă de reductbltate petru aceata d urmă Let ( T ) be a potve defte covarat par wth repect to a r- dcrete groupod G Ug a theorem of type Steprg we ca aocate to the par ( T ) a covarat repreetato ( U ) I th artcle we provde a eceary ad uffcet codto of rreducblty for the repreetato ( U ) Keyword: r-dcrete groupod potve defte covarat par covarat repreetato Itroducto For a utable ameable r-dcrete prcpal groupod G wth the ut pace G ad the emgroup I of t compact ad ope G-et we defe a covarat repreetato of I to be a par ( T ) where () a -repreetato of C ( G ) ( the C -algebra of complexvalued fucto o G that are cotuou ad vahg at fty) o a (complex ad eparable) Hlbert pace H; () T { T ( ) I} a famly of operator o H uch that T ( ) ad T ( ) T T ( for t I; () T ( ) ( ( a ) T ( ) for a C ( G ) ad I (ee Ob (v; ad (v) T ( ) ( ℵ ) for I uch that G (where ℵ deote the charactertc fucto of ) At Departmet of Mathematc II Uverty Poltehca of Bucharet ROMANIA

8 Roxaa Vdca Defto If T a fucto from I to B (H ) ad a - repreetato of C ( G ) o H the ( T ) a potve defte covarat par f the followg codto are atfed: () T ( ) ( ( a ) T ( ) for a C ( G ) I; () T ( ) T T ( for t I; () T ( ) ( ℵ ) for I wth G ; ad (v) for each fte collecto of pot I the operator matrx ( T ( actg o H H o-egatve (th mea that < T ( ) ξ ξ > H for ay ( ξ ξ ) H H ) Theorem Let ( T ) be a potve defte covarat par wth value B (H ) The there are a Hlbert pace H a covarat repreetato ( U ) of I o H ad a Hlbert pace omorphm V mappg H to H uch that ( a ) V ( V for a C ( G ) ad T V U V for t I e hall how a eceary ad uffcet codto of rreducblty for the covarat repreetato ( U ) from the above theorem term of the par ( T ) Prelmare The defto for the oto of: ameable r-dcrete prcpal groupod G- et C -algebra aocated wth a locally compact groupod ca be foud [] [] [3] [4] or [5] Now to udertadg more ealy the cotet of th paper we hall preet frt of all the mot mportat tage of the proof of the theorem from Itroducto Let H be the et of all fucto f : I H uch that () there a compact et K f (depedg of f ) uch that K f G ad f for t K f Ø ; ad () ft t I uch that t t the f t ) ( ) f ( ) (where d t ( ℵ d ( t ) t

Irreducble covarat repreetato aocated to a r-dcrete groupod 9 e defe o H a equlear fuctoal < > by the formula < f g > < T ( f g( ) > H where the um over ay two fte et { t }{ } I uch that K f t K g ad uch that t t k l Ø f k l Let N { f H < f f > } The equlear fuctoal < > from above pae to a er product o H / N ( For th er product we keep the otato < > ) Let H be the completo of H / N the aocated orm Next we defe V : H H by V ξ ξ + N where ξ ( t ) ( E( ℵ ξ ( E beg the codtoal expectato from C ( G) to t C ( G ) C ( G ) ; ee [4] p4) Th operator V a ometry Note that for f H ad t I f we deote f ( ) f ( t t ) I we obta f t H Now for q I the operator U o H / N defed by U ( q)( f + N) f q + N f H atfe the properte U ( q ) U ( q) ad U ( U ( ) U for q t I Fally for a C ( G ) we defe ( o H by the formula ( ( f ) ( a t ) f Deote ( a )( f + N) ( f + N a -repreetato of C ( G ) o H The par ( U ) ad the operator V have the properte aked the theorem Obervato () ( U ) ut a potve defte covarat par: let ( ξ ξ ) H H ad I; the T ( )ξ ξ U ( ) Vξ Vξ From here ug V ( H) H we deduce that ( U( a potve operator matrx

Roxaa Vdca () A V a ometry t reult V V I H Sce V a omorphm there ext V ad V V V V I hece H ( V V ) V The VH H mple V V Coequetly T V U V VT V U t I () If a o-degeerate repreetato of C ( G ) the the famly { T ( )} I o-degeerate ( e Sp{ T ( ) ξ Iξ H} H ) ad o { U ( )} alo: let η H ad { a } C ( G ){ ξ } H uch that I η lm ( a ) ξ ; fx { } I uch that : upp a The η lm ( ℵ a ) ξ lm ( ℵ ) ( a ) ξ lmt( c ) ( a ) ξ For the ecod part of the aerto : ce V a ometry V V are cotuou operator Therefore H V ( H ) V ( Sp( T ( I) H Sp( VT ( I)( H Sp( VT ( I) V ( H Sp( U ( I) H ) (v) If ( T ) a potve defte covarat par the T ( ) T ( ) I: from the proof of the theorem we kow that U U ( t ) t I hece T V U ( t ) V T ( t ) t I (v) If ( T ) a potve defte covarat par the T () I : let I ; the we have T () T ( ) T( ) T ( ) T ( ) T ( ) ( ℵ ) ℵ (v) For a C ( G ) ad I we have a ℵaℵ ( where from the rght de a G-et whle from the left de a otato for the fucto u d( u ) ; u the elemet x wth r ( x) u ) hece a ( a ) ( ℵ aℵ ) ( ℵℵ ) a( ℵℵ ) ℵ aℵ a Now: ( T ( ) ( T( ) T ( ) ( ( ℵ ) T ( ) ( a ( T ( ) ( T( ) ( a ) T ( ) T ( ) ( a 3 Irreducble repreetato of C ( ) ) Lema If ( T )( T ) are two potve defte covarat par of I o B (H ) wth o-degeerate repreetato of ( C G ) uch that N ad I: ( T ( ( T ( G

Irreducble covarat repreetato aocated to a r-dcrete groupod whle ( U H) repectvely ( U H ) are the correpodg covarat repreetato the there ext a cotracto B( H H) ( e ) atfyg: () V V ; () U ( ) U( ) I; ad () a ( ) ( a C ( G ) T ( a t ) T ( a t ) t I a C ( G ) Proof: () Let h h H ad I The: U ( ) Vh V U ( ) V h h T ( ) h h T ( ) h h U ( ) Vh Defg : H H by U Vh U( Vh we remark that a cotracto wth V V () For t I ad h H we have U ( ) U Vh U ( Vh U( Vh U( ) U( Vh U( ) U Vh () If a C ( G ) t I ad h H t follow a U t V h U t ( ) ( ) ( ) ( a t ) Vh U V ( a t ) h U( V ( a t O the other had a U t V h a U t V h U t ( ) ( ) ( ) ( ) ( ) ( a t ) V h U V ( a t ) h Hece a ( ) ( V U V ( a t ) h V U V ( a t ) h T ( a t ) h T ( a t ) h Defto Let H be a Hlbert pace ad U B(H ) U a potve operator ( ad we hall deote th by U ) f U U ad U ( x) x x H Notato 3 For a Hlbert pace H ad a et M uch that M B(H ) we wrte M for the commutat of M e M { U B( H ) V M : UV VU} ) h

Roxaa Vdca Lemma 4 Let ( T ) a covarat potve defte par o B (H ) wth a o-degeerate repreetato of C ( G ) ( U ) the correpodg covarat repreetato o B (H ) ad V the omorphm betwee H ad H uch that ( a ) V ( V a C ( G ) ad T V U V t I For U ( I) ad S ( C ( G we defe the applcato Φ : I B( H ) by Φ V U V t I ad Ψ S : C ( G ) B( H ) by S ( V S Ψ ( V a C ( G ) Uder thee codto the followg aerto are true: () the map Φ ad S ΨS are lear ad ectve; () f U ( I) ad the for N ad I the operator matrx ( Φ ( o-egatve; ad () f U ( I) uch that ad N I the operator matrx Φ ( o-egatve the ( Proof: () It uffce to how that the lear map Φ ectve For the ectvty of S ΨS the proof aalogou e aume that Φ Let t I ad h k H The: U Vh U ( ) Vk U ( ) U Vh Vk V U ( Vh k Φ ( h k hece (Oe apple Ob ( () e ue the obervato () from Prelmare ad the followg theorem: If A a volutve algebra H a Hlbert pace π a repreetato of A o H ad T B(H ) uch that T ad T π ( A) the there ext K B(H ) wth K K K T ad K π ( A) Thu for U ( I) wth there K B(H ) uch that K K K ad K U ( I) Let I ad ( ξ ξ ) H H The requred cocluo follow by Φ ( ) ξ ξ V U ( ) Vξ ξ K U ( ) Vξ Vξ U ( ) KVξ KVξ ( Oe ue ob ( () Let a the tatemet of the lemma For N I h h H ad k U( ) Vh + + U ( ) Vh we get

Irreducble covarat repreetato aocated to a r-dcrete groupod 3 hece k k V U ( ) U ( ) Vh h V U ( ) Vh h Φ ( ) h h Lemma 5 Let ( T ) ad ( T ) two potve defte covarat par wth o-degeerate repreetato of C ( G ) uch that N I ( T ( ( T ( ad : t I a C ( G ) : T ( a t ) T ( a t ) If ( U ) the covarat repreetato wth repect to ( T ) the there U ( I) ( C ( G wth the properte: I ( I : H H the detty operator I ( h) h h H ) T Φ ad Ψ Proof: Let V : H H the omorphm correpodg to the par ( T ) ( U ) the covarat repreetato ad V : H H the omorphm aocated wth the par ( T ) The lemma aure the extece of a cotracto : H H uch that V V U ( ) U ( ) I ad ( ( a C ( G ) e hall deote The ad for h H we have: h h h h h h hece I Chooe I The U ( ) U( ) ( U ( ) ) ( U( ) ) ( U ( U ( ) U ( ) mple U ( I) Aalogouly ( C ( G Suppoe that t I ad h k H Sce Φ h k U( Vh Vk U Vh Vk U Vh Vk ( U Vh Vk V U( Vh k T h k T Φ Smlarly Ψ Obervato 6 () If a G-et the are alo G-et ad at the ame tme they are ubet of G Moreover ( If x the x yu wth y ad u hece x y Coverely for x we have x xx x )

4 Roxaa Vdca () The operator from lemma 5 atfe a follow from T Φ T ( Φ ( t I Φ ( ) Φ T ( ) T T ( Φ ( V U ( ) VV U Vh V U ( Vh t I h H V U ( Vh V U ( Vh t I h H U ( Vh U ( Vh t I h H U ( ) h U ( ) h h H U h U h I ( ) ( ) I h H Propoto 7 Let ( T ) a potve defte covarat par The there a becto betwee the et A { U ( I ) ( C ( G I } ad the famly B of the potve defte covarat par ( S θ ) o B (H ) whch have the property N I : ( ( T ( ad ( S S ( a t ) S( θ ( a t ) a C ( G ) t I Proof: Aume A By lemma 4 for N ad t t I the operator matrx ( Φ ( t t o-egatve Becaue I we ca clam that N ad t t I : ( Φ I ( t t a o-egatve matrx hece Φ T Ug t clear that Φ ( Φ ( ) Φ t I ad that Ψ a repreetato of C ( G ) For I uch that G we have Φ ( ) V U ( ) V V ( ℵ ) V Ψ ( ℵ ) Next for a C ( G ) ad I t follow Φ Ψ a V U VV ( ) ( ) ( ) ( V V U ( ) ( V V ( a ) U ( ) V V ( a ) VV U ( ) V Ψ ( a ) Φ ( ) Fally f t I ad a C ( G ) the Φ ( a t ) V U V( a t ) ad Φ Ψ ( a t ) V U V V ( a t ) V V U( ( a t ) V V U( V ( a t ) Thu Φ ( a t ) Φ Ψ ( a t ) All thee mply ( Φ Ψ ) B By the lemma 4 the map ( Φ Ψ ) ectve It urectvty follow from lemma 5 ad obervato 6 ()

Irreducble covarat repreetato aocated to a r-dcrete groupod 5 Propoto 8 If ( T ) a o-ull potve defte covarat par uch that for every ( S θ ) B there λ C wth S λt the the covarat repreetato ( U ) aocated wth ( T ) rreducble ( the ee that oly the ubpace ad H are cloed ad varat wth repect to U (I) ad ( C ( G Proof: Let ( T ) ad ( S θ ) B a the tatemet of the propoto By lemma 5 we ca fd A uch that S Φ ad θ Ψ Coequetly there λ C uch that Φ λt It reult: V U( V λ T t I U λvt V t I U λ U t I λi Let K a cloed ubpace of H uch that K varat wth repect to U (I) ad ( C ( G For h H we coder the wrttg h h + h where h K ad h K e hall deote wth P K the proecto o K P K : H H P K h h It rema to how that P K A ( A from propoto 7) The t wll ext λ C uch that P K λi Th equvalet wth K or K H For k K h K ad I we have U ( ) k K hece U ( ) h k h U ( ) k from where U ( ) h K Thu f h H uch that h h + h wth h K ad h K we ca wrte PK U( ) h PK ( U( ) h + U( ) h U( ) h U( ) PK h e P K U( I) Aalogouly t reult that P ( ( K C G For h H wth the ame decompoto a above we fd PK h PK h h PK h P K h h h h + h h h ad ( P K I) h h h h h h h h h Hece P P ad P K I K K Propoto 9 If ( T ) a o-ull potve defte covarat par uch that a o-degeerate repreetato of C ( G ) ad ( U ) rreducble the for ( S θ ) B there λ C uch that S λt Proof: Chooe ( T ) ad ( U ) wth the properte from the tatemet ad ( S θ ) B By lemma 5 there a operator A uch that S θ ) ( Φ Ψ ) (

6 Roxaa Vdca Let K (H ) The ( H ) { h H k H aî h k} { h H h k} a lear cloed ubpace of H Moreover t varat wth repect to U (I) ad ( C ( G Becaue ( U ) rreducble we deduce that (H ) or H hece λi for ome λ C Fally t I : S( Φ V λ U V λt R E F E R E N C E S [] PMuhly BSolel Subalgebra of groupod C -algebra J ree ud agew Math 4 (989) pp4-75 [] P Muhly Coordate Operator Algebra 997 preprt [3] A Patero Groupod Ivere Semgroup ad ther Operator Algebra Progre Mathematc vol7 Brkhäuer 999 [4] J Reault A groupod Approach to C -algebra Lecture Note Mathematc vol793 Sprger Verlag 98 [5] Roxaa Vdca C -ubalgebra of thec -algebra aocated wth a r-dcrete groupod UPB Sc Bull Sere A vol 64 pp3-38 [6] m B Arveo Subalgebra of C -algebra Acta Math 3 (969) pp4-4 Th reearch wa upported by grat CNCSIS (Romaa Natoal Coucl for Reearch Hgh Educato) code A 65/6