STRUCTURED TRIANGULAR LIMIT ALGEBRAS

Transcription

1 STRUCTURED TRIANGULAR LIMIT ALGEBRAS DAVID R. LARSON nd BARUCH SOLEL [Received 5 Februry 1996 Revised 24 September 1996] 1. Introduction Suppose is unitl Bnch lgebr which contins n incresing chin n n 1 of finite-dimensionl sublgebrs, ll contining the unit I of, such tht is the closure of the union! n n. Then cn be viewed s direct limit of the lgebrs n. It frequently occurs in this setting tht there is sequence of bounded mppings n : 5 n which stisfy the conditionl expecttion property n ( CAB ) C n ( A ) B whenever C, B n, nd with the further property tht n converges to the identity mpping of 5 in the strong opertor topology (tht is, n ( A ) 5 A for A ). Indeed, if ll the n re C*-lgebrs, so is n AF (pproximtely finite-dimensionl) C*-lgebr, this is lwys the cse. This property holds in mny non-selfdjoint settings s well. In some specil cses, not only do the n exist but they cn be ten to be multiplicti e on. So the n re unitl homomorphisms of into itself. This implies very strong structurl properties of the limit lgebr. Non-trivil instnces of this type of highly - structured property re not possible in the C*-setting unless, for instnce, the lgebrs re belin. However, certin of the more interesting nd widely studied of the tringulr UHF (uniformly hyperfinite) lgebrs exhibit precisely this behviour. The purpose of this rticle is to investigte nd prtilly clssify this fmily. We begin by estblishing proper frmewor for this study. 2. Preliminries A unitl C*-lgebr is clled n AF lgebr if it cn be written s direct limit of finite-dimensionl C*-lgebrs lim( B Å 5, ). Here ech B is finite-dimensionl C*-lgebr nd : B 5 B 1 is unitl injective -homomorphism. If ech B is the lgebr of n n complex mtrices, denoted M n or M n ( ), then is clled UHF lgebr. Due to the wor of Strtil nd Voiculescu [ 11 ] we now tht, given n AF lgebr, we cn find B, defining s direct limit, nd fix system of mtrix units in ech B, s B is isomorphic to sum of full mtrix lgebrs, in such wy tht, for ech, mps the digonl mtrices of B into the digonl mtrices of B 1 nd, for every mtrix unit e in B, ( e ) is sum of mtrix units in B 1. In such cse the mps will be sid to be cnonicl nd lim( B Å 5, ) is cnonicl presenttion of. This presenttion will in generl be non-unique. Given cnonicl presenttion s bove, we cn set D lim( C Å 5, ) where C is the lgebr of ll digonl mtrices in B. Then D is ms in. Such ms The wor of the first uthor ws supported in prt by NSF grnt DMS The second uthor ws prticipnt in the worshop in Liner Anlysis nd Probbility, Texs A & M University. His reserch ws supported in prt by Fund for Promotion of Reserch t The Technion Mthemtics Subject Clssifiction : 47D25, 46J35. Proc. London Mth. Soc. (3) 75 (1997)

2 178 DAVID R. LARSON AND BARUCH SOLEL is clled cnonicl ms. We lso cll norm-closed sublgebr of cnonicl if it contins cnonicl ms D. If, in ddition * D, is sid to be cnonicl tringulr AF lgebr. In this pper we will del only with cnonicl lgebrs nd, therefore, every tringulr AF or tringulr UHF lgebr will be ssumed to be cnonicl, nd we shll simply write TAF or TUHF. In fct, except in the lst section when we consider TAF lgebrs, we will be interested only in TUHF lgebrs tht cn be written s lim( T Å 5 n, ) where lim( M n Å 5, ) is cnonicl presenttion of UHF lgebr with the property tht ech mps the upper tringulr mtrices in M n, denoted T n, into the upper tringulr mtrices in M n. These lgebrs re clled in the 1 literture strongly mximl tringulr in fctors (see [ 7, p. 105 ; 8, Exmple 2. 12]). Given n AF lgebr with cnonicl ms D, we cn express s C *( G ) for some r -discrete menble principl groupoid. In fct the groupoids of the ind we re considering here my be viewed s equivlence reltions on X, the mximl idel spce of D, hving countble equivlence clsses. Hence G cn be viewed s subset of X X with topology tht my be finer thn the product topology. For detils bout groupoids nd the ssocited C*-lgebrs see [ 10 ]. Groupoids for AF lgebrs re treted in [ 10, Chpter III, 1]. Elements of the AF lgebr re viewed s continuous functions on G nd the multipliction nd djoint opertions re defined s follows : ( fg )( u, ) w f ( u, w ) g ( w, ), ( w,u) G f *( u, ) f (, u ). For f we write supp f ( u, ) G : f ( u, ) 0. We hve D f : supp f where ( u, u ): u X. In fct we shll identify with X (nd u X with ( u, u ) G ). Given cnonicl sublgebr D, there is n open subset P G such tht (i) P ; (ii) P P P ; tht is, if ( x, y ), ( y, z ) re in P then ( x, z ) is in P nd f : supp f P (we then write ( P )) (see [ 4 ]). The lgebr ( ( P )) is tringulr if nd only if (iii) P P 1 (where P 1 ( u, ): (, u ) P ). For strongly mximl tringulr lgebr we lso hve (iv) P P 1 G [ 5, Theorem 1. 3]. Whenever (i) (iii) re stisfied, P induces prtil order on every equivlence clss (we write [ u ] for the equivlence clss of u X ). The condition (iv) mens tht this order is totl. When we write u it will lwys be ssumed tht ( u, ) P. Since we del mostly with the cse where is UHF lgebr, note tht in this cse ech equivlence clss is dense in X (tht is, the groupoid is miniml). Given cnonicl presenttion lim( M Å 5 n, ) s bove, we write the mtrix units of ( ) s e ij. Viewing s C *( G ), we cn view ech e ij ( ) s function on G. In M n

3 STRUCTURED TRIANGULAR LIMIT ALGEBRAS 179 fct it is chrcteristic function of some open nd closed subset ê ( ij ) G (if i j, ê ( ij ) cn be viewed s subset of X ). We lso write Ê for the support of E whenever E is sum of mtrix units. With this nottion ( e ij ( ) ) is sum of mtrix units in M n nd e 1 l,m is one of these mtrix units if nd only if ê l,m ê ( ij ). Given point ( u, ) G there re e ( i ),j with ê ( ) i,j ê i 1 j such tht ( u, ) 1 " ê ( i ) ( ),j, nd ice ers. Every such intersection with ê i,j ê i 1,j defines 1 point in G. For more detils bout tringulr AF lgebrs nd the ssocited prtil orders see [ 9 ]. Finlly, let us recll tht UHF lgebrs lim( M Å 5 n, ) re chrcterized by the supernturl number ssocited with the sequence n [ 2 ]. Given supernturl number (lso clled generlized integer) n, we write M ( n ) for the ssocited UHF lgebr. 3. Structured lgebrs In this section we study clss of tringulr UHF lgebrs. In order to define this clss we first introduce the following definitions. D E F I N I T I O N S. A mp : M ( ) 5 M n ( ) is clled cnonicl embedding if (1) it is n injective unitl -homomorphism; (2) ( T ) T n (where T n M n ( ) is the sublgebr of ll upper tringulr mtrices) ; (3) mps ech mtrix unit of M ( ) into sum of mtrix units in M n ( ). Then is clled structured cnonicl embedding if, in ddition, (4) there is contrctive conditionl expecttion from M n ( ) onto ( M ( )) such tht, restricted to T n, is multiplictive. L EMMA For cnonicl embedding : M ( ) 5 M n ( ), condition (4) is equi lent to (4 ) there is contrcti e mp g : M n ( ) 5 M ( ) such tht g id M nd g 3 T n is multiplicti e. Proof. Given s in (4) we define g 1 (here 1 : ( M ( )) 5 M ( )). It clerly stisfies (4 ). Conversely, given g s in (4 ) we let g nd then g g id g, so tht is indeed conditionl expecttion onto ( M ( )). If the bove holds, we cll g the bc mp of. L EMMA For cnonicl embedding : M ( ) 5 M n ( ), is structured ( tht is, stisfies condition (4) bo e ) if nd only if it stisfies (4 ) there is some 0 l n such tht, writing N i : l i l,

4 180 DAVID R. LARSON AND BARUCH SOLEL we h e, for e ery A M ( ), ( A ) p,m 0 A p l,m l ( Here N c is the complement of N. ) if ( p, m ) ( N N ) ( N c N c ), if ( p, m ) N N. If this is stisfied, we shll sy tht the identity is n inter l summnd of. Proof. Let : M ( ) 5 M n ( ) be cnonicl embedding stisfying (4). We shll show tht it stisfies (4 ). We now write e i j for the mtrix units of M ( ) nd f p q for the ones in M n ( ). Also set i j ( e i j ). Since i j is sum of some of the mtrix units of M n ( ), we cn define supp i j ( p, q ) 1,..., n 2 : f p q is one of the mtrix units of M n ( ) whose sum is i j. If ( i, j ) ( c, d ) then either e i j e c * d 0 or e c * d e i j 0, nd hence either i j c * d 0 or c * d i j 0. Hence, if ( i, j ) ( c, d ), then supp i j supp c d. For ( M ) T n, i j ( e i j ) T n. Hence i j T n whenever i j 0. Since ( T ) T n, this shows tht ( T ) ( M ) T n. Similrly ( D ) ( M ) D n (where D is the lgebr of digonl mtrices in M ( )). Since ech i j is prtil isometry with i j i * j i i nd i * j i j j j in D n, we see tht there re disjoint subsets J i 1,..., n, for 1 i, nd one-to-one functions τ i j : J i 5 J j such tht i j m J i f m, τ i j ( m ) nd τ i i ( m ) m for m J i. Let be the conditionl expecttion of (4). Then ( i j ) i j. Hence m J i ( f m, τ i j ( m ) ) J i f m, τ i j ( m ). (1) For i j we get m J i ( f m, m ) J i f m, m ( i i ( e i i )). But every ( f m, m ) is projection in ( M ) (since is multiplictive on D n nd selfdjoint) nd ( e i i ) is miniml projection in ( M ). Hence, for every i there is some m i J i such tht ( f m i m i ) ( e i i ) nd ( f j j ) 0 if j J i with j m i. Since is multiplictive on T n, we hve ( f p q ) ( f p p ) ( f p q ) ( f q q ) for p q (nd since is selfdjoint, for ll p, q ). Write N m i : 1 i. Then it follows tht, if ( p, q ) N N, we hve ( f p q ) 0. Using (1), we hve τ i j ( m i ) m j nd ( f p q ) 0 if ( p, q ) N N, ( e i j ) if p m i, q m j.

5 STRUCTURED TRIANGULAR LIMIT ALGEBRAS 181 Suppose p, q N nd p r q. Then 0 ( f p q ) ( f p r f r q ) ( f p r ) ( f r q ). Hence r N. We conclude tht N is subintervl of 1,..., n nd write N l 1,..., l (0 l n ). It is cler tht m i l i. Now te ( p, m ) N N c. Then p m i for some i nd then τ i j ( p ) m j m (s m N ). It follows tht ( p, m ) is not in! i, j supp i j. Hence for ll A M, ( A ) p,m 0. On the other hnd, if ( p, m ) N N, then p l i, m l j nd τ i j ( p ) m, so tht ( e i j ) p,m 1 nd, for A M, ( A ) p,m A i, j A p l,m l. This proves tht, for cnonicl embedding, (4) implies (4 ). For the other direction, suppose stisfies (4 ). Then we cn define g : M n ( ) 5 M ( ) by g ( A ) i j A l i,l j, for 1 i, j (with l s in (4 )). Then g id M nd g 3 T n is multiplictive since g ( AB ) i j ( AB ) l i,l j A l i, B,l j l i l j l i l j Lemm 3. 1 completes the proof. g ( A ) i, l g ( B ) l,j ( g ( A ) g ( B )) i j. We now define structured tringulr sublgebr of UHF lgebr to be n lgebr of the form A lim( T Å 5 n, ) where : T n 5 T n cn be extended to structured embedding 1 of M n ( ) into M n ( ) for every nd 1 Å lim( 5 M n, ). T HEOREM A sublgebr of UHF lgebr is structured if nd only if we h e sublgebrs B nd A nd contrcti e conditionl expecttions : 5 B such tht (i) B B 1,! B nd B is isomorphic to M n ( ) ; (ii) A B A 1,! A, nd the isomorphism of (i) mps A onto T n ; (iii) e ery mtrix unit of B n is sum of mtrix units of B n 1 ; here the mtrix units of n re the stndrd mtrix units induced by the isomorphism of (i); (iv) is surjecti e onto B, 1 nd 3 is multiplicti e. Proof. Assume tht nd stisfy the conditions of the theorem. Using (i) (iii) we cn write lim( M Å 5 n, ), lim( T Å 5 n, 3 T n ), for some cnonicl embeddings. More precisely, we hve isomorphisms : B 5 M n ( ) mpping A onto T n such tht 1 i 1 (where i : B 5 B 1 is the inclusion mp) is cnonicl embedding for every. Fix

6 182 DAVID R. LARSON AND BARUCH SOLEL nd define : M n 1 5 M n 1 by Then is contrctive conditionl expecttion onto 1 ( ( B 1 )) 1 ( ( 1 ( B ))) 1 ( B ) 1 ( B ) ( M n ) M n. 1 Since is multiplictive on nd 1 1 ( T n ) A 1 1, is multiplictive on T n. Hence 1 is structured for every. For the other direction, let B lim( M n Å 5, ) nd lim( T Å 5 n, 3 T n ) with structured embeddings. From Lemm 3. 1 we get mps g : M n 5 M 1 n stisfying g id M nd g n 3 T n is 1 multiplictive. Fix. For l we define g l, g g 1... g l : M n 5 M l 1 n. If r l then g r, r... l 1 l g l,. ( ) Now, the mps give rise to embeddings : M n 5 nd then ( ) shows tht, if we set g l, g l, l 1 1 : l 1 ( M n ) 5 l 1 n ( M n ), we get g * r, 3 l 1 ( M n ) g l 1 l,. Also g r, 1 for ll r. Hence, for given, the fmily g r, r defines mp : B 5 ( M n ) by 3 r 1 ( M n ) g r 1 r,. Write B ( M n ) nd A ( T n ). Then (i) (iii) re stisfied nd it is left to show tht is conditionl expecttion onto B which is multiplictive on. To show tht it is conditionl expecttion onto B it suf fices to show tht 3 B id. But this follows from the definition since 3 B 1 g, g, 1 1 g 1 1, nd for x ( ) nd M n, we hve 1 ( ( )) ( ) x ; hence ( ) 1 1 nd ( x ) ( g ( )) ( ) x. To show tht 3 is multiplictive, it is enough to use the fct tht g 3 T n 1 is multiplictive for every. Let be UHF lgebr nd D B be cnonicl ms. Let G be the equivlence reltion ssocited with B on the spce X, the mximl idel spce of D, s in 2. Given sublgebr D B let P G be the open prtil order supporting. Here we will del with lgebrs tht re tringulr (tht is, * D ) nd strongly mximl ( * is dense in ) nd then P becomes totl order. More precisely, for every u X, P induces totl order on the equivlence clss, [ u ], of u. For UHF lgebr G is miniml ; tht is, for every u X, [ u ] is dense in X. Excluding finite-dimensionl UHF lgebrs we find tht this implies tht [ u ] is infinite nd countble. If [ u ] hs the property tht, for every w in [ u ], the intervl z [ u ] : z w is finite, then we sy tht [ u ] is loclly finite. Tht mens tht [ u ] is order isomorphic to subset of. P ROPOSITION If is structured then there is some u X such tht [ u ] is loclly finite.

7 STRUCTURED TRIANGULAR LIMIT ALGEBRAS 183 Proof. We cn write lim( M Å 5 n, ) nd lim( T Å 5 n, ) nd ssume tht this presenttion is structured (tht is, is structured for ll ). Ech hs some l l ( ) such tht (4 ) holds (see Lemm 3. 2). Write E n i 1 e l ( ) i,l( ) i M n. Then 1 (1) for every e ( ) ij, ê ( ) ij ( Ê Ê ) ( X Ê ) ( X Ê ), (2) for every e ( ij ), ê ( ij ) ( Ê Ê ) ê l ( ) i,l( ) j. This follows from Lemm In prticulr, it follows tht for every nd 1 i n, ê l ( ) i,l( ) i ê ( i,i ). Hence ê (1) 1, 1 ê (2) l (1) 1,l(1) 1 ê (3) l (2) l (1) 1,l(2) l (1) 1... j 1 l ( j ), we cn define 1 u " 1 ê ( ) m ( ) 1,m ( ) 1. nd, if we write m ( ) From (2) bove we cn conclude tht for every nd every 1 i, j n, ê l ( ) i,l( ) j ê ( ij ) nd, thus, for every 1 i, j n 1, " 1 ê ( m ( ) ) i,m( ) j is point in G. Hence the points " 1 ê ( m ( ) ) i,m( ) i re ll equivlent in X. Write C 1 " ê ( m ( ) ) i,m( ) i : 1 i n 1 [ u ]. 1 Similrly we cn write, for p 1, C p " p ê ( m ( ) ) m ( p ) i,m( ) m ( p ) i : 1 i n p [ u ]. Note tht we hve u C p, by ting i m ( p ) 1. In fct, for p q, C p C q. To see this te w " p ê ( m ( ) ) m ( p ) i,m( ) m ( p ) i in C p. Write it s " q ê ( m ( ) ) m ( q ) j,m( ) m ( q ) j by setting j m ( q ) m ( p ) i. Thus! p 1 C p [ u ]. We shll show tht in fct [ u ]! p 1 C p. This will complete the proof becuse it is esy to see tht, for p q, C p forms n intervl in C q. Indeed, s ws shown bove, the points in C p correspond to vlues of j rnging from m ( q ) m ( p ) i to m ( q ) m ( p ) n p nd it is cler tht j S " q ê ( ) m ( ) m ( q ) j,m ( ) m ( q ) j is n order isomorphism of 1,..., n q onto C q. So to complete the proof we now te some w [ u ]. For some p, there re i, j with 1 i, j n p such tht ( u, w ) ê ( i,j p ). In prticulr, u ê ( ii p ) nd w ê ( jj p ). But

8 184 DAVID R. LARSON AND BARUCH SOLEL u ê ( p ) m ( p ) 1,m ( p ) 1. Hence i m ( p ) 1. Also u Ê p ; hence, by (1) bove, w Ê p. It then follows from (2) tht w ê jj ( p ) ( Ê p Ê p ) ê ( p 1) l ( p ) j,l( p ) j ( p 1) ê m ( p 1) m ( p ) j,m( p 1) m ( p ) j. Using (2) gin nd the fct tht w Ê p 1, becuse u Ê p 1, we get ( p 2) w ê m ( p 2) m ( p ) j,m( p 2) m ( p ) j. Continuing this wy we find tht w C p [ u ]. The proof of the proposition is constructive nd we shll refer to the equivlence clss obtined in this wy from given structured presenttion with choice of l ( ) s the structured equivlence clss ssocited with given presenttion nd choice of l ( ). If, for exmple, ech is the stndrd embedding (cf. [ 9 ]) then we cn choose l ( ) in mny wys nd dif ferent choices might give rise to dif ferent equivlence clsses. In fct, in this cse every equivlence clss will be ssocited with some choice of l ( ). On the other hnd, if ech is the refinement embedding (cf. [ 9 ]) then it is not structured nd, in fct, we do not hve ny loclly finite equivlence clss. However, the converse of Proposition 3. 4 is flse. As we show in the exmple below, the existence of loclly finite equivlence clss does not imply the structured property. E X A M P L E. We now present n exmple of strongly mximl tringulr AF lgebr tht hs loclly finite equivlence clss but the lgebr is not structured. The lgebr is sublgebr of M (2 3 ) nd is defined s lim( T Å 5 2 3, τ ) where τ : M M 2 3 re defined below. First define τ 1 1 : M 6 5 M 18 s in Fig. 1. This describes τ 1 on T 6 nd it cn be extended to M 6 in the obvious wy. Note tht, if we consider the mtrices s 2 2 bloc mtrices s shown in the digrm, then the blocs re invrint nd, on the (1, 1) bloc, the embedding is the stndrd embedding. On the (2, 2) bloc the embedding is mixture of the stndrd nd refinement embeddings. It cn be described s being the stndrd embedding on the 4th nd 5th digonl mtrix units nd stretching the 6th one. For τ 2, the mp on the (1, 1) bloc will be the stndrd embedding, nd the mp on the (2, 2) bloc will be the stndrd embedding on the 10th 17th digonl mtrix units nd it will stretch the 18th one. On the (1, 2) bloc, τ 2 will be defined in wy similr to the definition of τ 1 on the (1, 2) bloc. The embeddings τ for 3, re defined similrly. The forml definition of τ is s follows. (1) For the (1, 1) bloc : with 1 i, j 3, τ ( e ( ij ) ) e ( i,j ) e 3 i, 3 j e This is the stndrd embedding of M 3 into M i, 2 3 j.

9 STRUCTURED TRIANGULAR LIMIT ALGEBRAS K 16 K ÅÅ τ K K F IG. 1 (2) For the (1, 2) bloc : with 1 i, j 3, e τ ( e ( i,j ) i,j 2(3 3 ) 1) 3 1 e e i, e i 3, e (3) For the (2, 2) bloc : wtih 1 i, j 3, τ ( e ( ) i 3,j 3 ) e e e e 1 e i 3,j (3 1) 3 i 2 3,j , i 2(3 1) 3 1,j 2(3 1) 3 1 e i (3 1) 3 1,j (3 1) 3 i 2(3 1) 3 1, e i (3 1) 3 1, e 2 3 1,j 2(3 1) 3 1 e ,j (3 1) 3 1 e 2 3 1, e , e if j 3, if j 3. 1 e i 3 1,j 3 1 if i, j 3, i 3 1, if j 3 i, ,j 3 1 if i 3 j, , if i j 3. (4) The definition for the (2, 1) bloc cn be derived from the definition for the (1, 2) bloc. Since the blocs re invrint, the limit lgebr, hs the form, where 11 is isomorphic to the stndrd embedding lgebr in M (3 ) nd 22 is isomorphic to the sublgebr of M (3 ) defined by lim( T Å 5 n, τ n ) where τ n is τ n, restricted to the (2, 2) bloc. Note lso tht 1 1 F F nd 2 2 ( I F ) ( I F ),

10 186 DAVID R. LARSON AND BARUCH SOLEL where F e (1) 11 e (1) 22 e (1) 33 is projection in (Lt ). We shll show tht is not structured, lthough it hs loclly finite equivlence clss. We suspect tht this might be true lso for 22 but re unble to show it. We shll now write G for the groupoid ssocited with lim( M Å 5 2 3, τ ), P for the prtil order supporting nd X for the unit spce of G (equl to the spectrum of lim( D 2 3 Å 5, τ )). For every mtrix unit e ( i,j ) we shll write ê ( ij ) for the corresponding closed nd open subset of G (the support set of e ( ij ) ). If i j, then ê ( ii ) X. Also write X 1 for the subset of X ssocited with F, tht is, X 1! 3 i 1 ê (1) i,i. Let X 2 be X X 1. Note tht G ( X 1 X 1 ) nd P ( X 1 X 1 ) re the groupoid nd the prtil order of the stndrd embedding lgebr, nd G ( X 2 X 2 ) nd P ( X 2 X 2 ) re the ones for 22. Now fix u 1 " ê ( ) 3, 3 X 1 nd 1 " ê ( ) 3 1, 3 1 X 2. Note tht for every, e 3 1, τ ( e ( 3 ), 3 1 ). Hence ê 3 1, ê ( 3 ), 3 1 nd " ê ( 3 ), 3 1 defines point in P ( G ). This point is clerly ( u 1, 1 ) nd this shows tht [ u 1 ] [ 1 ]. It is nown tht [ u 1 ] X 1 is order isomorphic to nd n 1 is the mximl element there. This follows from the nlysis of equivlence clsses in the stndrd embedding lgebr. In fct, it is nown tht this is the only equivlence clss in X 1 tht hs mximl element. Note lso tht for every u X, [ u ] ([ u ] X 1 ) ([ u ] X 2 ) nd for every w 1 [ u ] X 1 nd w 2 [ u ] X 2, we hve ( w 1, w 2 ) P since ( X 2 X 1 ) P. Thus, if [ u ] is loclly finite, [ u ] X 1 should hve mximl element ; hence [ u ] [ u 1 ]. We conclude tht, if hs loclly finite equivlence clss, then it is [ u 1 ] ( [ 1 ]). We shll now show tht [ 1 ] X 2 is order isomorphic to nd 1 is the miniml element there. For this, fix [ 1 ] X 2. Then ( 1, ) ( X 2 X 2 ) G nd it cn be written s " ê ( 1 ) 3,j m for some j 3 (s 1 " ê ( 1 ) 3, 1 3 ) where ê 1 3 1,j ê ( ) 1 1 3,j ; tht is, e 1 3 1,j τ 1 ( e ( 1 ) 3,j ) for ll m. Exmining the definition of τ, on the (2, 2) bloc, one finds tht, for m, we hve either j m j 3 m (nd then j j 3, for ll m ) for some j or j m 3 m nd j m m 1 2. In the ltter cse we get j m 2 (3 m 1 2) 3 m 2 nd j (3 m 1 2)3 for ll m 1. Replcing m by m 1 we see tht, in either cse, " ê j ( ) 3,j 3 m for some j (in the second cse j 3 m 1 2). We shll write j for this element nd it is now cler tht X 2 [ 1 ] j : j 1

11 STRUCTURED TRIANGULAR LIMIT ALGEBRAS 187 nd j S j is n order isomorphism from onto X 2 [ 1 ]. Since [ 1 ] ( X 1 [ 1 ]) ( X 2 [ 1 ]) ( X 1 [ u 1 ]) ( X 2 [ 1 ]) nd since P ( X 2 X 1 ), we see tht [ 1 ] is order isomorphic to. In prticulr, it is loclly finite. It is now left to show tht [ 1 ] is not structured equivlence clss. This will complete the proof tht is not structured lgebr. Suppose, by negtion, tht is structured with structured clss [ 1 ]. First note tht for given structured presenttion of, sy Å lim( 5 T m, l ) with structured clss [ 1 ], there will be some K 0 such tht, for ll K 0, F is sum of digonl mtrix units in T m. Tht will enble us to write the mtrix lgebrs in bloc form. Considering Å lim( 5 F T m F, l 3 F T m F ), we get structured presenttion for 22. It is lso cler tht, s [ 1 ] is the structured clss for, [ 1 ] X 2 is structured clss for 22. It is left, therefore, to prove the following lemm. L EMMA The lgebr 22 does not h e structured presenttion with structured clss j : j 1. Proof. Assume, by negtion, tht there is such presenttion, nmely, 22 lim( T Å 5 3 n, ). When we refer to this presenttion we shll write f ( ij ) for the mtrix units. We lso hve the presenttion 22 lim( T Å 5 3, τ ) where τ is the restriction of τ, defined bove, to the (2, 2) bloc. For exmple, if i, j 3, τ ( e ( i,j ) ) e i 2(3 1),j 2(3 1) e i (3 1),j (3 1) e i,j. (1) The definition of τ for the other cses is similr modifiction of the definition of τ (tht is, drop 3 1 ). With this modifiction we now hve j " ê j (,j ) for j 1, (2) m j where m j is the smllest integer with 1 j 3 m j. Note tht given m nd j with 1 j 3 m, it is cler tht " m ê j (,j ) is well defined nd is the immedite successor of " m ê ( j ) 1,j 1 ; we thus get 1,..., 3 m 1. We now ssume tht j is the structured clss ssocited with the structured presenttion. Using the construction of the structured clss nd the fct tht here it hs miniml element ( 1 ), we see tht there is some 0 such tht for ll 0, nd every 1 i 3 n, ( ) i fˆ i,i (3) In fct, by strting the structured presenttion from M 3 n 0 we cn ssume tht for every, ( f ( ii ) ) f ii for 1 i 3 n.

12 188 DAVID R. LARSON AND BARUCH SOLEL In prticulr f ( ) ( f (1) 11 ) nd ( ) (1) fˆ 11 fˆ 11 for 1. (4) Note tht f (1) 11 is projection in the digonl of 22 ; hence it is digonl projection in T 3 for some lrge enough n (s e n 22 lim( T Å 5 3, τ n n )). Therefore it is the sum of some projections from the set e ( i,i n ) : 1 i 3 n. Since 1 ê ( n ) (1) 1, 1 fˆ 1, 1, we see tht, for n lrge enough, sy n N, ê ( n ) (1) 1, 1 fˆ 1, 1. (5) Note tht 3 " n m n 1 e ( m ) 3 n, 3 n (recll (2)) nd ( n 1) ê ê ( n ) 3 n, 3 n 1, 1 s τ n ( e ( n ) ( n 1) 1, 1 ) e (see (1) with i j 1). For n N we see tht 3 n, 3 n 3 ê ( n ) n 1, 1 f (1) 1, 1 ( ) (see (5)). On the other hnd, 3 n fˆ whenever n n 3 n, 3 n. (See (3).) Now fix l such tht n l N. Then, for n n l we hve (1) ( l ) 3 n fˆ 11 fˆ ( n n 3 n, 3 n l ). (6) But, since preserve the upper tringulr mtrices, it follows tht fˆ 3 n 1, 3 n 1 ( f ( ) 3 n, 3 n ) for ll. ( ) (1) ( l ) Hence fˆ 3 n 1, 3 n 1 fˆ 3 n, 3 n... fˆ 3 n 1, 3 n 1 nd for n n l, fˆ fˆ (1) 3 n, 3 n 3 n 1, 3 n 1, which contrdicts (6). The contrdiction completes the proof of the lemm. 4. The in erse limit ssocited with structured presenttion Let be structured lgebr nd let lim( T Å 5 n, ) be given structured presenttion. This mens tht we hve contrctive mps g : M n 5 M 1 n such tht g 3 T n is multiplictive, g 1 id M nd g n g 1 g ( g re the bc mps). We cn write T n 6 g ÅÅ 1 1 T n 6 g ÅÅ 2 2 T n 3 6 ÅÅ.... With such digrm we cn ssocite n inverse limit lgebr s indicted in the following proposition. P ROPOSITION Let ( T n, g ) be s bo e. Define C M A 1 M T n : g ( A 1 ) A. With the norm nd the lgebric structure inherited from M M n, C is Bnch lgebr. Moreo er, it hs the following property. Gi en Bnch lgebr E nd homomorphisms θ : E 5 T n stisfying θ 1 nd g θ 1 θ, there is contrcti e homomorphism θ : E 5 C such tht θ ( x ) M θ ( x ). Proof. The proof tht C is Bnch lgebr with the norm nd the product M A sup A ( M A )( M B ) M A B

13 STRUCTURED TRIANGULAR LIMIT ALGEBRAS 189 is strightforwrd nd uses the fct tht g is contrctive homomorphism on T n 1. Now ssume tht E nd θ re s in the sttement of the proposition nd define θ : E 5 C by θ ( x ) M θ ( x ). Then θ is clerly multiplictive since ech θ is, nd contrctive since θ ( x ) sup θ ( x ) x. Note tht if we define j : C 5 T n by j ( M A i ) A then j is contrctive homomorphism nd g j 1 j. In fct ( C, j ) hs the following uniqueness property. L EMMA Let C be Bnch lgebr nd j be contrcti e homomorphisms j : C 5 T n such tht g j 1 j for 1 nd " er j 0. Assume tht ( C, j ) hs the property indicted in Proposition Tht is, for e ery Bnch lgebr E nd contrcti e homomorphisms θ : E 5 T n with g θ 1 θ there is contrcti e homomorphism θ : E 5 C with j θ θ. Then there is n isometric isomorphism : C 5 C such tht j j for 1. Proof. If C is s bove, we use the properties of C nd C to get contrctive homomorphisms : C 5 C nd : C 5 C stisfying j j nd j j. Hence j j nd j j nd, since " er j j " er j 0, id C nd id C. Since 1 nd 1, is n isometric isomorphism. Now, let be structured lgebr with structured presenttion lim( T n Å 5, ) nd let C be the inverse limit ssocited with it nd with choice of bc mps. Write g for the contrctive homomorphism of onto T n s in the proof of Theorem 3. 3 ( g is 1 in the nottion there). Then we now tht g g 1 g ; hence, from Proposition 4. 1 we get contrctive homomorphism θ : 5 C stisfying θ ( x ) M g ( x ). Since " er g 0, θ is injective. In fct, x sup g ( x ) ; hence θ is n isometric embedding of into C. Note tht, given u X, with equivlence clss [ u ], there is representtion π u of B defined s follows. Let H l 2 ([ u ]) nd define π u ( f ) B ( H ) for f by ( π u ( f ) g )( ) f (, w ) g ( w ). w [ u ] Now let B be structured UHF lgebr nd fix structured presenttion lim( T Å 5 n, ) nd sequence l ( ), where l ( ) is ssocited with s in Lemm Then we get unique equivlence clss [ u ] ssocited with this dt s in Proposition In fct [ u ] hs been constructed s n incresing union [ u ]! 1 C of subintervls. With ech C we now ssocite projection F B ( H ) B ( l 2 [ u ])) which is the orthogonl projection onto spn e : C, where e : [ u ] is the stndrd orthonorml bsis in l 2 ([ u ]). Similrly, for

14 190 DAVID R. LARSON AND BARUCH SOLEL every subset Ω [ u ] we cn define F ( Ω ) to be the projection onto spn e : Ω. Let F ( Ω ) : Ω [ u ] is decresing. Then is nest of projections nd Alg ( T B ( H ) : (1 N ) TN 0 for ll N ) is the lgebr of opertors whose mtrix with respect to the bsis e is upper tringulr. L EMMA Let C be the in erse limit defined bo e. Then C is isometriclly isomorphic to Alg. Proof. Recll tht C M A 1 Let : Alg 5 C be defined by M T n : g ( A 1 ) A nd M A sup A. ( A ) M F AF. Here we use the projection F defined bove nd note tht we cn identify F AF with T n in the obvious wy since C [ u ] is n intervl of length n. Using this identifiction, we see tht the mp g (the bc mp for ) is simply F 1 AF 1 5 F F 1 AF 1 F F AF. Hence ( A ) C. Conversely, te some M A C. For ech we ssocite n opertor à F Alg F in the obvious wy. Tht is, its mtrix is à e, e w ( A ) w, if, w C nd 0 otherwise. Note tht by identifying C with 1,..., n in n order-preserving wy, we see tht ( A ) w, mes sense. Since g ( A 1 ) A, we hve F à 1 F Ã. Clerly, à sup j A j nd, thus, there is some subnet à tht converges -wely to some A B ( H ). Clerly, A Alg nd F j AF j lim F j A F j. But for j, F j A F j A j, nd hence F j AF j A j. It follows tht ( A ) M A nd is surjective onto C. Clerly A sup F AF sup A M A so tht is n isometry from Alg onto C. Since ech C is n intervl of [ u ], the mps A 5 F AF re homomorphisms on Alg nd, thus, is homomorphism. C OROLLARY θ π u. Proof. We hve 1 ( θ ( f )) 1 ( M g ( f )). For, w C the mtrix coef ficients of 1 θ ( f ) re ( 1 ( θ ( f ))),w ( g ( f )),w f (, w ). Since! C [ u ], we hve 1 ( θ ( f )) π u ( f ). Note tht π u is well-defined representtion for every u X. Wht we sw in the bove corollry is tht when [ u ] is structured, π u cn be fctored through sublgebr of M T n.

15 STRUCTURED TRIANGULAR LIMIT ALGEBRAS Residully finite - dimensionl tringulr lgebrs A Bnch lgebr is clled residully finite - dimensionl (or simply RFD) if for every x y in the lgebr there is finite-dimensionl contrctive representtion such tht ( x ) ( y ). For C*-lgebrs this property hs been studied in [ 1, 3, 6 ] nd elsewhere. If lim( T Å n, ) is structured lgebr with bc mps g nd ssocited mps g : 5 T n s in the discussion following Lemm 4. 2, then g re finite-dimensionl contrctive representtions tht seprte points in. Hence is RFD. In fct it suf fices to ssume tht the tringulr UHF lgebr hs loclly finite equivlence clss in order to conclude tht the lgebr is RFD. For this simply write this equivlence clss, sy [ u ], s union of incresing intervls, [ u ]! 1 L, nd consider the representtions π defined by compressing π u (defined in 4) into spn e : L. We now introduce necessry nd suf ficient condition, in terms of the supporting totl order P G, for tringulr AF lgebr to be RFD. In order to discuss contrctive representtions of strongly mximl TAF lgebr note first tht ech such representtion is completely contrctive [ 4, Theorem 5. 4] ; hence it is compression of some C*-representtion of the C*-lgebr, generted by, to semi-invrint subspce. From the wor of Renult [ 10, Theorem 1. 21] we now tht -representtion π of is the integrted form of representtion (, U, K ) of the equivlence reltion G ssocited with the AF lgebr. Here K K ( u ) u X is mesurble field of Hilbert spces, is qusi-invrint mesure on X (here qusi-invrint mens tht if we let θ ( u, ) (, u ) then θ ) nd U ( u, ) is n isometry from K ( ) to K ( u ) (for ( u, ) G ) stisfying U ( u, u ) I nd U ( u, ) U (, w ) U ( u, w ) lmost everywhere. To sy tht π is the integrted form of (, U, K ) mens tht, up to unitry equivlence, the spce of π is K X K ( u ) d ( u ) nd π ( f ) is defined on K by the formul π ( f ), f ( u, ) U ( u, ) ( ), ( u ) 1 / 2 ( u, ) d ( u, ), where d / ( d θ ) nd is the mesure on G ssocited with ; tht is, X d ( u ) where u is the counting mesure on (, u ) : [ u ]. If is contrctive representtion of then, s mentioned bove, there is some C*-representtion π ssocited with (, U, K ) s bove nd subspce H K K ( u ) d ( u ) tht is semi-invrint for π ( ) (tht is, is the dif ference of two invrint subspces) such tht ( ) P H π ( ) 3 H, where P H is the projection onto H. Since H is semi-invrint for π ( D ) (where D *), it is invrint for π ( D ). As π ( D ) is the lgebr of ll digonl opertors with respect to the decomposition K K ( u ) d ( u ), we cn decompose H H ( u ) d ( u ) where H ( u ) K ( u ) for ll u X. Now ssume tht is finite-dimensionl, nd hence dim H. Then H H ( u ) d ( u ) becomes H n H ( w i ) i 1

16 192 DAVID R. LARSON AND BARUCH SOLEL nd dim H ( w i ). If we write T i j for 1 2 ( w i, w j ) P H ( w i ) U ( w i, w j ) 3 H ( w j ), where P H ( w j ) is the orthogonl projection of K ( w j ) onto H ( w j ), then we get ( ( f ) ) j n f ( w j, w ) T j for f, H. 1 We cn write ( f ) ( f ( w j, w ) T j ). If ( w j, w ) G, we understnd f ( w j, w ) to be 0. Write T ( u, ) 1 2 ( u, ) P H ( u ) U ( u, ) 3 H ( ) nd Q ( ) ( u, ) : T ( u, ) 0. Then P Q ( ) P Q ( ) nd for every ( u, ) Q ( ), ( u, ) ( w i, w j ) for some 1 i, j n. Given ( u, ) Q ( ), we hve ( u, ) ( w i, w j ) nd for every u w, ( u, w ), ( w, ) re not in Q ( ) s P Q ( ) P Q ( ). Hence w w for some 1 n. It follows tht, if T ( u, ) 0, then the intervl [ u, ] is finite. We cn now prove the following. T HEOREM A tringulr AF lgebr is RFD if nd only if is dense in G.! J J : J is finite inter l in some equi lence clss Proof. First ssume tht the set! J J : J [ u ] is finite intervl is dense in G nd fix f. Since f is continuous function on G, there is some ( u, ) in this dense set such tht f ( u, ) 0. Since ( u, ) is in this set, there is some finite intervl in [ u ] contining both u nd. Hence the set w : u w is finite, sy of crdinlity n. Define : 5 M n ( ) by ( f ) ( f ( w i, w j )) where w : u w w 1,..., w n s ordered sets. Then is contrctive representtion of. In fct, it is compression of π u to the semi-invrint subspce of l 2 ([ u ]) ssocited with w : u w. Since ( f ) 0, we see tht for every f there is some contrctive finite-dimensionl representtion with ( f ) 0. This proves tht is RFD. For the other direction, ssume tht the set in the sttement of the theorem is not dense. Then there is some non-zero f such tht f ( u, ) 0 whenever [ u, ] ( w : u w ) is finite. We shll show tht for every contrctive finite-dimensionl representtion of, ( f ) 0. So fix such nd write ( ( f ) ) j n f ( w j, w ) T j 1 s in the discussion preceding the theorem. Here T j T ( w j, w ). As ws shown there, if T ( w j, w ) 0 then [ w j, w ] is finite. But then f ( w j, w ) 0 s ssumed bove nd, thus, ( f ) 0. This completes the proof. References 1. R. EXEL nd T. A. LORING, Finite dimensionl representtions of free product C*-lgebrs, Internt. J. Mth. 3 (1992) J. G L I M M, On certin clss of opertor lgebrs, Trns. Amer. Mth. Soc. 95 (1960) K. M. GOODEARL nd P. M E N A L, Free nd residully finite dimensionl C*-lgebrs, J. Funct. Anl. 90 (1990) P. MUHLY nd B. S O L E L, Sublgebrs of groupoid C*-lgebrs, J. Reine Angew. Mth. 402 (1989)

17 STRUCTURED TRIANGULAR LIMIT ALGEBRAS P. MUHLY nd B. S O L E L, On tringulr sublgebrs of groupoid C*-lgebrs, Isrel J. Mth. 71 (1990) V. G. P E S T O V, Opertor spces nd residully finite dimensionl C*-lgebrs, J. Funct. Anl. 123 (1994) J. P E T E R S, Y. T. POON, nd B. H. W A G N E R, Tringulr AF lgebrs, J. Opertor Theory 23 (1990) J. P E T E R S, Y. T. POON, nd B. H. W A G N E R, Anlytic TAF lgebrs, Cnd. J. Mth. 45 (1993) S. C. P O W E R, Limit lgebrs : n introduction to sublgebrs of C *- lgebrs, Pitmn Reserch Notes 278 (Longmn Scientific nd Technicl, Hrlow, Essex, 1992). 10. J. R E N A U L T, A groupoid pproch to C *- lgebrs, Lecture Notes in Mthemtics 793 (Springer, Berlin, 1980). 11. S. S T R A T I L A nd D. V O I C U L E S C U, Representtions of AF lgebrs nd of the group U ( ), Lecture Notes in Mthemtics 486 (Springer, Berlin, 1975). Deprtment of Mthemtics Deprtment of Mthemtics Texs A&M Uni ersity The Technion College Sttion Hif Texs Isrel U.S.A.