In this paper we give a criterion for an ideal of a TAF algebra to be meet
irreducible. We show that an ideal $J$ of $A$ is meet irreducile if and only if
the C$^*$-envelope of the quotient $A/J$ is primitive. In that case, A/J admits
a nest representation which extends to a *-representation of the C$^*$-envelope
for $A/J.$ We also characterize the meet irreducible ideals as the kernels of
nest representations; this settles the question of whether the n-primitive and
meet irreducible ideals coincide.Comment: 8 pages; accepted in JF

Davidson, Kenneth R.

Katsoulis, Elias

Peters, Justin

English

arXiv

AbstractIn this paper we give criteria for an ideal J of a TAF algebra A to be meet-irreducible. We show that J is meet-irreducible if and only if the C∗-envelope of A/J is primitive. In that case, A/J admits a faithful nest representation which extends to a ∗-representation of the C∗-envelope for A/J. We also characterize the meet-irreducible ideals as the kernels of bounded nest representations; this settles the question of whether the n-primitive and meet-irreducible ideals coincide

Davidson, Kenneth R

Elsevier - Publisher Connector 

Journal of Functional Analysis 200 (2003) 23–30
Meet-irreducible ideals and representations
of limit algebras
Kenneth R. Davidson,a,1 Elias Katsoulis,b,2 and Justin Petersc,*
aPure Mathematics Department, University of Waterloo, Waterloo, Ont., Canada N2L 3G1
bMathematics Department, East Carolina University, Greenville, NC 27858 USA
cMathematics Department, Iowa State University of Science and Technology, 400 Carver Hall,
Ames, IA 50011-2064 USA
Received 19 March 2002; accepted 17 May 2002
Abstract
In this paper we give criteria for an ideal J of a TAF algebraA to be meet-irreducible. We
show that J is meet-irreducible if and only if the Cn-envelope of A=J is primitive. In that
case, A=J admits a faithful nest representation which extends to a *-representation of the
Cn-envelope for A=J: We also characterize the meet-irreducible ideals as the kernels of
bounded nest representations; this settles the question of whether the n-primitive and meet-
irreducible ideals coincide.
r 2002 Elsevier Science (USA). All rights reserved.
MSC: 47L80
1. Introduction
Representation theory of operator algebras is still in its infancy. While for Cn-
algebras the fundamentals of representation theory have long been known, for
nonself-adjoint algebras there are hardly any results of a general nature. For
‘triangular operator algebras’ (a term which we leave undeﬁned), intuition suggests
that the fundamental building blocks for representation theory should be nest
*Corresponding author.
E-mail addresses: krdavids@uwaterloo.ca (K.R. Davidson), KatsoulisE@mail.ecu.edu (E. Katsoulis),
peters@iastate.edu (J. Peters).
1Partially supported by an NSERC Grant.
2Research partially supported by a Grant from ECU.
0022-1236/03/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved.
PII: S 0 0 2 2 - 1 2 3 6 ( 0 2 ) 0 0 0 5 8 - 7
representations [5–7]. In the category of Cn-algebras, the nest representations are
precisely the irreducible representations.
Recall that a nest representation is a representation for which the closed, invariant
subspaces form a nest (i.e., are linearly ordered). In his study of nonself-adjoint
crossed products, Lamoureux introduced the notion of n-primitive ideal. An ideal is
n-primitive if it is the kernel of a nest representation. Lamoureux has shown that in
various contexts in nonself-adjoint algebras the n-primitive ideals play a role
analogous to the primitive ideals in Cn-algebras. Thus, one can give the set of n-
primitive ideals the hull-kernel topology, and for every (closed, two-sided) ideal I in
the algebra, I is the intersection of all n-primitive ideals containing I; in other
words, I ¼ kðhðIÞÞ:
An ideal J of an algebra A is meet-irreducible if, for any ideals I1 and I2
containing J; the relation I1-I2 ¼ J implies that either I1 ¼ J or I2 ¼ J: In
the case of Tn; the algebra of upper triangular n  n matrices, meet-irreducible ideals
are obtained by ‘cutting a wedge’ from the algebra: let 1pi0pj0pn: The ideal
I ¼ fðaijÞ : aij ¼ 0; i0pipjpj0g
is meet-irreducible, and every meet-irreducible ideal of Tn has this form.
The relationship between meet-irreducible and n-primitive ideals is studied in a
variety of examples in [7], and in [3] meet-irreducible ideals in strongly maximal
triangular AF-algebras are characterized by sequences of matrix units and also in
terms of groupoids. In that paper it is shown that every meet-irreducible ideal is n-
primitive. This is done by constructing a nest representation. The converse question,
whether every n-primitive ideal is meet-irreducible, was left open.
In a recent work [2], the ﬁrst two authors examined the Cn-envelope of a quotient
A=J of a strongly maximal TAF algebra by an ideal J: They showed the Cn-
envelope is an AF Cn-algebra, even though the quotient A=J is not in general a
TAF algebra. It turns out that the Cn-envelope ofA=J is sensitive enough to detect
the meet irreducibility of J: In Theorem 2.3 we show that J is meet-irreducible if
and only if CnenvðA=JÞ; the Cn-envelope of A=J; is primitive. The theory of Cn-
envelopes provides the natural framework for studying results of this type. In
Theorem 2.4 we show that for a meet-irreducible ideal J; there exists a faithful and
irreducible *-representation of C
n
envðA=JÞ; whose restriction on A=J is a nest
representation. Since the converse is easily seen to be true, Theorem 2.4 provides
a characterization of meet-irreducible ideals in terms of the representation theory
for A:
The question of whether the kernel of a nest representation is a meet-irreducible
ideal emerged at the Ambelside, UK conference in summer, 1997. Subsequently
some progress was made. In [4] a partial result was obtained: if the TAF algebra A
has totally ordered spectrum, or if the nest representation p has the property that the
von Neumann algebra generated by pðA-AnÞ contains an atom, then kerðpÞ is
meet-irreducible. The solution presented in Theorem 2.6 is self-contained and does
not make use of the results of [3] or [4]. Thus the question is now settled for strongly
maximal TAF algebras.
K.R. Davidson et al. / Journal of Functional Analysis 200 (2003) 23–3024
Despite the fact that evidence at hand is limited, it nonetheless seems worthwhile
to ask.
Question 1.1. Are there any operator algebras for which the n-primitive ideals, and
the meet-irreducible ideals do not coincide?
2. The main results
We begin by recalling a result of Lamoureux [7].
Lemma 2.1. Let I be a closed, two-sided ideal in a separable Cn-algebra A: Then the
following are equivalent:
(i) I is n-primitive,
(ii) I is primitive,
(iii) I is prime,
(iv) I is meet-irreducible.
One can actually characterize when an AF Cn-algebra is primitive in terms
of its Bratteli diagram. Let A ¼ li
-
mðAi;jiÞ be an AF Cn-algebra and assume
that each Ai decomposes as a direct sum Ai ¼"j Aij of ﬁnite-dimensional
full matrix algebras Aij: A path G for A ¼ li-mðAi;jiÞ is a sequence fAijig
N
i¼1 so
that for each pair of nodes ðði; jiÞ; ði þ 1; jiþ1ÞÞ there exist an arrow in the
Bratteli diagram for A ¼ li
-
mðAi;jiÞ which joins them. It is known that A is
primitive iff there is a path G for A ¼ li
-
mðAi;jiÞ so that each summand of
Ai is eventually mapped into a member of G: We call such a path G an essential path
for A:
Beyond Cn-algebras, a meet-irreducible ideal need not be primitive. In [3], a
description of all meet-irreducible ideals of a TAF algebra was given in terms of
matrix unit sequences.
Deﬁnition 2.2. LetA ¼ li
-
mðAi;jiÞ be a TAF algebra. A sequence ðeiÞiXN of matrix
units from A will be called an mi-chain if the following two conditions are satisﬁed
for all iXN:
(A) eiAAi:
(B) eiþ1AIdiþ1ðeiÞ;
where Idiþ1ðeiÞ denotes the ideal generated by ei in Aiþ1:
If ðeiÞiXN is an mi-chain for A ¼ li-mðAi;jiÞ; let J be the join of all ideals which
do not contain any matrix unit ei from the chain. In [3, Theorem 1.2] it is shown
that for a TAF algebra A ¼ li
-
mðAi;jiÞ; given an mi-chain ðeiÞiXN ; the ideal
J associated with ðeiÞiXN is meet-irreducible. Conversely, every proper
K.R. Davidson et al. / Journal of Functional Analysis 200 (2003) 23–30 25
meet-irreducible ideal inA ¼ li
-
mðAi;jiÞ is induced by some mi-chain, chosen from
some contraction of this representation.
In this paper we give a characterization of the meet-irreducible ideals of TAF
algebras in terms of Cn-envelopes of quotient algebras. We need to recall the
notation and machinery from [2].
Let A ¼ li
-
mðAi;jiÞ be the enveloping Cn-algebra for a TAF algebra A ¼
li
-
mðAi;jiÞ: Let JDA be a closed ideal, and let Ji :¼ J-Ai: For each iX1; Si
denotes the collection of all diagonal projections p which are semi-invariant for Ai;
are supported on a single summand of Ai and satisfy ðpAipÞ-J ¼ f0g: We form
ﬁnite dimensional Cn-algebras
Bi :¼
X
pASi
"BðRan pÞ;
where BðRan pÞ denotes the bounded operators on Ran p: Of course, BðRan pÞ is
isomorphic to Mrank p: Let si be the map from Ai into Bi given by siðaÞ ¼P"
pASi papjRan p: The map sijAi factors as riqi where qi is the quotient map of Ai
onto Ai=Ji and ri is a completely isometric homomorphism of Ai=Ji into Bi:
Notice that Bi equals the Cn-algebra generated by riðAi=JiÞ:
We then consider unital embeddings pi of Bi into Biþ1 deﬁned as follows. For
each qASiþ1 we choose projections pASi which maximally embed into q under the
action of ji: This way, we determine multiplicity one embeddings of BðRan pÞ into
BðRan qÞ: Taking into account all such possible embeddings, we determine the
embedding pi of Bi into Biþ1:
Finally we form the subsystem of the directed limit B ¼ li
-
mðBi; piÞ corresponding
to all summands which are never mapped into a summand BðRan pÞ where p is a
maximal element of some Si: Evidently, this system is directed upwards. It is also
hereditary in the sense that if every image of a summand lies in one of the selected
blocks, then it clearly does not map into a maximal summand and thus already lies in
our system. By Davidson [1, Theorem III.4.2], this system determines an ideal I of
B: The quotient B0 ¼ B=I is the AF algebra corresponding to the remaining
summands and the remaining embeddings; it can be expressed as a direct limit
B0 ¼ li
-
mðB0i;p0iÞ; with the understanding that B0i ¼"jBij for these remaining
summands Bij of Bi: It can be seen that the quotient map is isometric onA=J and
that B0 is the Cn-envelope of A=J:
Theorem 2.3. Let A be a TAF algebra and let JDA be an ideal. Then J is meet-
irreducible if and only if the algebra CnenvðA=JÞ is primitive.
Proof. Assume that B0 ¼ CnenvðA=JÞ is primitive and let G ¼ ðBijiÞNi¼1 an essential
path for B0: Let ei in Biji be the characteristic matrix units for Biji ; i.e., the ones on
the top right corner of Biji :
K.R. Davidson et al. / Journal of Functional Analysis 200 (2003) 23–3026
Assume that there exist ideals I1 andI2; properly containingJ: Since I1 andI2
properly contain J; there exist matrix units fkAIk with fkeJ; k ¼ 1; 2: So the
images of the fk appear in the presentation for the C
n-envelope in perhaps different
summands. However, the existence of an essential path G implies that some
subordinates for the fk will appear in some member of G; say Biji ; for i sufﬁciently
large, and so eiAI1-I2: However, eieJ and so J is properly contained in
I1-I2: It follows J is meet-irreducible.
Conversely, assume that J is meet-irreducible. In light of Lemma 2.1 and the
subsequent comments, it sufﬁces to show that the trivial ideal f0g is meet-irreducible
in the Cn-envelope B0:
By way of contradiction assume that there are nontrivial ideals I1 andI2 of B
0 so
that I1-I2 ¼ f0g: We claim that ðA=JÞ-Ikaf0g; k ¼ 1; 2: Indeed, any
nontrivial summand of Ik will eventually be mapped into a direct summand Biji
of B0 corresponding to some maximal element of Si: Hence all matrix units in Biji
belong to Ik; including the characteristic one. This one however also belongs to
A=J and therefore in the intersection ðA=JÞ-Ik:
The claim shows that the zero ideal is not meet-irreducible in A=J: By
considering the pullback, this implies that J is not meet-irreducible in A; which is
the desired contradiction.
Notice that the sequence ðeiÞNi¼1 associated with the path G in the proof above
satisﬁes conditions (A) and (B) of Deﬁnition 2.2 and is therefore an mi-chain for the
ideal J:
Theorem 2.4. If A is a TAF algebra and J an ideal of A; then the following are
equivalent:
(i) There exists a faithful representation t : CnenvðA=JÞ-BðHÞ so that tðA=JÞ is
weakly dense in some nest algebra.
(ii) J is meet-irreducible.
Proof. Assume that (i) is valid and let t : CnenvðA=JÞ-BðHÞ be a faithful
representation so that tðA=JÞ weakly dense in some nest algebra AlgN: By way
of contradiction assume thatJ is not meet-irreducible. Theorem 2.3 and Lemma 2.1
imply the existence of nonzero closed ideals I1 and I2 in C
n
envðA=JÞ so that
I1I2 ¼ f0g: The nonzero subspaces ½tðIiÞH must be mutually orthogonal.
However they are both invariant under tðA=JÞ; and therefore belong to N; a
contradiction.
Conversely, assume that (ii) is valid and so, by Theorem 2.3, we know that
CnenvðA=JÞ is primitive. Retain the notation established in the paragraphs preceding
Theorem 2.3. Hence
CnenvðA=JÞ ¼ B0 ¼ li-mðB
0
i; p
0
iÞ;
K.R. Davidson et al. / Journal of Functional Analysis 200 (2003) 23–30 27
where B0i ¼"jBij for the remaining summands Bij of Bi: Let G ¼ ðBijiÞNi¼1 be the
essential path for B0:
Each Bij is a full matrix algebra and therefore contains the algebra Tij of upper
triangular matrices. Form the ﬁnite-dimensional algebrasT0i ¼"jTij and consider
the direct limit algebra
T0 ¼ li
-
mðT0i; p0iÞ;
where p0i is as earlier. Clearly, T
0 is a TAF algebra whose enveloping Cn-algebra is
B0: Moreover, T0 contains A=J:
We deﬁne a state o on B0 as follows. Let ðpiÞNi¼1 be a sequence of diagonal
projections with piATiji so that piþ1 is a subordinate of pi; iAN: We deﬁne
oi :B
0
i-C to be the compression on pi and we let o to be the direct limit o ¼ li-moi:
Consider the GNS triple ðt;H; gÞ associated with the state o; i.e., t is a
representation of B0 on H and gAH so that oðaÞ ¼ /tðaÞg; gS; aAB0: Since o
is pure, t is irreducible. Moreover, piABiji ; iAN and so t is also faithful.
An alternative presentation for ðt;H; gÞ was given in [8, Proposition II.2.2]. Since
o is multiplicative on the diagonal T0-ðT0Þn; one considers H to be L2ðX; mÞ;
where X is the Gelfand spectrum of T0-ðT0Þn and m the counting measure on the
orbit of o in X: With these identiﬁcations, given any matrix unit e; tðeÞ is the
translation operator on X deﬁned in the paragraphs preceding [8, Theorem II.1.1].
In [8, Proposition II.2.2] it is shown that t mapsT0 onto a weakly dense subset of
some nest algebra. The proof of the theorem will follow if we show that the weak
closure of tðA=JÞ contains tðT0Þ:
A moment’s reﬂection shows that given any contraction aAT0i j and matrix units
e1; e2;y; en and f1; f2;y; fn in B
0; there exists a contraction aˆAA=J so that
oðf nk aˆekÞ ¼ oðf nk aekÞ
and therefore
/tðaˆÞtðekÞg; tðfkÞgS ¼ /tðaÞtðekÞg; tðfkÞgS
for all k ¼ 1; 2;y; n: However the collection of all vectors of the form tðeÞg; where e
ranges over all matrix units of B0; forms a dense subset of H and so the desired
density follows.
Remark 2.5. (1) Implication (i) ) (ii) also follows from Theorem 2.6.
(2) There exists a faithful representation t of CnenvðA=JÞ inBðHÞ so that tðA=JÞ
is weakly dense in a nest algebra if and only if there is a faithful irreducible
representation t of CnenvðA=JÞ in BðHÞ so that tðA=JÞ is weakly dense in a nest
algebra.
K.R. Davidson et al. / Journal of Functional Analysis 200 (2003) 23–3028
Theorem 2.6. Let A be a strongly maximal TAF algebra, and let p be a bounded
nest representation of A on a Hilbert space H: Then kerðpÞ is a meet-irreducible
ideal.
Proof. Since a bounded representation of the diagonal masa A-An is completely
bounded, cf. [9, Theorem 8.7], and a completely bounded representation is similar to
a completely contractive representation by Paulsen [9, Theorem 8.1], we may assume
that the restriction of p to the diagonal masa is completely contractive. It fol-
lows that the restriction of p to the diagonal masa is a star representation. Let
J ¼ kerðpÞ; and J1;J2 be ideals in A properly containing J: We need to show
that J1-J2 properly contains J:
Since p is a nest representation, we have (after possibly interchanging J1 and J2)
that
ð0Þa½pðJ1ÞHD½pðJ2ÞH;
where ½X denotes the closed subspace generated by XCH: Fix nAN and let u be a
matrix unit in J1-An\J: Choose a vector hAH be such that jjpðuÞhjj ¼ 1: There
exist mXn; NX1; matrix units vtAJ2-Am and vectors htAH for 1ptpN such
that
pðuÞh 
XN
t¼1
pðvtÞht



o14:
In particular, jjPNt¼1 pðvtÞhtjj43=4: We may assume that pðvtÞa0 for all t:
Deﬁne a projection E ¼ WNt¼1 pðetÞ ¼ pðWNt¼1 etÞ; where et ¼ vtvnt are diagonal
matrix units (which need not be distinct). For all s; t;
pðesÞpðvtÞ ¼ pðvsvns vtÞ ¼
pðvtÞ if vsvns ¼ vtvnt ;
0 otherwise:
(
So we have E
P
pðvtÞht ¼
P
pðvtÞht: Now
EpðuÞh 
XN
t¼1
pðvtÞht



 ¼ E pðuÞh 
XN
t¼1
pðvtÞht
 !



p pðuÞh 
XN
t¼1
pðvtÞht



o14:
Therefore jjEpðuÞhjj41=2: In particular, there exists at least one t; 1ptpN; such
that pðetÞpðuÞa0:
Embed uAAn+Am and decompose it as a sum u ¼
P
us of matrix units in Am:
Then etu ¼ us; for some s; and so it follows pðusÞa0; i.e., useJ: Thus we have
identiﬁed matrix units usAJ1\J and vtAJ2\J ofAm with the same ﬁnal projection.
K.R. Davidson et al. / Journal of Functional Analysis 200 (2003) 23–30 29
Say us ¼ eðm;rÞij and vt ¼ eðm;rÞik : We now distinguish three cases:
If j ¼ k; then us ¼ vtAJ1-J2\J;
If jok; then vt ¼ useðm;rÞjk AJ1-J2\J;
If j4k; then us ¼ vteðm;rÞkj AJ1-J2\J:
It follows that in all three cases J1-J2 properly contains J: Thus J is meet-
irreducible.
Acknowledgments
We would like to thank Alan Hopenwasser for several suggestions, including one
leading to the ﬁnal version of Theorem 2.6.
References
[1] K. Davidson, C-algebras by example. Fields Institute Monographs, American Mathematical Society,
Providence, RI, 1996.
[2] K. Davidson, E. Katsoulis, Primitive limit algebras and C-envelopes, Adv. Math., to appear.
[3] A. Donsig, A. Hopenwasser, T. Hudson, M. Lamoureux, B. Solel, Meet irreducible ideals in direct
limit algebras, Math. Scand. 87 (2000) 27–63.
[4] A. Hopenwasser, J. Peters, S. Power, Nest representations of TAF algebras, Canad. J. Math. 52 (2000)
1221–1234.
[5] M. Lamoureux, Nest representations and dynamical systems, J. Funct. Anal. 114 (1993) 345–376.
[6] M. Lamoureux, Ideals in some continuous nonselfadjoint crossed product algebras, J. Funct. Anal.
142 (1996) 211–248.
[7] M. Lamoureux, Some triangular AF algebras, J. Operator Theory 37 (1997) 91–109.
[8] J. Orr, J. Peters, Some representations of TAF algebras, Paciﬁc. J. Math. 167 (1995) 129–161.
[9] V.I. Paulsen, Completely Bounded Maps and Dilations, Longman Scientiﬁc, New York, Wiley, 1986.
K.R. Davidson et al. / Journal of Functional Analysis 200 (2003) 23–3030


Meet Irreducible Ideals and Representations of Limit Algebras

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