J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
67Ae71.
CHARACTERISATION OF PRIMITIVE IDEALS OF
TOEPLITZ ALGEBRAS OF QUOTIENTS
Rizky Rosjanuardi Department of Mathematics Education.
Universitas Pendidikan Indonesia (UPI) Jl.
Dr.
Setiabudhi No.
229 Bandung 40154 rizky@upi.
Abstract.
Let e be a totally ordered abelian group, the topology on primitive ideal space of Toeplitz algebras Prim T .
can be identified through the upwards-looking topology if and only if the chain of order ideals is well-ordered.
Let I be an order ideal of such that the chain of order ideals of e/I is not well-ordered, we show that for any order ideal J ' I , the topology on primitive ideal space can be identified through the upwards-looking topology.
Also we discuss the closed sets in Prim T .
with the upwards-looking topology and characterize maximal primitive ideals.
Key words: Toeplitz algebra, totally ordered group, primitive ideal, quotient, characterisation.
Abstrak.
Misalkan e adalah grup abel terurut total, topologi pada ruang ideal primitif dari aljabar Toeplitz Prim T .
dapat diidentifikasi melalui topologi upwards-looking jika dan hanya jika rantai dari ideal urutan adalah terurut dengan rapi .
ell-ordere.
Misalkan I adalah sebuah ideal urutan sedemikian sehingga rantai dari ideal urutan dari e/I tidak terurut dengan rapi, diperlihatkan bahwa untuk sembarang ideal J ' I , topologi pada ruang ideal primitif dapat diidentifikasi melalui topologi upwards-looking.
Pada paper ini juga dibahas himpunan-himpunan tutup di Prim T .
di bawah topologi upwards-looking, dan karakterisasi dari ideal primitif maksimal.
Kata kunci: Aljabar Toeplitz, grup terurut total, ideal primitif, kuosien, karakterisasi 2000 Mathematics Subject Classification: 46L55.
Received: 28-10-2011, revised: 30-05-2012, accepted: 30-05-2012.
Rosjanuardi Introduction Suppose e is a totally ordered abelian group.
Let .
be the chain of order ideals of e, and X.
denotes the disjoint union
F I
{I : I OO .
} = {(I, ) : I OO .
OO I}.
Adji and Raeburn shows that every primitive ideal of Toeplitz algebra T .
of e is of the form Oe1 ker QI e where I is an order ideal of e and OO eC.
They also showed [?.
Theorem 3.
that there is a bijection L of X.
onto the primitive ideal space Prim T .
of Toeplitz algebra T .
given by L(I, ) := ker QI e Oe1 where OO eC satisfies |I = .
Adji and Raeburn [?] introduced a topology in X.
which is called the upwards-looking topology.
When .
is isomorphic with a subset of N O {O}, the bijection L is a homeomorphism [?.
Proposition 4.
, so the usual hull-kernel topology of Prim T .
can be identified through the upwards-looking topology in X.
Later.
Raeburn and his collaborators [?] showed that L is a homeomorpism if and only if .
is well-ordered, in the sense that every nonempty subset has a least element.
More recently.
Rosjanuardi and Itoh [?] characterised maximal primitive ideals of T .
A series of analysis on subsets of .
implies that any singleton set {} which consists of a character in eC is closed.
This implies that every maximal primitive ideal of T .
is of the following form L.
, ) = ker Qe e Oe1 Given a totally ordered abelian group e and an order ideal I.
In this paper, we apply the method in [?] and [?] to characterise maximal primitive ideal of Toeplitz algebra T .
/J) of quotient e/J when the chain of order ideal .
/I) is not well-ordered.
Upwards-looking Topology Let e be a totally ordered abelian group.
The Toeplitz algebra T .
of e is the C*-subalgebra of B(`2 .
)) generated by the isometries {Tx = Txe : x OO e } which are defined in terms of the usual basis by Tx .
y ) = ey x .
This algebra is universal for isometric representation of e [?.
Theorem 2.
e/I Let I be an order ideal of e.
Then the map x 7OeIe Tx I is an isometric representation of e in T.
/I).
Therefore by the universality of T .
, there is a e/I homomorphism QI : T .
OeIe T .
/I) such that QI (Tx ) = Tx I , and that QI is Suppose C.
I) denotes the ideal in T .
generated by {Tu TuO Oe Tv TvO :
Characterisation of Primitive Ideals v Oe u OO I } and IndeCI Ou (T .
/I), e/I ) is the closed subalgebra of C.
T .
/I)) sate/I Oe1 isfying f .
= h .
) for x OO eC, h OO I Ou .
It was proved in [?.
Theorem 3.
that there is a short exact sequence of C O -algebras:
0 Ie C.
I) Ie T .
Ie IndeCI Ou (T .
/I), e/I ) Ie 0.
in which II .
() = QI .
)Oe1 .
for a OO T .
OO eC, and is dual action of eC on T .
characterized by e (Tx ) = .
Tx .
The identity representation T e/I of T .
/I) is irreducible [?], it follows from [?.
Proposition 6.
that ker QI .
)Oe1 is a primitive ideal of T .
If X.
denotes the disjoint union
F I
{I : I OO .
} = {(I, ) : I OO .
OO I}, it was showed in [?.
Theorem 3.
that L(I, ) := ker QI e Oe1 where OO eC satisfies |I = , .
is a bijection of X.
onto Prim T .
Using the bijection L.
Adji and Raeburn describe a new topology on X which corresponds to the hull-kernel topology on Prim T .
This new topology, is later called the upwards-looking topology.
They topologise X by specifying the closure operation as stated in the following definition.
Definition 2.
[?] The closure F of a subset F of X is the set consisting of all pairs (J, ) where J is an order ideal and OO JI such that for every open I there exists I OO .
and N OO N for which I OC J and neighbourhood N of in J, (I.
N|I ) OO F .
Example 2.
[?.
Example .
We are going to discuss some description of sets in X.
by considering specific cases of e.
An observation on e := Z Oilex Z gives interesting results.
Let I be the ideal {.
, .
: n OO Z}, since I is the only ideal, we have X.
= 0Ct IIt eC.
Suppose 0 is a character in II defined by .
, .
7Ie e2Ain , and let F = .
Next we consider a character in eC defined by .
, .
7Ie e2Ai.
It is clear that |I = 0 .
Then OO FE , because every open neighbourhood N of in eC contains an element .
hich is nothing but it self ) such that its restriction on I gives a character in F .
It is clear that 6OO F , hence F is not closed in the upwards-looking topology for X.
Adji and Raeburn [?] proved that this is the correct topology to identify the hull-kernel topology of Prim T .
when e is a group such that the set .
of order ideal is order isomorphic to a subset of N O {O}.
In [?].
Raeburn and his collaborators extended the results in [?].
Their main theorem, says that Prim T .
is homeomorphic to X.
with the upwards-looking topology if and only if the totally ordered set .
is well-ordered in the sense that every non-empty subset has a least element.
Their technique uses classical Toeplitz operators as well as the Rosjanuardi universal property of T .
which was the main tool in [?].
Then they described Prim T .
when parts of .
are well-ordered.
Rosjanuardi in [?] improved the results in [?] to the case when .
is not well In [?.
Proposition .
it is stated that when .
is isomorphic to a subset of {OeO} O Z O {O}, then we can use the upwards-looking topology on X.
/I) to identify the topology on Prim T .
/I).
For general totally abelian group e, as long as there is an order ideal I such that every order ideal J ON I has a successor, the upwards-looking topology is the correct topology for Prim T .
/I) [?.
Proposition In [?.
Theorem .
it was proved that for any quotient e/I such that the chain .
/I) is isomorphic to a subset {OeO} O Z O {O}, for any order ideal J % I, the upwards-looking topology on .
/J) is the correct topology for Prim T .
/J).
Characterisation of Primitive Ideals Example ?? implies that any closed set in the point wise topology is not necessarily closed in the upwards-looking topology.
When it is applied to any complement F C of a set F , it arrives to a conclusion that any open set in the point wise topology, is not necessarily open in the upwards-looking topology.
This example motivated Rosjanuardi and Itoh [?] to prove more general cases.
Combining results in [?] with ones in [?] give characterisation results for more general cases than in [?].
Proposition 3.
Suppose that e is a totally ordered abelian group such that the chain .
of order ideals in e is isomorphic to a subset of {OeO} O Z O {O}.
For any I OO .
, the maximal primitive ideals of T .
/I) are of the form ker Qe/I .
/I )Oe1 .
Proof.
Let I OO .
The chain of order ideals in e/I is I OC J1 /I OC J2 /I OC .
where Ji OO .
and I OC Ji OC Ji 1 for all i.
Hence .
/I) is well ordered.
Give F d the set X.
/I) := {J/I : J OO .
I OC J} the upwards-looking topology, hence Le/I is a homeomorphism of X.
/I) onto Prim T .
/I) by Theorem 3.
1 of [?].
Proposition 6 of [?], then implies that X.
/I) is homeomorphic with Prim(T .
Theorem 11 of [?] then gives the result.
Proposition 3.
Suppose that e is a totally ordered abelian group, and let I be an order ideal in e such that every oreder ideal J ON I has a successor.
Then the maximal primitive ideals of T .
/I) are of the form ker Qe/I .
/I )Oe1 .
Proof.
Let I OO .
such that every order ideal J ON I has a successor.
Since each nontrivial element of .
/I) is of the form J/I for J OO .
and J ) I, every element of .
/I) has a successor.
This implies that .
/I) is well ordered.
Give F d the set X.
/I) := {J/I : J OO .
I OC J} the upwards-looking topology, hence Characterisation of Primitive Ideals Le/I is a homeomorphism by Theorem 3.
1 of [?].
The result then follows from Theorem 9 of [?].
Theorem 3.
Suppose that e is a totally ordered abelian group, and I OO .
such that .
/I) O = {OeO} O Z O {O}.
Let J OO .
such that J % I.
Then the maximal primitive ideals of T .
/J) are of the form ker Qe/J .
/J )Oe1 .
Proof.
Since every nontrivial ideal of e/I is of the form J/I where J OO .
and J ! I and for ideals J1 .
J2 such that J1 /I OI J2 /I implies J1 OI J2 , then may write .
/I) := {I = JOeO OI .
OI Jk /I OI Jk 1 /I OI .
OI e = JO }.
Now consider the subset I := {I = JOeO OI .
OI Jk OI Jk 1 OI .
OI JO = .
If J 6= I is an element of I, i.
e J OO .
such that J % I, the set .
/J) = {J OI K1 /J OI K2 /J.
e/J is well ordered.
Hence L is a homeomorphism of X.
/J) onto Prim T .
/J) by Theorem 3.
1 of [?].
The result is then follow from Theorem 9 of [?].
Acknowledgement.
This paper was completed when the author visited the Department of Mathematics.
Gunma University.
Maebashi-Japan, 2011.
Supported by Gunma University and UPI research fund.
The author would thank to these References