Algebraic sets of types A, B, and C coincide

0.22 MB·

by  Torben Maack Bisgaard

6
1
0 Algebraic sets of types A, B, and C coincide
2

n Torben Maack Bisgaard and Jan Stochel
a
J

3 Abstract. It is proved that the definition of an algebraic set of type A (a
2 notionrelatedtothemultidimensionalHamburgermomentproblem)doesnot
depend onthe choiceofapolynomialdescribingthealgebraicsetinquestion

] and that an algebraic set of type B is always of type A. This answers in the
A
affirmativetwoquestionsposedin1992bythesecondauthor. Itisalsoshown
F thatanalgebraicsetisoftypeAifandonlyifitisoftypeC(anotionlinked
. toorthogonalityofpolynomialsofseveralvariables). This,inturn,enablesus
h toanswerthreequestionsposedin2005byCichon´,Stochel,andSzafraniec.
t
a
m
[

1 1. Introduction
v
ItiswellknownthateachHamburgermomentn-sequenceonanalgebraicsub-
4
6 setV of Rn is positive definite andsatisfies a certainsystem ofalgebraicequations
2 generated by V. If the converse holds, the algebraic set V is called of type A or B,
6 depending on the system of equations (cf. [39]). The knowledge of various classes
0 ofalgebraicsets oftype Aseems to be ofgreatimportance becauseofapplications,
.
1 not only in moment problems, but also in the theory of orthogonal polynomials
0 in several variables (cf. [12]). We now provide a brief overview of some known
6 classes of algebraic sets of type A. First, note that any algebraic subset V of R is
1
of type A. Indeed, the case of V = R follows from Hamburger’s theorem (cf. [4]),
:
v while the case of V R can be inferred from [12, Lemma 49] (or [12, Theorem
Xi 43]). In turn, each compact algebraic subset of Rn is of type A (see [12, Theorem
43] which heavily depends on [37, Theorem 1]). By [39, Theorem 5.4], every al-
r
a gebraic subset of R2 induced by a non-zero polynomial P R[X ,X ] of degree
1 2
at most 2 is of type A. However, as shown in [39, Theorem∈ 6.3], there exists an
algebraic subset V of R2 induced by a polynomial P R[X ,X ] of degree 3 (e.g.
1 2
P =X (X X2)) such that V is not of type A (usin∈g a quotient technique from
2 2− 1
Section6,thecarefulreadercandeducefrom[29]thatthepolynomialP =X3 X2
induces an algebraic subset of R2 which is not of type A). The question of1w−hen2
an algebraic subset of Rn induced by a polynomial P R[X ,...,X ] of the form
1 n
∈
1991 Mathematics Subject Classification. Primary44A60,14P05;Secondary13F20, 42C05.
Key words and phrases. Polynomial ideal, realradical, positive definite functional, multidi-
mensionalmomentproblem,(real)algebraicsetoftypeA,B,C,orthogonalpolynomialsinseveral
variables.
RunningexpensesofthefirstauthorwerecoveredbytheCarlsbergFoundation. Thesecond
authorhasbeensupportedbytheMNiSzWgrantN20102632/1350.
1
2 TORBENMAACKBISGAARDANDJANSTOCHEL
P = Xα1 Xαn Xβ1 Xβn (α ,...,α and β ,...,β are nonnegative inte-
gers) is1of··ty·penA−has1be·e·n· conmplet1ely ansnwered in1 [6]. Tnhe interested reader is
referredto [34, 22, 23, 18, 11, 1, 36, 28, 21, 39, 40, 6, 41, 42, 12] for further
information concerning this subject including other classes of sets of type A (see
also [2, 27, 10, 37, 17, 33, 13, 14, 31, 38, 19, 30, 15, 16] for moments on
semi-algebraic sets and related problems).
In this paper we show that the definition of type A is correct and that types
A and B coincide (cf. Theorems 3.1 and 3.2 as well as Theorems 4.4 and 4.5).
This answers in the affirmative two questions posed in [39]. The main tool in our
considerations is the Real Nullstellensatz, the celebrated theorem of real algebraic
geometry. Asubstantialpartofthepaperisdevotedtothestudyofpositivedefinite
functionals vanishing on polynomial ideals.
The notion of type C was invented in [12] to distinguish a class of polyno-
mial ideals which have the property that each sequence of polynomials in several
variablessatisfyinganappropriatethreetermrecurrencerelationmodulothe poly-
nomial ideal is orthogonal with respect to some positive measure. We prove that
an algebraic subset V of Rn is of type A if and only if the set ideal c(V) is of
type C (cf. Theorem 7.4), which answers in the affirmative QuestionI2 posed in
[12]. In fact, Theorem7.4enables us to answerin the negativetwo morequestions
(Questions 3 and 4) posed in [12]. Namely, we show that there exists a non-zero
set ideal of C[X ,...,X ] (n > 2) which is not of type C and that the zero ideal
1 n
of C[X ,...,X ] (n > 2) is not of type C. Let us point out that [12, Question 4]
1 n
was answeredin the negativefor N =2 by Friedrich [20]; he constructeda strictly
positive definite linear functional on C[X ,X ] which is not a moment functional,
1 2
showing that the zero ideal of C[X ,X ] is not of type C. The proof of Theorem
1 2
7.4 depends on a quotient technique for finitely generated commutative (real or
complex) -algebras with unit worked out in Section 6.
Types∗A,B,andCareshowntobeequivalentalsointhecontextofthemultidi-
mensionalcomplexmomentproblem(cf.Section8). ThepaperendswithAppendix
which contains complex counterparts of Lemmata 2.2 and 2.3.
Let us point out that in Sections 2 to 5 we use the notation (P), (Z) and
I
only in the case of the real algebra R[X ,...,X ], while in Sections 6, 7, and 8
Λ 1 n
I
the same notation is used also in the case of the complex -algebra C[X ,...,X ].
1 n
∗
Since it is clear from the context which algebra is considered, there is no clash of
notation. In Sections 4 and 5, and Appendix A the complexified notation (P)c,
c
Ic(Z) and IΛ is employed in order to distinguish it from the real one in the case
when both are used simultaneously.
2. Positive definite functionals vanishing on ideals
In what follows R and C stand for the fields of real and complex numbers
respectively. Let n be a positive integer. Denote by R[X ,...,X ] the real alge-
1 n
bra of all polynomials in n commuting indeterminates X ,...,X with real coeffi-
1 n
cients. Given a polynomial P R[X ,...,X ], a subset Z of Rn and an ideal I of
1 n
∈
R[X ,...,X ], we write
1 n
(P)= the ideal of R[X ,...,X ] generated by P,
1 n
(Z)= P R[X ,...,X ]: P(x)=0 for every x Z ,
1 n
I { ∈ ∈ }
(I)= x Rn: P(x)=0 for every P I . (2.1)
Z { ∈ ∈ }
ALGEBRAIC SETS OF TYPES A, B, AND C COINCIDE 3
The following properties of () and () are easy and elementary to prove.
Z · I ·
Lemma 2.1. If I and J are ideals of R[X ,...,X ] such that I J, then
1 n
⊂
I ( (I)) ( (J)). If Z Rn, then Z ( (Z)) and (Z)= ( ( (Z))).
⊂I Z ⊂I Z ⊂ ⊂Z I I I Z I
An ideal I of R[X ,...,X ] is said to be a set ideal if there exists a subset
1 n
Z of Rn such that I = (Z). It is worth noting that there are ideals which are
I
not set ideals (e.g. I = (Xs), where s > 2 is an integer). A set of the form (I),
1 Z
where I is an ideal of R[X ,...,X ], is called a (real) algebraic set. It is known
1 n
thatforeachalgebraicsubsetV ofRn thereexistsapolynomialP R[X ,...,X ]
1 n
∈
such that V = := x Rn: P(x) =0 (indeed, by Hilbert’s basis theorem the
P
Z { ∈ }
ideal I is generatedby finitely many polynomials Q ,...,Q R[X ,...,X ] and
1 m 1 n
∈
so (I) = with Q = Q2 +...+Q2 ). If V = , then we say that V is an
Z ZQ 1 m ZP
algebraic set induced by P. For fundamentals of the theory of real algebraic sets
we recommend the monographs [3, 9].
Let I be an ideal of R[X ,...,X ]. We say that I is real if for every fi-
1 n
nite sequence Q ,...,Q R[X ,...,X ] such that Q2 + +Q2 I, we have
1 s ∈ 1 n 1 ··· s ∈
Q ,...,Q I. Denote by √RI the real radical of I, that is the intersection of all
1 s
∈
realprimeidealsofR[X ,...,X ]containingI (ifthereisnosuchprimeideal,then
1 n
we put √RI =R[X ,...,X ]).
1 n
We now recall a useful description of the real radical.
Lemma 2.2 ([9, Proposition 4.1.7]). Let I be an ideal of R[X ,...,X ]. Then
1 n
√RI isthesmallestrealidealofR[X ,...,X ]containingI. Moreover, apolynomial
1 n
P R[X ,...,X ]belongsto √RI ifandonlyifthereexistfinitelymanypolynomials
1 n
∈
Q ,...,Q R[X ,...,X ]andanintegerm>0suchthatP2m+Q2+ +Q2 I.
1 s ∈ 1 n 1 ··· s ∈
The following variant of the Real Nullstellensatz plays a crucial role in the
present paper.
Lemma 2.3 ([9, Corollary 4.1.8]). If I is an ideal of R[X ,...,X ], then
1 n
( (I))= √RI.
I Z
A linear functional Λ: R[X ,...,X ] R is said to be positive definite if
1 n
→
Λ(P2) > 0 for every P R[X ,...,X ]. Note that every positive definite linear
1 n
∈
functional Λ: R[X ,...,X ] R satisfies the Schwarz inequality
1 n
→
Λ(PQ) 6Λ(P2)1/2Λ(Q2)1/2, P,Q R[X ,...,X ]. (2.2)
1 n
| | ∈
Given a positive definite linear functional Λ: R[X ,...,X ] R, we write
1 n
→
= P R[X ,...,X ]: Λ(P2)=0 .
Λ 1 n
I { ∈ }
We now state three basic properties of the set .
Λ
I
Lemma 2.4. If Λ: R[X ,...,X ] R is a positive definite linear functional,
1 n
→
then1
(i) is the greatest ideal of R[X ,...,X ] contained in the kernel of Λ,
Λ 1 n
I
(ii) = √R ,
Λ Λ
I I
(iii) is a set ideal.
Λ
I
1Itfollowsfrom(ii)that Λ=√ ΛbecauseI √I R√IforeveryidealIofR[X1,...,Xn],
I I ⊂ ⊂
√I beingtheradicalofI (see[24]forthebasicpropertiesoftheradical).
4 TORBENMAACKBISGAARDANDJANSTOCHEL
Proof. (i)SincethemappingP Λ(P2)1/2 isaseminormonR[X ,...,X ],
1 n
7→
we deduce that is a vector space. Using (2.2), we get
Λ
I
Λ((PQ)2) 6Λ(P2)1/2Λ(P2Q4)1/2 =0, P , Q R[X ,...,X ],
Λ 1 n
| | ∈I ∈
which shows that is an ideal of R[X ,...,X ]. Applying (2.2) to P and
Λ 1 n Λ
I ∈ I
Q = 1, we see that is contained in the kernel kerΛ of Λ. Moreover, if J is an
Λ
I
ideal of R[X ,...,X ] such that J kerΛ, then for every P J, P2 J kerΛ
1 n
⊂ ∈ ∈ ⊂
and consequently P ; hence J , which proves (i).
Λ Λ
∈I ⊂I
(ii)ByLemma2.2,itisenoughtoshowthattheideal isreal. IfQ ,...,Q
Λ 1 s
I ∈
R[X ,...,X ] are such that Q2+ +Q2 , then by (i) we have
1 n 1 ··· s ∈IΛ
0=Λ(Q2+ +Q2)=Λ(Q2)+ +Λ(Q2),
1 ··· s 1 ··· s
which,whencombinedwithpositivedefinitenessofΛ,impliesthatQ ,...,Q .
1 s Λ
∈I
(iii) is a direct consequence of (ii) and Lemma 2.3. (cid:3)
The next lemma constitutes the main tool for our further investigations.
Lemma 2.5. Let I be an ideal of R[X ,...,X ] and let Λ: R[X ,...,X ] R
1 n 1 n
→
be a positive definite linear functional. Then Λ vanishes on I if and only if it
vanishes on ( (I)).
I Z
Proof. Since I ( (I)), it suffices to prove the “only if” part of the con-
⊂ I Z
clusion. If Λ vanishes on I, then, by Lemma 2.4(i), we get I . This, together
Λ
⊆I
with Lemma 2.1 and Lemma 2.4 (iii) and (i), implies that
( (I)) ( ( ))= kerΛ,
Λ Λ
I Z ⊆I Z I I ⊆
which completes the proof. (cid:3)
3. Types A and B coincide
A functional Λ: R[X ,...,X ] R is called a moment functional on a closed
1 n
→
subset F of Rn if there exists a positive Borel measure µ on F such that2
Λ(Q)= Q(x) dµ(x), Q R[X ,...,X ]. (3.1)
Z ∈ 1 n
F
A positive Borel measure µ on F satisfying (3.1) is called a representing measure
of Λ. We shall abbreviate the expression “moment functional on Rn” simply to
“moment functional”. A representing measure of a moment functional may not be
unique(cf.[4]). Obviously,eachmomentfunctionalonaclosedsubsetF ofRn isa
moment functional (but not conversely) and each moment functional is a positive
definite linear functional (but not conversely except for n=1, cf. [4, 20, 7, 8]).
Consider a polynomial P R[X ,...,X ]. In view of [39, Proposition 2.1], a
1 n
∈
moment functional Λ is a moment functional on if and only if Λ vanishes on
P
Z
the principal ideal (P), or equivalently on the ideal ( ). Hence, each moment
P
I Z
functional on is a positive definite linear functional vanishing on ( ), and
P P
consequently oZn (P). Following [39], we say that P is of type A (respIecZtively B)
if each positive definite linear functional Λ: R[X ,...,X ] R vanishing on (P)
1 n
→
(respectively ( )) is a moment functional; any such functional is automatically
P
a moment funIctZional on . Clearly, each polynomial of type A is of type B.
P
Z
2We tacitly assume that all polynomials are absolutely integrable with respect to µ; in
particular,µhastobefinite.
ALGEBRAIC SETS OF TYPES A, B, AND C COINCIDE 5
Theorem 3.1. Let P,Q R[X ,...,X ] be such that = . Then P is of
1 n P Q
type A if and only if Q is of t∈ype A. Z Z
Proof. Suppose that P is of type A. Set I =(P) and J =(Q). Then
( (I))= ( )= ( )= ( (J)). (3.2)
P Q
I Z I Z I Z I Z
If a positive definite linear functional Λ: R[X ,...,X ] R vanishes on J, then
1 n
→
by (3.2) and Lemma 2.5, the functional Λ vanishes on ( (I)) I. Since P is of
type A, Λ is a moment functional. Hence Q is oftype AI. BZy sym⊃metry the proof is
complete. (cid:3)
LetusrecalltheoriginaldefinitionsofalgebraicsetsoftypesAandBfrom[39].
Definition. An algebraic subset V of Rn induced by a polynomial P
R[X ,...,X ] is of type A (respectively B) if P is of type A (respectively B). ∈
1 n
Theorem 3.1 implies that the definition of algebraic sets of type A is correct;
thatis,itshowsthatthemeaningofthe statement“V is oftype A”is independent
ofthe choiceof a polynomial P whichinduces the algebraicset V. This answersin
the affirmativeaquestionposedin[39,Section4]. Certainly,the definitionoftype
B is correct.
The next theorem is an immediate consequence of Lemma 2.5 (with I =(P)).
It answers in the affirmative another question posed in [39, Section 4].
Theorem 3.2. An algebraic subset of Rn is of type A if and only if it is of
type B.
4. Complexification
Inarecentpaper[12],typesAandBhavebeen“complexified”. Thereasonfor
this modification comes from the spectral theory of Hilbert space operators which
is well developed in the complex case. We now discuss this in more detail.
In what follows, members of := C[X ,...,X ] and := R[X ,...,X ]
n 1 n n 1 n
P R
will be identified with complex and real polynomial functions on Rn respectively.
The set is a -algebra with involution (P +iQ)∗ :=P iQ for P,Q . We
n n
say thatPa comp∗lex linear functional Λ: C is positive−definite if Λ(∈PR∗P)> 0
n
P →
for all P . If Λ is a positive definite complex linear functional on , then
n n
∈P P
∗
Λ(P )=Λ(P), P (equivalently: Λ( ) R),
n n
∈P R ⊂
and so Λr := Λ|Rn is a positive definite real linear functional on Rn. The map-
ping Λ Λr is a bijection between the set of all positive definite complex linear
7→
functionals on and the set of all positive definite real linear functionals on .
n n
P R
Given a positive definite complex linear functional Λ on and Z Rn, we write
n
P ⊂
c ∗
= P : Λ(P P)=0 ,
IΛ { ∈Pn } (4.1)
c(Z)= P n: P(x)=0 for every x Z .
I { ∈P ∈ }
Moreover, we denote by (P)c the ideal of n generated by P n. We point out
P ∈ P
that the sets , (Z) and (P) defined in Section 2 are ideals of , the first two
Λ n
I I c R
being real, and that the sets IΛ and Ic(Z) defined in (4.1) are ideals of Pn that
maynotbereal( istheonlyrealidealof ). Thefollowingsimplefactisstated
n n
P P
without proof (we write = x Rn: P(x)=0 also for P ).
P n
Z { ∈ } ∈P
6 TORBENMAACKBISGAARDANDJANSTOCHEL
Lemma 4.1. Let P . A positive definite complex linear functional Λ on
n
n vanishes on (P)c (re∈spPectively c( P)) if and only if Λr vanishes on (P∗P)
P(respectively ( )). Moreover, ifIP Z , then (P∗P) can be replaced in the
P n
I Z ∈ R
above equivalence by (P).
We now formulate “complexified” variants of Lemmata 2.4 and 2.5 (see also
Appendix for the complex counterparts of Lemmata 2.2 and 2.3).
Lemma 4.2. If Λ is a positive definite complex linear functional on , then
n
P
(i) c is the greatest ideal of contained in the kernel of Λ,
(ii) IIΛΛc is a set ideal (i.e. IΛc =PInc(Z) for some Z ⊂Rn).
Proof. (i) is proved in [12, pages 24 and 42].
(ii) By Lemma 2.4, there exists Z Rn such that = (Z). This in turn
implies that IΛc =IΛr +iIΛr =I(Z)+i⊂I(Z)=Ic(Z). IΛr I (cid:3)
Lemma 4.3. If I is a -ideal of and Λ is a positive definite complex linear
n
∗ P
functionalon n,thenΛvanishes onI ifandonlyifitvanishes on c( (I)), where
P I Z
(I) is defined in (2.1).
Z
Proof. The set Ir := P I: P∗ = P is an ideal of n and I = Ir +iIr.
{ ∈ } R
Since (I)= (Ir) and c(Z)= (Z)+i (Z) for each subset Z of Rn, we get
Z Z I I I
c( (I))= ( (Ir))+i ( (Ir)). (4.2)
I Z I Z I Z
Thus, we have
Lemma2.5 (4.2)
Λ|I =0 ⇐⇒ Λr|Ir =0 ⇐⇒ Λr|I(Z(Ir)) =0 ⇐⇒ Λ|Ic(Z(I)) =0,
which completes the proof. (cid:3)
Take P . Following [12, Section 10], we say that P is of type A
n n
(respectively∈B)Pifeachpositivedefinitecomplexlinearfunctional∈ΛoPn vanishing
n
P
on (P)c (respectively c( P)) is a moment functional. Clearly, each polynomial of
type A is of type B. BIy LZemma 4.1, the “complexified” definitions of types A and
B coincide with the “real” ones for every P . Applying again Lemma 4.1,
n
we see that P is of type A (respective∈lyRB) if and only if P∗P is of type
n
A (respectively∈B)P. This fact and the equality S = S∗S, S n, imply that
Z Z ∈ P
Theorem 3.1 remains valid for all P,Q .
n
∈P
Theorem 4.4. Let P,Q be such that = . Then P is of type A if
n P Q
and only if Q is of type A. ∈ P Z Z
Knowing this, we can define correctly type A for an algebraic subset of Rn
induced by a polynomial P . Finally, we see that Theorem 3.2 remains valid
n
∈ P
in the “complexified” setting as well.
Theorem 4.5. An algebraic subset of Rn induced by a polynomial P is
n
of type A if and only if it is of type B. ∈ P
Corollary 4.6. Let V and W be algebraic subsets of Rn such that V W.
If W is of type A, then so is V. ⊂
Proof. Apply [12, Proposition 48] and Theorem 4.5. (cid:3)
ALGEBRAIC SETS OF TYPES A, B, AND C COINCIDE 7
5. Type C
Following[12,Proposition42],wesaythata -idealI of isoftypeCifeach
n
∗ P c
positive definite complex linear functional Λ on satisfying the equality = I
is a momentfunctional. An algebraicsubset V ofPRnn is of type Cif the idealIΛc(V)
is of type C. Each algebraic set of type A is automatically of type C. In SeIction
7 we will show that the converse implication is true as well. In view of the proof
of Lemma 4.2(ii), we see that an algebraic subset V of Rn is of type C if and
only if each positive definite real linear functional Λ on satisfying the equality
n
R
= (V) is a moment functional.
Λ
I I
We now complete [12, Proposition 41].
Proposition 5.1. If I is a proper -ideal of , then the following three con-
n
∗ P
ditions are equivalent
(i) I is a set ideal,
(ii) thereexists apositive definitecomplex linear functional Λon suchthat
n
c =I, P
IΛ
(iii) there exists a rigid I-basis Q m of composed of real column poly-
{ k}k=0 Pn
nomials (with Q =1) and a complex linear functional Λ on such that
0 n
Λ(Q Q∗) is the zero matrix for all k =l, Λ(Q Q∗) is the idePntity matrix
k l 6 k k
for all k, and I kerΛ.
⊂
Proof. (i) (ii) See the proof of [12, Proposition 41(ii)].
⇒
(ii) (i) Apply Lemma 4.2(ii).
⇒ c
(ii) (iii) Since = I and the ideal I is proper, the functional Λ must be
⇒ IΛ
non-zero. Employing [12, Proposition 32(i) (ii)], [12, Remark 33] and the fact
c ⇒
that kerΛ, we get (iii).
IΛ ⊂
(iii) (ii) Use [12, Theorem 36]. (cid:3)
⇒
In fact, in view of the proofof [12, Proposition41(ii)], if I is a setideal of ,
n
P
then there always exists a (complex linear) moment functional Λ on such that
n
c P
=I (this is much more than required in Proposition 5.1(ii)).
IΛ
6. Semiperfect -algebras
∗
Our next goal is to study a class of -algebras in which positive definite linear
∗
functionals are automatically moment functionals. The results we obtain will be
used in Section 7. In what follows K stands either for R or C. In this section A is
assumedtobeafinitelygeneratedcommutative -algebraoverKwithaunite(the
∗
unitandthe zeroelementsofAarealwaysassumedto be different). One canshow
thatifK=R,thenAisthedirectsumoftwovectorspacesA := a A: a∗ =a
R
and a A: a∗ = a ,thefirstofwhichisafinitelygeneratedunita{l ∈-algebrawith}
{ ∈ − } ∗
the identity mapping as involution. If K = C, then A always has a finite set of
selfadjoint generators (if necessary replace the old generators a ,...,a of A by
1 n
selfadjoint ones a +a∗ n i(a a∗) n ). Denote by ∆∗(A) the set of all
{ k k}k=1 ∪{ k − k }k=1
multiplicativelinearfunctionals γ: A Ksuchthatγ(e)=1andγ(a∗)=γ(a)for
every a A. Equip ∆∗(A) with the to→pology of pointwise convergence. For a A,
we defin∈e the continuous mapping aˆ: ∆∗(A) K by aˆ(γ) := γ(a) for γ ∆∗∈(A).
→ ∈
If a:=(a ,...,a ) are generatorsof A (which are assumed to be selfadjoint in the
1 n
case of K=C), then the mapping
Φa: ∆∗(A) γ (γ(a1),...,γ(an)) (Φa) Rn (6.1)
∋ 7→ ∈R ⊂
8 TORBENMAACKBISGAARDANDJANSTOCHEL
is a homeomorphic embedding of ∆∗(A) onto a closed subset (Φa) of Rn. This
implies that the σ-algebra of all Borel subsets of ∆∗(A) is theRsmallest σ-algebra
of subsets of ∆∗(A) with respect to which all the mappings aˆ, a A, are mea-
surable. We say that A is ∆∗-separative if ∆∗(A) = ∅ and ∈ kerγ = 0
6 γ∈∆∗(A) { }
(equivalently:themappingA a aˆ K∆∗(A) isinjective).TIfAis∆∗-separative,
then the mapping a aˆ Φ−a∋1 is7→a -∈algebra isomorphism between -algebras A
7→ ◦ ∗ ∗
and Aa, where Aa consists of all functions f: (Φa) K for which there exists
R →
P K[X1,...,Xn] such that f(x) =P(x) for all x (Φa) (Aa is equipped with
∈ ∈R
pointwisedefinedalgebraicoperationsandinvolutionf∗(x)=f(x)forx (Φa)).
A linear functional Λ: A K such that Λ(a∗a) > 0 for all a A∈iRs called
→ ∈
positive definite. AlinearfunctionalΛ: A Kissaidtobestrictlypositive definite
if Λ(a∗a)> 0 for all a A 0 . Denote→by (A) the set of all positive definite
∈ \{ } PD
linear functionals on A and equip it with the topology of pointwise convergence.
Clearly,∆∗(A)isaclosedsubsetof (A). WesaythatAissemiperfectif∆∗(A)=
∅andforeveryΛ (A)thereisPaDpositiveBorelmeasureµon∆∗(A)suchtha6 t
∈PD
Λ(a)= aˆdµ, a A. (6.2)
Z∆∗(A) ∈
By (6.1), such µ is automatically regular (cf. [35, Theorem 2.18]). The functional
Λ of the form (6.2) is called a moment functional. Applying the identification
(6.1) and the measure transporttheorem, one can deduce from the Riesz-Haviland
characterizationof Hamburger moment problem on (Φa) (cf. [34, 22, 23]) that
R
if ∆∗(A) = ∅, then a linear functional Λ: A K is a moment
functional6 if and only if Λ(a)>0 for every a →A such that aˆ>0. (6.3)
∈
It follows directly from (6.3) that
if ∆∗(A) = ∅ and Λ : A K, k > 1, is a sequence of moment
k
functional6s which is pointw→ise convergentto a function Λ: A K, (6.4)
then Λ is a moment functional. →
Remark 6.1. If I is a proper -ideal of A, then the quotient -algebra A/I is
finitely generated and the set γ ∗∆∗(A): I kerγ is a closed∗subset of ∆∗(A)
which is homeomorphic to ∆∗({A/∈I) via the m⊂apping}γ γ , where γ (a+I) :=
I I
7→
γ(a) for a A. Similarly, the set Λ (A): I kerΛ is a closed subset
∈ { ∈ PD ⊂ }
of (A) which is homeomorphic to (A/I) via the mapping Λ Λ , where
I
PD PD 7→
Λ (a+I):=Λ(a) for a A.
I
∈
In what follows, we regard K[X ,...,X ] as the -algebra over K with the
1 n
unique involution such that X∗ = X for all j = 1,...,∗n. Thus the involution of
j j
R[X ,...,X ] is the identity mapping. Note that the -algebra A which has self-
1 n
∗
adjoint generators is -isomorphic to a quotient -algebra K[X ,...,X ]/I, where
1 n
∗ ∗
I is a proper -idealofK[X ,...,X ](apply the homomorphismtheoremto the -
1 n
∗ ∗
algebraepimorphismK[X ,...,X ] P P(a ,...,a ) A,wherea ,...,a are
1 n 1 n 1 n
∋ 7→ ∈
selfadjointgeneratorsofA). Otherwise,K=RandA =A. Inthisparticularcase,
R
6
the -algebraAis -isomorphictoaquotient -algebraR[X ,...,X ,Y ,...,Y ]/I,
1 n 1 n
∗ ∗ ∗
where I is a proper -ideal of the -algebra R[X ,...,X ,Y ,...,Y ] which is
1 n 1 n
equipped with the uniq∗ue involution s∗uch that X∗ =Y for all j =1,...,n.
j j
Wenowdiscuss∆∗-separativityofthemodel -algebraK[X ,...,X ]/I leaving
1 n
∗
thecaseR[X ,...,X ,Y ,...,Y ]/I aside(itismorecomplicatedandnotessential
1 n 1 n
for the main purpose of this paper).
ALGEBRAIC SETS OF TYPES A, B, AND C COINCIDE 9
Lemma6.2. LetI beaproper -idealofA=K[X ,...,X ]. Thenthefollowing
1 n
∗
assertions hold:
(i) ∆∗(A/I)=∅ if and only if (I)=∅, where (I) is defined in (2.1),
6 Z 6 Z
(ii) if (I)=∅, then the mapping
Z 6
∗
Ψ : ∆ (A/I) χ (χ(X +I),...,χ(X +I)) (I)
I 1 n
∋ 7→ ∈Z
is a well defined homeomorphism such that
∗
χ(P +I)=P(Ψ (χ)), P A, χ ∆ (A/I), (6.5)
I
∈ ∈
(iii) A/I is ∆∗-separative if and only if I is a set ideal.
Proof. (i)&(ii)Supposethat∆∗(A/I)=∅. Itiseasilyseenthat(6.5)holds.
Moreover, Ψ (χ) (I) for every χ ∆∗(A/6I), because
I
∈Z ∈
(6.5)
P(Ψ (χ)) = χ(P +I)=χ(0+I)=0, P I.
I
∈
Conversely, if x (I), then the mapping γ : A/I P +I P(x) K is well
x
defined and γ ∈∆Z∗(A/I). What is more, Ψ (γ ) =∋x. It foll7→ows from∈(6.5) that
x I x
γ = χ for a∈ll χ ∆∗(A/I), which means that Ψ−1(x) = γ for all x (I).
ΨI(χ) ∈ I x ∈ Z
That Ψ is a homeomorphism is now obvious.
I
(iii) Suppose that I = (Z):= P A: x Z P(x)=0 , where Z Rn. If
I { ∈ ∀ ∈ } ⊂
P A I,thenthereexistsx Z (I)suchthatP(x)=0. Henceγ (P+I)=0,
x
wh∈ich\means that A/I is ∆∗-∈sepa⊂raZtive. Conversely, if A6/I is ∆∗-separative, t6hen
kerχ = 0 . In view of (i) and (ii), this equality implies that every
χ∈∆∗(A/I) { }
PT A which vanishes on (I)(= ∅) belongs to I. In other words, ( (I)) I.
∈ Z 6 I Z ⊂
Since the reverse inclusion is always true, we get I = ( (I)). (cid:3)
I Z
Referring to Lemma 6.2, note that if I is a set ideal of K[X ,...,X ] of the
1 n
formI = (Z),whereZ Rn,then (I)istheclosureofZ intheZariskitopology,
I ⊂ Z
and I = ( (I)) (use Lemma 2.1 and consult [12, Section 9]).
WesIhoZwthatifthe -algebraAhasselfadjointgeneratorsand∆∗(A)=∅,then
∗
(A)= 0 , which means that A can be thought of as a semiperfect -algebra.
PD { } ∗
Proposition 6.3. Let A be a finitely generated commutative unital -algebra
∗
over K, which in the case of K = R is equipped with the identity involution. Then
∆∗(A)=∅ if and only if (A)= 0 .
PD { }
Proof. The“if”partoftheconclusionisobvious. Supposenowthat∆∗(A)=
∅. Asweknow,the -algebraAis -isomorphictoaquotient -algebraB/I,where
∗ ∗ ∗
I is a proper -ideal of B =K[X ,...,X ]. By Lemma 6.2(i), we have (I)=∅.
1 n
∗ Z
This implies that ( (I))=B. Take Ξ (B/I) and set Λ(Q)=Ξ(Q+I) for
I Z ∈PD
Q B. Then Λ (B) and Λ = 0. This, together with Lemma 2.5 (the real
I
∈ ∈ PD |
case)andLemma 4.3(the complex case),implies thatthe functional Λvanishes on
( (I))=B. Hence, Ξ =0, which completes the proof. (cid:3)
I Z
Below,wecharacterizesemiperfectnessofthemodel -algebraK[X ,...,X ]/I.
1 n
∗
Proposition 6.4. If I is a -ideal of A = K[X ,...,X ], then the following
1 n
∗
two conditions are equivalent:
(i) (I)=∅ and every positive definite linear functional Λ: A K vanish-
Z 6 →
ing on I is a moment functional on (I),
Z
(ii) the ideal I is proper and the -algebra A/I is semiperfect.
∗
10 TORBENMAACKBISGAARDANDJANSTOCHEL
Proof. (i) (ii) As (I) =∅, we see that I = A and, by Proposition 6.2(i),
∆∗(A/I)=∅. T⇒akeΞ Z (A6 /I)andsetΛ(Q)=6 Ξ(Q+I)forQ A. ThenΛ
6 ∈PD ∈ ∈
(A) and Λ =0. Hence Λ is a moment functional on (I) with a representing
I
PD | Z
measure µ. By Lemma 6.2(ii) and the measure transport theorem, we have
Ξ(Q+I)= Q(x) dµ(x) = Q(Ψ (χ)) d(µ Ψ )(χ)
ZZ(I) Z∆∗(A/I) I ◦ I
(6.5)
= χ(Q+I) d(µ Ψ )(χ), Q A,
Z∆∗(A/I) ◦ I ∈
where µ Ψ (σ)=µ(Ψ (σ)) for a Borel subset σ of ∆∗(A/I). This gives (ii).
I I
◦
(ii) (i)ByProposition6.2(i),wehave (I)=∅. IfΛ (A)vanishesonI,
⇒ Z 6 ∈PD
then the mapping Ξ: A/I Q+I Λ(Q) K is well defined and Ξ (A/I).
Hence there exists a positiv∋e Borel7→measure∈ν on ∆∗(A/I) such that ∈PD
(6.5)
Λ(Q)= χ(Q+I) dν(χ) = Q(Ψ (χ)) dν(χ)
Z Z I
∆∗(A/I) ∆∗(A/I)
= Q(x) d(ν Ψ−1)(x), Q A,
Z ◦ I ∈
Z(I)
which completes the proof. (cid:3)
Corollary 6.5. Let I and J be proper -ideals of A = K[X ,...,X ] such
1 n
∗
that (I)= (J) and for every Λ (A), Λ =0 if and only if Λ =0. Then
I J
Z Z ∈PD | |
the -algebra A/I is semiperfect if and only if the -algebra A/J is semiperfect.
∗ ∗
Recall that (P) is the ideal of K[X ,...,X ] generated by a polynomial P
1 n
∈
K[X ,...,X ] and := x Rn: P(x)=0 .
1 n P
Z { ∈ }
Corollary 6.6. If P A = K[X ,...,X ], P = P∗ and = ∅, then the
1 n P
∈ Z 6
ideals (P) and ( ) are proper and the following two conditions are equivalent:
P
I Z
(i) the polynomial P is of type A (respectively B),
(ii) the -algebra A/(P) (respectively A/ ( )) is semiperfect.
P
∗ I Z
Proof. Since ((P)) = and ( ( )) = , we can apply Proposition
P P P
Z Z Z I Z Z
6.4 to -ideals (P) and ( ) (use also [39, Proposition 2.1]). (cid:3)
P
∗ I Z
It may happen that K[X ,...,X ]/I is semiperfect but not ∆∗-separative.
1 n
Example 6.7. The proper -ideal I := (1+X2+X2) of A = K[X ,X ] has
the property that (I ) = ∅, w∗hich by L1emma 6.2(1i) me2ans that ∆∗(A1/I )2= ∅.
1 1
Z
In turn, if the polynomial P A is of the form P(X ,X ):=X2, then the -ideal
∈ 1 2 1 ∗
I := (P) of A is proper and (I ) = ∅. By Lemma 6.2(i), this implies that
2 2
∆∗(A/I )=∅. Since I is not aZset id6eal, the -algebraA/I is not ∆∗-separative
2 2 2
(cf.Lemma6 6.2(iii)). However,by[39,Theorem∗5.4],thepolynomialP isoftypeA
(as a member ofA) andhence, by Corollary6.6,the -algebraA/I is semiperfect.
2
∗
Recall that if A=K[X ,...,X ] and Λ (A), then the set
1 n
∈PD
∗
:= P A: Λ(P P)=0
Λ
I { ∈ }
is the greatest ideal of A contained in the kernel of Λ (see Lemmata 2.4 and 4.2).
Proposition6.8. LetI bea -idealofA=K[X ,...,X ]suchthat (I)=∅.
1 n
∗ Z 6
Then the ideal I is proper and the following two conditions are equivalent: