STRUCTURED TRIANGULAR LIMIT ALGEBRAS
DAVID R. LARSON and BARUCH SOLEL
[Received 5 February 1996—Revised 24 September 1996]
1 . Introduction
Suppose ! is a unital Banach algebra which contains an increasing chain
!n ‘ !n 11 of finite-dimensional subalgebras, all containing the unit I of ! , such
that ! is the closure of the union !n !n . Then ! can be viewed as a direct limit
of the algebras !n . It frequently occurs in this setting that there is a sequence of
bounded mappings c n : ! 5 !n which satisfy the conditional expectation property c n (CAB ) 5 Cc n (A)B whenever C , B P !n , and with the further property
that hc n j converges to the identity mapping of ! 5 ! in the strong operator
topology (that is, c n (A) 5 A for A P ! ). Indeed, if all the ! n are C*-algebras, so
! is an AF (approximately finite-dimensional) C*-algebra, this is always the case.
This property holds in many non-selfadjoint settings as well. In some special
cases, not only do the hc n j exist but they can be taken to be multiplicatiy e on ! .
So the c n are unital homomorphisms of ! into itself. This implies very strong
structural properties of the limit algebra. Non-trivial instances of this type of
highly -structured property are not possible in the C*-setting unless, for instance,
the algebras are abelian. However, certain of the more interesting and widely
studied of the triangular UHF (uniformly hyperfinite) algebras exhibit precisely
this behaviour. The purpose of this article is to investigate and partially classify
this family. We begin by establishing a proper framework for this study.
2. Preliminaries
A unital C*-algebra @ is called an AF algebra if it can be written as a
direct limit of finite-dimensional C*-algebras @ 5 lim(Bk , wk ). Here each Bk
Å5
is a finite-dimensional C*-algebra and wk : Bk 5 Bk11 is a unital injective
p -homomorphism. If each Bk is the algebra of nk 3 nk complex matrices, denoted
Mnk or Mnk (C) , then @ is called a UHF algebra. Due to the work of Stratila and
Voiculescu [11] we know that, given an AF algebra @ , we can find Bk , wk defining
@ as a direct limit, and fix a system of matrix units in each Bk , as Bk is isomorphic
to a sum of full matrix algebras, in such a way that, for each k , wk maps the
diagonal matrices of Bk into the diagonal matrices of Bk11 and, for every matrix
unit e in Bk , w (e ) is a sum of matrix units in Bk11 . In such a case the maps wk will
be said to be canonical and @ 5 lim(Bk , wk ) is canonical presentation of @ . This
Å5
presentation will in general be non-unique.
Given a canonical presentation as above, we can set D 5 lim(Ck , wk ) where Ck
Å5
is the algebra of all diagonal matrices in Bk . Then D is a masa in @ . Such a masa
The work of the first author was supported in part by NSF grant DMS-9401544. The second author
was a participant in the workshop in Linear Analysis and Probability, Texas A & M University. His
research was supported in part by a Fund for Promotion of Research at The Technion.
1991 Mathematics Subject Classification : 47D25, 46J35.
Proc. London Math. Soc. (3) 75 (1997) 177 – 193.
178
DAVID R. LARSON AND BARUCH SOLEL
is called a canonical masa. We also call a norm-closed subalgebra ! of @
canonical if it contains a canonical masa D. If, in addition ! > ! * 5 D , ! is said
to be a canonical triangular AF algebra. In this paper we will deal only with
canonical algebras and, therefore, every triangular AF or triangular UHF algebra
will be assumed to be canonical, and we shall simply write TAF or TUHF. In fact,
except in the last section when we consider TAF algebras, we will be interested
only in TUHF algebras ! that can be written as
! 5 lim(Tnk , wk )
Å5
where lim(Mnk , wk ) is a canonical presentation of a UHF algebra @ with the
Å5
property that each wk maps the upper triangular matrices in Mnk , denoted Tnk ,
into the upper triangular matrices in Mnk11 . These algebras are called in the
literature strongly maximal triangular in factors (see [7, p. 105; 8, Example 2.12]).
Given an AF algebra @ with a canonical masa D ‘ @ , we can express @ as
C *(G ) for some r -discrete amenable principal groupoid. In fact the groupoids of
the kind we are considering here may be viewed as equivalence relations on X ,
the maximal ideal space of D , having countable equivalence classes. Hence G can
be viewed as a subset of X 3 X with a topology that may be finer than the
product topology. For details about groupoids and the associated C*-algebras see
[10]. Groupoids for AF algebras are treated in [10, Chapter III, § 1]. Elements of
the AF algebra @ are viewed as continuous functions on G and the multiplication
and adjoint operations are defined as follows:
( fg )(u , y ) 5
O
f (u , w )g (w , y ) ,
w
(w ,u )PG
f *(u , y ) 5 f (y , u ).
For f P @ we write supp f 5 h(u , y ) P G : f (u , y ) ? 0j. We have D 5 h f P
@: supp f ‘ Dj where D 5 h(u , u ): u P X j. In fact we shall identify D with X (and
u P X with (u , u ) P D ‘ G ).
Given a canonical subalgebra D ‘ ! ‘ @ , there is an open subset P ‘ G such
that
(i) D ‘ P ;
(ii) P + P ‘ P ; that is, if (x , y ) , ( y , z ) are in P then (x , z ) is in P and
! 5 h f P @: supp f ‘ P j (we then write ! 5 ! (P )) (see [4]).
The algebra ! (5 ! (P )) is triangular if and only if
(iii) P > P 21 5 D (where P 21 5 h(u , y ): (y , u ) P P j).
For a strongly maximal triangular algebra we also have
(iv) P < P 21 5 G [5, Theorem 1.3].
Whenever (i) – (iii) are satisfied, P induces a partial order on every equivalence
class (we write [u ] for the equivalence class of u P X ). The condition (iv) means
that this order is total. When we write u < y it will always be assumed that
(u , y ) P P.
Since we deal mostly with the case where @ is a UHF algebra, note that in this
case each equivalence class is dense in X (that is, the groupoid is minimal). Given
a canonical presentation @ 5 lim(Mnk , wk ) as above, we write the matrix units of
Å5
Mnk as he ij(k)j. Viewing @ as C *(G ) , we can view each e ij(k ) as a function on G. In
STRUCTURED TRIANGULAR LIMIT ALGEBRAS
179
fact it is a characteristic function of some open and closed subset ˆe (ijk ) ‘ G (if i 5 j ,
ˆ for the support of E
ˆe (ijk ) can be viewed as a subset of X ). We also write E
whenever E is a sum of matrix units. With this notation wk (e ij(k)) is a sum of matrix
k 11)
k 11)
units in Mnk11 and e (l ,m
is one of these matrix units if and only if eˆ (l,m
‘ eˆ (ijk ).
(k )
(k )
(k 11)
Given a point (u , y ) P G there are e ik ,jk with eˆ ik,jk “ eˆ ik11 jk11 such that h(u , y )j 5
"k eˆ (ikk,j)k , and y ice y ersa . Every such intersection with eˆ (ikk,j)k “ eˆ i(kk111 ,j1)k11 defines a
point in G. For more details about triangular AF algebras and the associated
partial orders see [9].
Finally, let us recall that UHF algebras lim(Mnk , wk ) are characterized by the
Å5
supernatural number associated with the sequence hnk j [2]. Given a supernatural
number (also called a generalized integer) n , we write M (n ) for the associated
UHF algebra.
3 . Structured algebras
In this section we study a class of triangular UHF algebras. In order to define
this class we first introduce the following definitions.
DEFINITIONS. A map w : Mk (C) 5 Mn (C) is called a canonical embedding if
(1) it is an injective unital p -homomorphism;
(2) w (Tk ) ‘ Tn (where Tn ‘ Mn (C) is the subalgebra of all upper triangular
matrices);
(3) w maps each matrix unit of Mk (C) into a sum of matrix units in Mn (C).
Then w is called a structured canonical embedding if, in addition,
(4) there is a contractive conditional expectation c from Mn (C) onto w (Mk (C))
such that c , restricted to Tn , is multiplicative.
LEMMA 3.1. For a canonical embedding w : Mk (C) 5 Mn (C) , condition (4) is
equiy alent to
(49) there is a contractiy e map g : Mn (C) 5 Mk (C) such that g + w 5 idMk and
g 3Tn is multiplicatiy e.
Proof. Given c as in (4) we define g 5 w 21 + c (here w 21: w (Mk (C)) 5 Mk (C)).
It clearly satisfies (49). Conversely, given g as in (49) we let c 5 w + g and then
c + c 5 w + g + w + g 5 w + id + g 5 c , so that c is indeed a conditional expectation
onto w (Mk (C)).
If the above holds, we call g the back map of w.
LEMMA 3.2. For a canonical embedding w : Mk (C) 5 Mn (C) , w is structured (that
is , satisfies condition (4) aboy e ) if and only if it satisfies
(40) there is some 0 < l < n such that , writing
N 5 hi : l , i < l 1 k j ,
180
DAVID R. LARSON AND BARUCH SOLEL
we hay e , for ey ery A P Mk (C) ,
w (A)p ,m 5
H0A
if ( p , m ) ¸ (N 3 N ) < (N c 3 N c),
if ( p , m ) P N 3 N.
p 2l ,m 2l
(Here N c is the complement of N.)
If this is satisfied, we shall say that the identity is an intery al summand of w.
Proof. Let w : Mk (C) 5 Mn (C) be a canonical embedding satisfying (4). We
shall show that it satisfies (40). We now write heij j for the matrix units of Mk (C)
and h fpq j for the ones in Mn (C). Also set y ij 5 w (eij ). Since y ij is a sum of some of
the matrix units of Mn (C) , we can define supp y ij 5 h( p , q ) P h1 , .. . , n j2: fpq is one
of the matrix units of Mn (C) whose sum is y ij j. If (i , j ) ? (c , d ) then either
eij ec*d 5 0 or ec*d eij 5 0 , and hence either y ij y c*d 5 0 or y c*dy ij 5 0 . Hence, if
(i , j ) ? (c , d ) , then supp y ij > supp y cd 5 [. For a P w (Mk ) > Tn , a 5 o aij w (eij ) P
Tn . Hence y ij P Tn whenever aij ? 0 . Since w (Tk ) ‘ Tn , this shows that w (Tk ) 5
w (Mk ) > Tn . Similarly w (Dk ) 5 w (Mk ) > Dn (where Dk is the algebra of diagonal
matrices in Mk (C)).
Since each y ij is a partial isometry with y ij y i*j 5 y ii and y i*jy ij 5 y jj in Dn , we see
that there are disjoint subsets Ji ‘ h1 , . . . , n j , for 1 < i < k , and one-to-one
functions τij : Ji 5 Jj such that
y ij 5
Of
m PJi
m ,τ ij (m )
and τii (m ) 5 m for m P Ji . Let c be the conditional expectation of (4). Then
c (y ij ) 5 y ij . Hence
O c(f
mPJi
For i 5 j we get
O c(f
m PJi
m ,m
)5
m ,τ ij (m )
)5
Of
m ,τ ij (m )
.
(1)
Ji
Of
m ,m
(5 y ii 5 w (eii )).
Ji
But every c ( fm ,m ) is a projection in w (Mk ) (since c is multiplicative on Dn and
selfadjoint) and w (eii ) is a minimal projection in w (Mk ). Hence, for every i there is
some mi P Ji such that
c ( fmimi) 5 w (eii )
and
c ( fjj ) 5 0 if j P Ji with j ? mi .
Since c is multiplicative on Tn , we have
c ( fpq ) 5 c ( fpp )c ( fpq )c ( fqq )
for p < q (and since c is selfadjoint, for all p , q ). Write N 5 hmi : 1 < i < k j. Then
it follows that, if ( p , q ) ¸ N 3 N , we have c ( fpq ) 5 0 . Using (1), we have
τij (mi ) 5 mj and
0
if ( p , q ) ¸ N 3 N ,
c ( fpq ) 5
w (eij ) if p 5 mi , q 5 mj .
H
STRUCTURED TRIANGULAR LIMIT ALGEBRAS
181
Suppose p , q P N and p , r , q. Then
0 ? c ( fpq ) 5 c ( fpr frq ) 5 c ( fpr )c ( frq ).
Hence r P N. We conclude that N is a subinterval of h1, ..., n j and write
N 5 hl 1 1 , .. . , l 1 k j (0 < l , n ). It is clear that mi 5 l 1 i. Now take ( p , m ) P
N 3 N c. Then p 5 mi for some i and then τij ( p ) 5 mj ? m (as m ¸ N ). It follows
that ( p , m ) is not in !i , j supp y ij . Hence for all A P Mk , w (A)p ,m 5 0. On the other
hand, if ( p , m ) P N 3 N , then p 5 l 1 i , m 5 l 1 j and τ ij ( p ) 5 m , so that
w (eij )p ,m 5 1 and, for A P Mk , w (A)p ,m 5 Ai ,j 5 Ap 2l ,m 2l.
This proves that, for a canonical embedding, (4) implies (40). For the other
direction, suppose w satisfies (40). Then we can define g : Mn (C) 5 Mk (C) by
g (A)ij 5 Al 1i ,l 1j , for 1 < i , j < k (with l as in (40)). Then g + w 5 idMk and g 3Tn is
multiplicative since
g (AB )ij 5 (AB )l 1i ,l 1j 5
5
O
l 1i <k <l 1j
O
l 1i <k <l 1j
Al 1i ,kBk ,l1j
g (A)i ,k 2l g (B )k 2l ,j 5 (g (A)g (B ))ij .
Lemma 3.1 completes the proof.
We now define a structured triangular subalgebra of a UHF algebra @ to be an
algebra of the form
A 5 lim(Tnk , wk )
Å5
where wk : Tnk 5 Tnk11 can be extended to a structured embedding w
˜ k of Mnk (C)
into Mnk11(C) for every k and @ 5 lim(Mnk , w
˜ k ).
Å5
THEOREM 3.3. A subalgebra ! of a UHF algebra @ is structured if and only if
we hay e subalgebras Bk ‘ @ and Ak ‘ ! and contractiy e conditional expectations
c k : @ 5 Bk such that
(i) Bk ‘ Bk 11 , ! Bk 5 @ and Bk is isomorphic to Mnk (C);
(ii) Ak ‘ Bk > Ak 11 , ! 5 ! Ak , and the isomorphism of (i) maps Ak onto Tnk ;
(iii) ey ery matrix unit of Bn is a sum of matrix units of Bn 11; here the matrix
units of @n are the standard matrix units induced by the isomorphism of (i);
(iv) c k is surjectiy e onto Bk , c k + c k 11 5 c k and c k 3! is multiplicatiy e.
Proof. Assume that ! and @ satisfy the conditions of the theorem. Using
(i) – (iii) we can write
@ 5 lim(Mnk , wk ) ,
Å5
! 5 lim(Tnk , wk 3Tnk ) ,
Å5
for some canonical embeddings hwk j. More precisely, we have isomorphisms
1
h k : Bk 5 Mnk (C) mapping Ak onto Tnk such that wk 5 h k 11 + ik + h 2
(where
k
ik : Bk 5 Bk11 is the inclusion map) is a canonical embedding for every k. Fix k
182
DAVID R. LARSON AND BARUCH SOLEL
1
and define c : Mnk11 5 Mnk11 by c 5 h k 11 + c k + h 2
k 11 . Then c is a contractive
conditional expectation onto
h k 11(c k (Bk 11)) 5 h k 11(c k (c k 11(B ))) 5 h k 11 + c k (B ) 5 h k 11(Bk )
5 wk (Mnk ) ‘ Mnk11 .
1
Since c k is multiplicative on ! and h 2
k 11(Tnk11) 5 Ak 11 ‘ ! , c is multiplicative
on Tnk11 . Hence wk is structured for every k. For the other direction, let B 5
lim(Mnk , wk ) and ! 5 lim(Tnk , wk 3Tnk ) with wk structured embeddings. From
Å5
Å5
Lemma 3.1 we get maps gk : Mnk11 5 Mnk satisfying gk + wk 5 idMnk and gk 3Tnk11 is
multiplicative. Fix k. For l > k we define gl ,k 5 gk + gk 11 + . .. + gl : Mnl11 5 Mnk . If
r . l > k then
gr,k + wr + . .. + wl 11 + wl 5 gl ,k .
(p)
Now, the maps hwk j give rise to embeddings w
˜ k : Mnk 5 @ and then ( p ) shows
that, if we set
1
gl9,k 5 w
˜ k + gl ,k + w
˜2
˜ l 11(Mnl11) 5 w
˜ n (Mnk ) ,
l 11 : w
we get g *r ,k3w
˜ l11(Mnl11) 5 gl9,k . Also i gr9,k i < 1 for all r > k. Hence, for a given k ,
the family hgr9,k j`r5k defines a map
ck : B 5 w
˜ k (Mnk )
by c k 3w
˜ r11(Mnr11) 5 gr9,k . Write Bk 5 w
˜ k (Mnk ) and Ak 5 w
˜ k (Tnk). Then (i) – (iii) are
satisfied and it is left to show that c k is a conditional expectation onto Bk which is
multiplicative on ! . To show that it is a conditional expectation onto Bk it suffices
to show that c k 3Bk 5 id. But this follows from the definition since
1
1
c k 3Bk11 5 gk9 ,k 5 w
˜ k + gk ,k + w
˜2
˜ k + gk + w
˜2
k 11 5 w
k11 ,
and for x 5 w
˜ k (a ) and a P Mnk , we have w
˜ k 11(wk (a )) 5 w
˜ k (a ) 5 x ; hence wk (a ) 5
1
w
˜2
k 11 and
c k (x ) 5 w
˜ k (gk + wk (a )) 5 w
˜ k (a ) 5 x .
To show that c k 3a is multiplicative, it is enough to use the fact that gk 3Tnk11 is
multiplicative for every k.
Let @ be a UHF algebra and D ‘ B be a canonical masa. Let G be the
equivalence relation associated with B on the space X , the maximal ideal space of
D , as in § 2. Given a subalgebra D ‘ ! ‘ B let P ‘ G be the open partial order
supporting ! . Here we will deal with algebras ! that are triangular (that is,
! > ! * 5 D ) and strongly maximal (! 1 ! * is dense in @ ) and then P becomes
a total order. More precisely, for every u P X , P induces a total order on the
equivalence class, [u ] , of u. For a UHF algebra G is minimal; that is, for every
u P X , [u ] is dense in X. Excluding finite-dimensional UHF algebras we find that
this implies that [u ] is infinite and countable. If [u ] has the property that, for
every y < w in [u ] , the interval hz P [u ]: y < z < w j is finite, then we say that [u ]
is locally finite. That means that [u ] is order isomorphic to a subset of Z.
PROPOSITION 3.4. If ! is structured then there is some u P X such that [u ] is
locally finite.
STRUCTURED TRIANGULAR LIMIT ALGEBRAS
183
Proof. We can write
@ 5 lim(Mnk , wk ) and ! lim(Tnk , wk )
Å5
Å5
and assume that this presentation is structured (that is, wk is structured for all k ).
Each wk has some l 5 l (k ) such that (40) holds (see Lemma 3.2). Write
Ek 5 oni5k 1 e (l(kk1)11)i ,l (k )1i P Mnk11 . Then
ˆk3E
ˆ k ) < (X \ E
ˆ k ) 3 (X \ E
ˆ k) ,
(1) for every e (ijk ) , ˆe (ijk ) ‘ (E
ˆk3E
ˆ k ) 5 eˆ (l(kk1)11)i ,l (k )1j.
(2) for every e (ijk ) , ˆe (ijk ) > (E
This follows from Lemma 3.2. In particular, it follows that for every k and
1 < i < nk ,
eˆ (l(kk1)11)i ,l (k )1i ‘ eˆ (i ,ik ).
Hence ˆe (1)
ˆ (2)
ˆ (3)
1 ,1 “ e
l (1)11 ,l (1)11 “ e
l (2)1l (1)11 ,l (2)1l (1)11 “ . . . and, if we write m (k ) 5
k 21
oj51 l ( j ) , we can define
`
u 5 " eˆ (mk()k )11 ,m (k )11.
k 51
From (2) above we can conclude that for every k and every 1 < i , j < nk ,
eˆ l((kk1)11)i ,l (k)1j ‘ eˆ (ijk )
and, thus, for every 1 < i , j < n 1 , "`k51 eˆ (mk()k )1i ,m (k )1j is a point in G. Hence the
points "`k51 eˆ (mk()k )1i ,m (k )1i are all equivalent in X. Write
H
J
`
C 1 5 " eˆ (mk()k )1i ,m (k )1i: 1 < i < n 1 ‘ [u ].
k51
Similarly we can write, for p > 1 ,
H
J
`
Cp 5 " eˆ (mk()k )2m ( p )1i ,m (k )2m ( p)1i : 1 < i < np ‘ [u ].
k 5p
Note that we have u P Cp , by taking i 5 m ( p ) 1 1. In fact, for p < q , Cp ‘ Cq . To
see this take w 5 "`k5p eˆ (mk()k )2m( p )1i ,m (k )2m ( p )1i in Cp . Write it as
"`k5q eˆ (mk()k )2m (q)1j ,m (k )2m(q )1j by setting j 5 m (q ) 2 m ( p ) 1 i. Thus !`p51 Cp ‘
[u ]. We shall show that in fact [u ] 5 !`p51 Cp . This will complete the proof
because it is easy to see that, for p < q , Cp forms an interval in Cq . Indeed, as was
shown above, the points in Cp correspond to values of j ranging from
m (q ) 2 m ( p ) 1 i to m (q ) 2 m ( p ) 1 np and it is clear that
`
j S " eˆ (mk()k )2m(q )1j ,m (k)2m (q )1j
k 5q
is an order isomorphism of h1 , . .. , nq j onto Cq .
So to complete the proof we now take some w P [u ]. For some p , there are i , j
with 1 < i , j < np such that (u , w ) P ˆe (i,jp ). In particular, u P eˆ (iip ) and w P eˆ (jjp ). But
184
DAVID R. LARSON AND BARUCH SOLEL
( p)
ˆ
ˆ
u P eˆ m
( p )11 ,m ( p )11 . Hence i 5 m ( p ) 1 1 . Also u P E p ; hence, by (1) above, w P E p .
It then follows from (2) that
ˆ p3E
ˆ p ) 5 ˆe l((pp1)11)j ,l( p )1j
w P eˆ jj( p ) > (E
5 eˆ (mp(1p1)
11)2m ( p )1j ,m ( p 11)2m ( p )1j .
ˆ p 11 , because u P E
ˆ p 11 , we get
Using (2) again and the fact that w P E
( p 12)
w P eˆ m
( p 12)2m ( p )1j ,m ( p 12)2m ( p )1j .
Continuing this way we find that w P Cp ‘ [u ].
The proof of the proposition is constructive and we shall refer to the
equivalence class obtained in this way from a given structured presentation with a
choice of hl (k )j as the structured equivalence class associated with a given
presentation and a choice of hl (k )j. If, for example, each wk is the standard
embedding (cf. [9]) then we can choose l (k ) in many ways and different choices
might give rise to different equivalence classes. In fact, in this case every
equivalence class will be associated with some choice of hl (k )j. On the other hand,
if each wk is the refinement embedding (cf. [9]) then it is not structured and, in
fact, we do not have any locally finite equivalence class.
However, the converse of Proposition 3.4 is false. As we show in the example
below, the existence of a locally finite equivalence class does not imply the
structured property.
EXAMPLE. We now present an example of a strongly maximal triangular AF
algebra that has a locally finite equivalence class but the algebra is not structured.
The algebra is a subalgebra of M (2 ? 3`) and is defined as
# 5 lim(T2 ? 3k , τk )
Å5
where τk : M2 ? 3k 5 M2 ? 3k11 are defined below. First define τ 1: M6 5 M18 as in Fig.
1. This describes τ 1 on T6 and it can be extended to M6 in the obvious way.
Note that, if we consider the matrices as 2 3 2 block matrices as shown in the
diagram, then the blocks are ‘invariant’ and, on the (1, 1) block, the embedding is
the standard embedding. On the (2, 2) block the embedding is a mixture of the
standard and refinement embeddings. It can be described as being the standard
embedding on the 4th and 5th diagonal matrix units and ‘stretching’ the 6th one.
For τ 2 , the map on the (1, 1) block will be the standard embedding, and the map
on the (2, 2) block will be the standard embedding on the 10th – 17th diagonal
matrix units and it will ‘stretch’ the 18th one. On the (1, 2) block, τ 2 will be
defined in a way similar to the definition of τ 1 on the (1, 2) block. The
embeddings τk for k > 3 , are defined similarly. The formal definition of τ k is as
follows.
(1) For the (1, 1) block: with 1 < i , j < 3k ,
11)
(k 11)
τk (e (ijk )) 5 e (i,jk ) 1 e (3kk1
i ,3k 1j 1 e 2 ? 3k 1i ,2 ? 3k 1j .
This is the standard embedding of M3k into M3k11 .
185
STRUCTURED TRIANGULAR LIMIT ALGEBRAS
K
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??
a 22 a 23 ? a 24 a 25 a 26
??
a ? a
a
a
. . . . . . . . . . .3.3 . .?? . . .34. . . . 3. 5. . . . 3.6.
? a 44 a 45 a 46
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a 55 a 56
?
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?
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a 12
K
τ1
£ÅÅ5
?
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a 33 ? a 34 a 35 0
a 36 0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .??. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0
0
0
0
a 46 0
? a 44 a 45 0
??
0
0
0
0
a 55 0
a 56 0
?
??
0
a 44 a 45 0
0 a 46 0
?
??
0
0 a 56 0
a 55 0
?
??
a 44 a 45
0
0 a 46
?
??
0
0 a 56
a 55
?
??
a 66 0
0
?
??
a 66 0
?
??
a 66
?
?
a 11
a 12
a 13
0
0
0
0
0
0
0
0
0
0
0
0
FIG. 1
K
(2) For the (1, 2) block: with 1 < i , j < 3k ,
)
τk (e (i ,jk1
3k) 5
Hee
(k 11)
(k 11)
(k 11)
i ,j 12(3k 21)13k11
i 13k ,j 1(3k 21)13k11
i 12 ? 3k ,j 13k11
(k 11)
(k 11)
(k 11)
k
k
1
1
k
k
1
1
i 13 ,2 ? 3
21
112 ? 3 ,2 ? 3
22
i ,2 ? 3k11
1e
1e
1e
1e
if j ? 3k ,
if j 5 3k.
(3) For the (2, 2) block: wtih 1 < i , j < 3k ,
k)
τk (e (i1
3k ,j 13k) 5
5
k 11)
(k 11)
(k 11)
e (i1
if i , j ? 3k ,
2(3k 21)13k11 ,j 12(3k 21)13k11 1 e i 1(3k 21)13k11 ,j 1(3k 21)13k11 1 e i 13k11 ,j 13k11
(k 11)
(k 11)
(k 11)
e i12(3k 21)13k11 ,2 ? 3k11 1 e i1(3k 21)13k11 ,2 ? 3k1121 1 e i13k11 ,2 ? 3k1122
if j 5 3k . i ,
e (2k?13k1)11,j 12(3k 21)13k11 1 e (2k?13k1)1121 ,j 1(3k 21)13k11 1 e (2k?13k1)1122 ,j 13k11
if i 5 3k . j ,
(k 11)
(k 11)
(k 11)
e 2 ? 3k11,2 ? 3k11 1 e 2 ? 3k1121 ,2 ? 3k1121 1 e 2 ? 3k1122 ,2 ? 3k1122
if i 5 j 5 3k.
(4) The definition for the (2, 1) block can be derived from the definition for the
(1, 2) block.
Since the blocks are invariant, the limit algebra, # has the form
#5
S#0
11
D
#12
,
#22
where #11 is isomorphic to the standard embedding algebra in M (3`) and #22 is
isomorphic to the subalgebra of M (3`) defined by lim(Tn , τ˜ n ) where τ˜ n is τ n ,
Å5
restricted to the (2, 2) block. Note also that
#11 5 F#F
and #22 5 (I 2 F )# (I 2 F ) ,
186
DAVID R. LARSON AND BARUCH SOLEL
(1)
(1)
where F 5 e (1)
11 1 e 22 1 e 33 is a projection in (Lat # ) > # . We shall show that # is
not structured, although it has a locally finite equivalence class. We suspect that
this might be true also for #22 but are unable to show it.
We shall now write G for the groupoid associated with lim(M2 ? 3k , τ k ) , P for the
Å5
partial order supporting # and X for the unit space of G (equal to the spectrum
(k )
of lim(D2 ? 3k , τk )). For every matrix unit e i,j we shall write eˆ (ijk ) for the
Å5
corresponding closed and open subset of G (the support set of e (ijk )). If i 5 j , then
(k )
ˆe ii ‘ X. Also write X 1 for the subset of X associated with F , that is,
X 1 5 !3i51 eˆ (1)
i ,i . Let X 2 be X \ X 1 . Note that G > (X 1 3 X 1) and P > (X 1 3 X 1)
are the groupoid and the partial order of the standard embedding algebra, and
G > (X 2 3 X 2) and P > (X 2 3 X 2) are the ones for #22.
Now fix
u 1 5 " eˆ (3kk,)3k P X 1
k
)
and y 1 5 " ˆe (3kk1
1 ,3k 11 P X 2 .
k
(k )
Note that for every k , e (3kk1111)
ˆ (3kk1111)
ˆ (3kk ,)3k 11 and
,3k1111 < τ k (e 3k ,3k 11) . Hence e
,3k1111 ‘ e
(k )
"k eˆ 3k ,3k11 defines a point in P (‘G ). This point is clearly (u 1 , y 1) and this
shows that [u1] 5 [y 1].
It is known that [u1] > X 1 is order isomorphic to 2N and n 1 is the maximal
element there. This follows from the analysis of equivalence classes in the
standard embedding algebra. In fact, it is known that this is the only equivalence
class in X 1 that has a maximal element. Note also that for every u P X ,
[u ] 5 ([u ] > X 1) < ([u ] > X 2)
and for every w1 P [u ] > X 1 and w2 P [u ] > X 2 , we have (w1 , w2) P P since
(X 2 3 X 1) > P 5 [. Thus, if [u ] is locally finite, [u ] > X 1 should have a maximal
element; hence [u ] 5 [u 1]. We conclude that, if # has a locally finite equivalence
class, then it is [u1] (5 [y 1]). We shall now show that [y 1] > X 2 is order isomorphic
to N and y 1 is the minimal element there. For this, fix y P [y 1] > X 2 . Then
(y 1 , y ) P (X 2 3 X 2) > G and it can be written as
" ˆe (1k1)3k,jk
k >m
k11 ,j
for some jk > 3k (as y 1 5 " ˆe (1k1)3k,113k) where eˆ (1k1131)
‘ eˆ (1k1)3k,jk; that is,
k 11
e (1k1131)
< τ k (e (1k1)3k,jk)
k11 ,j
k 11
for all k > m.
Examining the definition of τk , on the (2, 2) block, one finds that, for k 5 m , we
have either jm 5 j 1 3m (and then jk 5 j 1 3k , for all k > m ) for some j or jm 5 3m
and jm11 5 2 ? 3m11 2 2. In the latter case we get jm 12 5 (3m11 2 2) 1 3m12 and
jk 5 (3m11 2 2)3k for all k > m 1 1 .
Replacing m by m 1 1 we see that, in either case,
y 5 " eˆ j(1k)3k ,j13k
k >m
for some j (in the second case j 5 3m11 2 2). We shall write y j for this element and
it is now clear that
X 2 > [y 1] 5 hy j : j > 1j
STRUCTURED TRIANGULAR LIMIT ALGEBRAS
187
and j S y j is an order isomorphism from N onto X 2 > [y 1]. Since
[y 1] 5 (X 1 > [y 1]) < (X 2 > [y 1])
5 (X 1 > [u 1]) < (X 2 > [y 1])
and since P > (X 2 3 X 1) 5 [ , we see that [y 1] is order isomorphic to Z. In
particular, it is locally finite.
It is now left to show that [y 1] is not a structured equivalence class. This will
complete the proof that # is not a structured algebra. Suppose, by negation, that
# is structured with a structured class [y 1]. First note that for a given structured
presentation of # , say lim(Tmk , lk ) with a structured class [y 1] , there will be some
Å5
K0 such that, for all k > K0 , F is a sum of diagonal matrix units in Tmk . That will
enable us to write the matrix algebras in a block form. Considering
lim(F 'Tmk F ' , lk 3F 'Tmk F ') ,
Å5
we get a structured presentation for #22. It is also clear that, as [y 1] is the
structured class for # , [y 1] > X 2 is a structured class for #22. It is left, therefore,
to prove the following lemma.
LEMMA 3.5. The algebra #22 does not hay e a structured presentation with
structured class hy j : j > 1j.
Proof. Assume, by negation, that there is such a presentation, namely,
#22 5 lim(T3nk , s k ).
Å5
When we refer to this presentation we shall write f (ijk ) for the matrix units. We
also have the presentation
# 22 5 lim(T3k , τ˜ k )
Å5
where τ˜ k is the restriction of τk, defined above, to the (2, 2) block. For example, if
i , j ? 3k ,
k 11)
(k 11)
(k 11)
τ˜ k (e (i,jk )) 5 e (i1
(1)
.
2(3k 21) ,j 12(3k 21) 1 e i 1(3k 21) ,j 1(3k 21) 1 e i ,j
The definition of τ˜ k for the other cases is a similar modification of the definition
of τk (that is, ‘drop ‘‘13k11’’ ’). With this modification we now have
y j 5 " eˆ j(,jk) for j > 1 ,
k >mj
(2)
where mj is the smallest integer with 1 < j , 3mj . Note that given m and j with
1 < j , 3m , it is clear that "k >m eˆ j(,jk ) is well defined and is the immediate successor
k)
of "k>m eˆ (j2
1 ,j 21 ; we thus get y 1 , . . . , y 3m 21 .
We now assume that hy j j is the structured class associated with the structured
presentation. Using the construction of the structured class and the fact that here
it has a minimal element (y 1) , we see that there is some k0 such that for all
k > k 0 , and every 1 < i < 3nk ,
y i P ˆf i(,ik)
(3)
In fact, by starting the structured presentation from M3nk0 we can assume that for
every k ,
s k ( f (iik )) > f (iik 11) for 1 < i < 3nk .
188
DAVID R. LARSON AND BARUCH SOLEL
In particular f (11k ) < s k + .. . + s 1( f (1)
11 ) and
ˆf (k ) < ˆf (1)
for k > 1.
11
11
(4)
Note that f
is a projection in the diagonal of #22; hence it is a diagonal
projection in T3n for some large enough n (as e22 5 lim(T3n , τ˜ n )). Therefore it is
Å5
the sum of some projections from the set he (i,in ): 1 < i < 3n j. Since y 1 P eˆ (1n,1) > ˆf (1)
1 ,1 ,
we see that, for n large enough, say n > N ,
(1)
11
eˆ (1n,1) ‘ ˆf (1)
1 ,1 .
(5)
Note that y 3n 5 "m >n 11 e (3mn ,3)n (recall (2)) and
eˆ (3nn,13n1) ‘ eˆ (1n,1)
as τ˜ n (e (1n,1)) > e (3nn ,13n1) (see (1) with i 5 j 5 1). For n > N we see that y 3n P eˆ (1n,1) ‘ f (1)
1 ,1
(see (5)). On the other hand, y 3n P ˆf (3kn,)3n whenever n < nk . (See (3).)
Now fix l such that nl > N. Then, for n 5 nl we have
ˆ (l )
y 3n P ˆf (1)
(n 5 nl ).
11 > f 3n ,3n
(6)
But, since hs k j preserve the upper triangular matrices, it follows that
ˆf (knk1111) nk11 < s k ( f (knk) nk) for all k.
3
,3
3 ,3
ˆ (l )
ˆ (1)
Hence ˆf (3knk1111),3nk11 ‘ ˆf (3knk),3nk ‘ .. . ‘ ˆf (1)
3n1 ,3n1 and for n 5 nl , f 3n ,3n ‘ f 3n1 ,3n1 , which contradicts (6). The contradiction completes the proof of the lemma.
4. The iny erse limit associated with a structured presentation
Let ! be a structured algebra and let
! 5 lim(Tnk , wk )
Å5
be a given structured presentation. This means that we have contractive maps
gk : Mnk11 5 Mnk such that gk 3Tnk11 is multiplicative, gk + wk 5 idMnk and gk + gk11 5 gk
(gk are the back maps). We can write
g1
g2
Tn1 6ÅÅ Tn2 6ÅÅ Tn3 6ÅÅ .. . .
With such a diagram we can associate an inverse limit algebra as indicated in the
following proposition.
4.1. Let
PROPOSITION
(Tnk , gk )
be
as
aboy e.
Define
C 5 h M Ak P
o`k51 M Tn : gk (Ak11) 5 Ak j. With the norm and the algebraic structure inherited
from o M Mn , C is a Banach algebra. Moreoy er , it has the following property.
k
k
Giy en a Banach algebra E and homomorphisms θ k : E 5 Tnk satisfying i θ k i < 1
and gk + θ k 11 5 θ k , there is a contractiy e homomorphism θ : E 5 C such that
θ (x ) 5 o M θ k (x ).
Proof. The proof that C is a Banach algebra with the norm
i M Ak i 5 sup i Ak i
and the product
( M Ak )( M Bk ) 5 M Ak Bk
STRUCTURED TRIANGULAR LIMIT ALGEBRAS
189
is straightforward and uses the fact that gk is a contractive homomorphism on
Tnk11 .
Now assume that E and hθ k j are as in the statement of the proposition and
define θ : E 5 C by
θ (x ) 5
O M θ (x).
k
Then θ is clearly multiplicative since each θ k is, and contractive since i θ (x ) i 5
supk i θ k (x ) i < i x i .
Note that if we define jk : C 5 Tnk by jk ( M Ai ) 5 Ak then jk is a contractive
homomorphism and gk + jk 11 5 jk . In fact (C , h jk j) has the following uniqueness
property.
LEMMA 4.2. Let C 9 be a Banach algebra and h jk9 j be contractiy e homomorphisms
jk9 : C 9 5 Tnk such that gk + jk9 11 5 jk9 for k > 1 and " ker jk9 5 h0j. Assume that
(C 9 , h jk9 j) has the property indicated in Proposition 4 .1. That is , for ey ery Banach
algebra E and contractiy e homomorphisms θ k : E 5 Tnk with gk + θ k 11 5 θ k there is
a contractiy e homomorphism θ : E 5 C 9 with jk9 + θ 5 θ k . Then there is an isometric
isomorphism c : C 9 5 C such that jk + c 5 jk9 for k > 1.
Proof. If C 9 is as above, we use the properties of C and C 9 to get contractive
homomorphisms c : C 9 5 C and c 9: C 5 C 9 satisfying jk + c 5 jk9 and jk9 + c 9 5 jk .
Hence jk + c + c 9 5 jk and jk9 + c 9 + c 5 jk9 and, since " ker jj 5 " ker jk9 5 h0j ,
c + c 9 5 idC and c 9 + c 5 idC9 . Since i c i < 1 and i c 9 i < 1 , c is an isometric
isomorphism.
Now, let ! be a structured algebra with a structured presentation ! 5
lim(Tnk , wk ) and let C be the inverse limit associated with it and with a choice of
Å5
back maps. Write g˜ k for the contractive homomorphism of ! onto Tnk as in the
1
proof of Theorem 3.3 (g˜ k is w
˜2
k + c k in the notation there). Then we know that
g˜ k + gk 11 5 g˜ k ; hence, from Proposition 4.1 we get a contractive homomorphism
θ : ! 5 C satisfying
θ (x ) 5
O M g˜ (x).
k
Since " ker g˜ k 5 h0j , θ is injective. In fact, i x i 5 sup i g˜ k (x ) i ; hence θ is an
isometric embedding of ! into C .
Note that, given u P X , with equivalence class [u ] , there is a representation πu
of B defined as follows. Let H 5 l 2([u ]) and define π u ( f ) P B (H ) for f P @ by
(πu ( f )g )(y ) 5
O f (y , w)g(w).
w P[u ]
Now let ! ‘ B be a structured UHF algebra and fix a structured presentation
! 5 lim(Tnk , wk ) and a sequence hl (k )j , where l (k ) is associated with wk as in
Å5
Lemma 3.2. Then we get a unique equivalence class [u ] associated with this data
as in Proposition 3.4. In fact [u ] has been constructed as an increasing union
[u ] 5 !`k51 Ck of subintervals. With each Ck we now associate a projection
Fk P B (H ) 5 B (l 2[u ])) which is the orthogonal projection onto spanhey : y P Ck j ,
where hey : y P [u ]j is the standard orthonormal basis in l 2([u ]). Similarly, for
190
DAVID R. LARSON AND BARUCH SOLEL
every subset Ω ‘ [u ] we can define F (Ω) to be the projection onto spanhey : y P Ωj.
Let
1 5 hF (Ω): Ω ‘ [u ] is decreasingj.
Then 1 is a nest of projections and Alg 1 (5 hT P B (H ): (1 2 N )TN 5 0 for all
N P 1 j) is the algebra of operators whose matrix with respect to the basis hey j is
upper triangular.
LEMMA 4.3. Let C be the iny erse limit defined aboy e. Then C is isometrically
isomorphic to Alg 1 .
Proof. Recall that
H
C 5 M Ak P
O M T : g (A
`
k 51
nk
k
k 11
) 5 Ak
J
and
i M Ak i 5 sup i Ak i .
Let r : Alg 1 5 C be defined by
r (A) 5 M Fk AFk .
Here we use the projection Fk defined above and note that we can identify Fk AFk
with Tnk in the obvious way since Ck ‘ [u ] is an interval of length nk . Using this
identification, we see that the map gk (the back map for wk ) is simply
Fk11 AFk11 5 Fk Fk11 AFk11 Fk 5 Fk AFk . Hence r (A) P C. Conversely, take some
˜ k P Fk Alg 1 Fk in the obvious
M Ak P C. For each k we associate an operator A
˜ k ey , ew l 5 (Ak )w ,y if y , w P Ck and 0 otherwise. Note
way. That is, its matrix is kA
that by identifying Ck with h1, ..., nk j in an order-preserving way, we see that
˜ k 11 Fk 5 A
˜ k . Clearly,
(Ak )w ,y makes sense. Since gk (Ak 11) 5 Ak , we have Fk A
˜ k i < supj i Aj i and, thus, there is some subnet A
˜ k 9 that converges s -weakly to
iA
some A P B (H ). Clearly, A P Alg 1 and Fj AFj 5 limk 9 Fj Ak 9 Fj . But for k 9 > j ,
Fj Ak 9 Fj 5 Aj , and hence Fj AFj 5 Aj . It follows that r (A) 5 M Ak and r is
surjective onto C. Clearly i A i 5 sup i Fk AFk i 5 sup i Ak i 5 i M Ak i so that r is
an isometry from Alg 1 onto C . Since each Ck is an interval of [u ] , the maps
A 5 Fk AFk are homomorphisms on Alg 1 and, thus, r is a homomorphism.
COROLLARY 4.4. r 21 + θ 5 πu .
Proof. We have r 21(θ ( f )) 5 r 21( M g˜ k ( f )).
coefficients of r 21θ ( f ) are
For
y , w P Ck the matrix
(r 21(θ ( f )))y ,w 5 (g˜ k ( f ))y ,w 5 f (y , w ).
Since ! Ck 5 [u ] , we have r 21(θ ( f )) 5 π u ( f ).
Note that πu is a well-defined representation for every u P X. What we saw in
the above corollary is that when [u ] is structured, π u can be factored through a
subalgebra of o M Tnk .
STRUCTURED TRIANGULAR LIMIT ALGEBRAS
191
5. Residually finite -dimensional triangular algebras
A Banach algebra is called residually finite -dimensional (or simply RFD) if for
every x ? y in the algebra there is a finite-dimensional contractive representation
r such that r (x ) ? r ( y ). For C*-algebras this property has been studied in [1, 3, 6]
and elsewhere. If ! 5 lim(Tnk , wk ) is a structured algebra with back maps gk and
Å5
associated maps g˜ k : ! 5 Tnk as in the discussion following Lemma 4.2, then h g˜ k j
are finite-dimensional contractive representations that separate points in ! .
Hence ! is RFD.
In fact it suffices to assume that the triangular UHF algebra has a locally finite
equivalence class in order to conclude that the algebra is RFD. For this simply
write this equivalence class, say [u ] , as a union of increasing intervals,
[u ] 5 !`k51 Lk , and consider the representations π k defined by compressing π u
(defined in § 4) into spanhey : y P Lk j.
We now introduce a necessary and sufficient condition, in terms of the
supporting total order P ‘ G , for a triangular AF algebra to be RFD.
In order to discuss contractive representations of a strongly maximal TAF
algebra ! note first that each such representation r is completely contractive [4,
Theorem 5.4]; hence it is a compression of some C*-representation of the
C*-algebra @ , generated by ! , to a semi-invariant subspace.
From the work of Renault [10, Theorem 1.21] we know that a p -representation
π of @ is the integrated form of a representation (m , U , K ) of the equivalence
relation G associated with the AF algebra @ . Here K 5 hK (u )ju PX is a
measurable field of Hilbert spaces, m is a quasi-invariant measure on X (here
quasi-invariant means that if we let θ (u , y ) 5 (y , u ) then m , m + θ ) and U (u , y ) is
an isometry from K (y ) to K (u ) (for (u , y ) P G ) satisfying U (u , u ) 5 I and
U (u , y )U (y , w ) 5 U (u , w ) almost everywhere. To say that π is the integrated form
of (m , U , K ) means that, up to unitary equivalence, the space of π is K 5
e%
X K (u ) dm (u ) and π ( f ) is defined on K by the formula
kπ ( f )j , h l 5
E f (u, y )kU(u, y )j (y ), h (u)lD
21/2
(u , y ) d… (u , y ) ,
where D 5 dm / (dm + θ ) and … is the measure on G associated with m ; that is,
… 5 eX l m dm (u ) where l u is the counting measure on h(y , u ): y P [u ]j.
If r is a contractive representation of ! then, as mentioned above, there is
some C*-representation π associated with (m , U , K ) as above and a subspace
H ‘ K 5 e% K (u ) dm (u ) that is semi-invariant for π (! ) (that is, is the difference
of two invariant subspaces) such that
r (a ) 5 PH π (a )3H ,
where PH is the projection onto H. Since H is semi-invariant for π (D ) (where
D 5 ! > ! *) , it is invariant for π (D ). As π (D ) is the algebra of all diagonal
operators with respect to the decomposition K 5 e% K (u ) dm (u ) , we can decompose H 5 e% H (u ) dm (u ) where H (u ) ‘ K (u ) for all u P X.
Now assume that r is finite-dimensional, and hence dim H , ` . Then H 5
e% H (u ) dm (u ) becomes
O H(w )
n
H5
i 51
i
192
DAVID R. LARSON AND BARUCH SOLEL
and dim H (wi ) , ` . If we write Tij for D2 –2 (wi , wj )PH(wi)U (wi , wj )3H (wj ) , where
PH(wj) is the orthogonal projection of K (wj ) onto H (wj ) , then we get
1
O f (w , w )T j
n
(r ( f )j )j 5
k 51
j
k
jk k
for f P ! , j P H.
We can write r ( f ) 5 ( f (wj , wk )Tjk ). If (wj , wk ) ¸ G , we understand f (wj , wk ) to be
1
0. Write T (u , y ) 5 D2 –2(u , y )PH(u )U (u , y )3H (y ) and Q (r ) 5 h(u , y ): T (u , y ) 5 0j.
Then P + Q (r ) + P ‘ Q (r ) and for every (u , y ) ¸ Q (r ) , (u , y ) 5 (wi , wj ) for some
1 < i , j < n. Given (u , y ) ¸ Q (r ) , we have (u , y ) 5 (wi , wj ) and for every u < w <
y , (u , w ) , (w , y ) are not in Q (r ) as P + Q (r ) + P ‘ Q (r ). Hence w 5 wk for some
1 < k < n. It follows that, if T (u , y ) ? 0 , then the interval [u , y ] is finite. We can
now prove the following.
THEOREM 5.1. A triangular AF algebra is RFD if and only if
! hJ 3 J : J is a finite intery al in some equiy alence class j
is dense in G.
Proof. First assume that the set ! hJ 3 J : J ‘ [u ] is a finite intervalj is dense in
G and fix f P !. Since f is a continuous function on G , there is some (u , y ) in this
dense set such that f (u , y ) ? 0 . Since (u , y ) is in this set, there is some finite
interval in [u ] containing both u and y . Hence the set hw : u < w < y j is finite, say
of cardinality n. Define r : ! 5 Mn (C) by
r ( f ) 5 ( f (wi , wj ))
where hw : u < w < y j 5 hw1 , . . . , wn j as ordered sets. Then r is a contractive
representation of ! . In fact, it is a compression of πu to the semi-invariant
subspace of l 2([u ]) associated with hw : u < w < y j. Since r ( f ) ? 0 , we see that for
every f P ! there is some contractive finite-dimensional representation r with
r ( f ) ? 0. This proves that ! is RFD.
For the other direction, assume that the set in the statement of the theorem is
not dense. Then there is some non-zero f P ! such that f (u , y ) 5 0 whenever
[u , y ] (5 hw : u < w < y j) is finite. We shall show that for every contractive
finite-dimensional representation r of ! , r ( f ) 5 0. So fix such r and write
O f (w , w )T j
n
(r ( f )j )j 5
k 51
j
k
jk k
as in the discussion preceding the theorem. Here Tjk 5 T (wj , wk ). As was shown
there, if T (wj , wk ) ? 0 then [wj , wk ] is finite. But then f (wj , wk ) 5 0 as assumed
above and, thus, r ( f ) 5 0. This completes the proof.
References
1. R. EXEL and T. A. LORING, ‘Finite dimensional representations of free product C*-algebras’,
Internat. J. Math. 3 (1992) 469 – 476.
2. J. GLIMM, ‘On a certain class of operator algebras’, Trans. Amer. Math. Soc. 95 (1960) 318 – 340.
3. K. M. GOODEARL and P. MENAL, ‘Free and residually finite dimensional C*-algebras’, J. Funct.
Anal. 90 (1990) 391 – 410.
4. P. MUHLY and B. SOLEL, ‘Subalgebras of groupoid C*-algebras’, J. Reine Angew. Math. 402 (1989)
41 – 75.
STRUCTURED TRIANGULAR LIMIT ALGEBRAS
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Department of Mathematics
Texas A&M Uniy ersity
College Station
Texas 77843-3368
U.S.A.
Department of Mathematics
The Technion
Haifa 34000
Israel