Math. Nachr. 146 (1990) 111-117

Partially Commutative Moment Problems

Dedicated to the and to the 501* birthday of a. Lafiner

By H. J. BORCHERS of Gottingen and JAKOB YNQVASON of Iceland

birthday of A . Uhlmann

(Received May 3, 1989)

Abstract. Partially commutative tensor algebras occur naturally in the algebraic formulation of WIGETYAN field theory. A state on an algebra of this type leads via GNS-construction to a par- tially commutative family of hermitean operators on HILBEET space. We d i ~ c u ~ e the question when these operators can be extended to self adjoint operators preserving the commutation pro- perties and state a necessary and sufficient condition for the existence of such an extension in terms of a positivity property of the state.

1. Introduction

I n the algebraic approach to WIQHTBUN quantum field theory initiated in [3] and [8] the field corresponds to a representation of the tensor algebra of an appropriate space of test functions. Local commutativity of the (unbounded) field operators means that the representation annihilates a two sided ideal generated by commutators of test functions with space-like separated supports. The relevant algebra i0 thus the quotient of the full tensor algebra and thie ideal. Algebras of this type have been called partially commutative tensor aZgebras in [9] and studied further in [l] and [ll].

In [l] it was shown that the partially symmetric tensor algebras occuring in WIQFIT- MAN field theory admit a large family of C*-norms. This property suggests a new approach to an old problem of axiomatic field theory wiz. to establish a connection between the WIQHTMAX formalism and the ARAEI-HAAQ-KASTLER theory of local nets of C*- or W*-algebras [2], [9]. Applying a general method due to DUBOIS-VIOLETTE [6] one can use the family of C*-norms on the partially symmetric tensor algebras to define a quasi-local C*-algebra. A atate on the tensor algebra extends to a state on thia C*-algebra if and only if the state is strongly positive, which means that it is positive on the closure of the positive cone in the tensor algebra with respect to the topology defined by the family of C* norms [6] , [l].

The extension of a state from a partially symmetric tensor algebra to the correspond- ing DUBOIS-VIOLETTE C*-algebra can be regarded as a noncommutative generalization of the classical Hamburger moment problem. In the commutative cage the problem ie

112 Math. Nachr. 145 (1990)

to write the elements of a given sequence of numbers as moments of a positive measure on (a subset of) Rn. The sequence defines a linear functional on the algebra of poly- nomials, i.e. on a totally symmetric tensor algebra, and the measure corresponds to a state on a C*-algebra of bounded continuous functions on (a subset of) Rn.

There is, however, an alternative way of looking at the classical moment problem. The state on the polynomial algebra defines via GNS-construction a commutative algebra generated by (unbounded) hermitian operators on a HILBERT space. A solution to the moment problem is equivalent to a simultanous extension of these hermitean operators to self adjoint operators, possibly on an enlarged HILBERT space, such that the self adjoint operators commute in the strong sense that their bounded functions commute. The partially commutative version of this problem is to extend a partially commutative family of hermitean operators to a, family of self adjoint operators having the same commutation properties as the original operators. Every solution of the DUBOIS-VIOLETTE problem leads automatically to a solution of this latter problem, but the converse is in general not true. This is best illustrated by an example.

1.1 Example. Let !$ denote the *-algebra over C defined by generators and relations in the following way: '@ is generated by four hermitean elements, X1, . . ., X4, and the relations are

From these relations follows that [XI, X3] = [X,, X,] commutes with X,, k = 1, . . ., 4, and thus belongs to the centre of !$. In every bounded, irreducible representation these commutators are therefore a multiple of 1. Since the canonical commutation relations have no bounded representations, these commutators are annihilated by any C*-semi- norm on '@. It follows that only abelian representations of '@ extend to the Dwo~s- VIOLETTE C*-algebra. On the other hand one has e.g. the unbounded, noncommutative representation on L2(Ra) defined by Xi I+ xi, j = 1,2 ; X, I+ ialaz,, X4 H i a l a ~ , . Here all linear combinations of X, are represented by (essentially) self adjoint operators and the required commutation relations are satisfied.

The partially symmetric ten8or algebras considered in [l] and [ll] are of a special kind because commutation relations are only specified between elements of a basis of generators. (Such relations were called simple relations in [lo].) These algebras always admit an abundance of nontrivial C*-seminorms, and it seems quite possible that the two ways mentioned above of defining the noncommutative moment problem (via C*- algebras or via self adjoint extensions) are equivalent in this case. This has not yet been generally proved, however.

In the present paper we suggest an approach to noncommutative moment problems that does not rely on the existence of C*-seminom and can be considered as an abstract variant of the DWOIS-VIOLETTE formalism. Common to both approaches is an em- bedding of the tensor algebra into a larger algebra, that contains bounded functions of the generators of the tensor algebra. But whereas this extended algebra is in DUBOIS- VIOLETPE'S approach defined as the completion of the original algebra with respect to a family of C*-seminorms, it is in our approach simply defined as an abstract algebra

Borchers/Yngvason, Partially Commutative Moment Problems 113

with generators and relations. In the same way aa in [6] a standard extension theorem for positive linear functionals leads then to a necessary and sufficient criterium for the solvability of the moment problem, An extension of the criterion to non-cyclic representations and a discussion of special cases will be presented elsewhere [4].

2. Partially Commutative Algebras and Self Adjoint Extensions

Suppose E is a complex vector space with involution X I+ X* and let T ( E ) denote the tensor algebra over E. Let Eh denote the hermitem part of E and let e be a relation on Eh, i.e. a subset of Eh x En. Let 9, denote the two-sided ideal in T ( E ) generated by the commutators [ X , Y ] = X @ Y - Y @ X with ( X , Y) E e. The quotient algebra S, (E) = T ( E ) / J , is called the partially Bymmetric tensor aZgebra over E corresponding to the relation e. The algebra B,(E) is in a natural way a *-algebra since the involution can be extended from E to T ( E ) and J , is a *-invariant ideal.

2.1 Remark. The ideal 3, and hence the algebra S,(E) clearly do not change if we replace e by the relation p on Eh that is defined in the following way: ( X , Y ) E p if and only if [X, Y] E lin. span ([X, Y'] I (X' , Y') E e}. We call g the completion of e and say that e is complete if e = g. The relation g has the properties

(1) ( X , X) E i! (2) if ( X , Y ) E g, then also (Y, X) E g (3) if ( X , Y ) E gand ( X , Z ) E g, then (X,arY + B X ) E @forallar,B E R.

It is easy to show that if e is complete, then X , Y E Eh commute in S,(E) if and only .if ( X , Y) E e.

In the following we shall consider the relation e as fixed, and when convenient we may assume it is complete. The algebra S,(E) = T ( E ) / 3 , will simply be denoted by 8, indicating that its element8 can be regarded as polynomials in the partially commutative variables from Eh.

A linear functional w on p that is positive on all elements of the form p*p with p E '$, gives rise via GNS-construction to a cyclic representation nu of 3 with cyclic vector a, in a HILBEET space X,. The operators n,(X) with X E En are symmetric on the domain 9), := nu('$) 8,.

We shall be concerned with the following question: Suppose Q is a subset of EP When is it possible to define a &BEET space 9, that contains Xu as a subspace, and self adjaint extensions, h, (X) of the symmetric operators n, (X) , X E Q with dense domains &X) in 2, such that all bounded functions of h,(X) and ft,(Y) commute, if ( X , Y ) E e?

2.2 Remark. Even in the case Q = Eh we do not require that R , defines a represen- tation of '$; in general h,(X + Y ) =+ R,(X) + R,(Y) and h , ( X Y ) is not defined.

To deal with this question we define an abstract *-algebra 3 with unit by its gen- erators and relations in the following way. The set of generators is E augmented by additional elements, denoted by R+(X) and R-(X) , for each X E a. The relations for

8 Math. Nechr., Bd. 146

114 Math. Nachr. 146 (1990)

the generators in E are the same as in 3, but in addition we require

(1) ( X + i) R+(x) = R+(x) ( X + i) = 1 for all X E Gi

(2 ) ( X - i) R-(X) = E ( X ) ( X - i) = 1 for all X E G (3) &,(XI %(Y) = RAY) U X ) if ( X , Y ) E e. The algebra '$ is embedded into 8 in a natural way as the subalgebra generated by

E. The element R,(X) is the inverse of ( X f i) in 8, and we shall also denote it by ( X f i)-1. The *-operation is extended from 3 to 8 by defining R+(X)* = R-(X). We note also that instead of R J X ) we can just as well u8e the hermitean generators B+(X) &(X) = (X2 + 1)-1, because B,(X) = ( X i) (Xa + l)-l. The hermitean part of 8 reap '$ is denoted by 8 h reap. '@A.

Next we define a subalgebra 9 c 8 as the algebra with unit generated by element8 of the form X"(X2 + l)-" with X E Q; n, m E N, n < 2m. The hermitean part of B is denoted by bh. Define

and let 5 denote the order relation on bh defined by this cone. We chim that every element in b h is bounded in this order by a multiple of the unit element. This implies that every *-representation of 23 is a representation by bounded operators.

2.3 Proposition. For every b E bh, there is a nonnegative number B such that -p1 d b

Proof.Theelement bcanbewrittenasasumof elementsof theformb,...b, + b:...b: with each bk in a subalgebra generated by a single element x k E a. we use induction over n and suppose first that b = bl = q ( X ) (X2 + l)-" where q is a real polynomial of degree 5 2m and X E Q. Then there exists a nonnegative number B such that +(z2 + 1)" I q(z) S B(z2 + 1)" for all z E R. Since every positive polynomial is a sum of squares and (X2 + l ) -m = ( X + i)*-" ( X + a?-", we then also have

5 p.

-/?1 g q ( X ) (Xa + 1)- 5 p1

in the order of b. Thus the statement i8 proven for n = 1. Next define b' = b1 -..b,l. By induction assumption b'* + b' and (b'* - b')/i are bounded by a multiple of 1. We now write

b = b'b, + b:b'* = (1, b,*) - cl* :)-(:) . and use the following lemma (cf. [6, Lemma 3.41):

2.4 Lemma. There are constants a i k E C, i = 1 , 2 ; k = 1, . .., 4, and hermitian ele- ments b; E lin span (a', b'*), k = 1, . . ., 4 such that

Borchers/Yngvason, Partially Commutative Moment Problems 115

The lemma is eadly checked with e.g.

= = 6119 = a14 = 1 , all = 1 , acZ2 = i azB = -1 0cZ4 = -i,

and bi = (b' + b'*)/4, bi = (b' - b'*)/4i, bj = -bi , bi = -bi .

4

k = l Given the lemma we can write b = b'b, + b:b'* = z Since &bL 5 P'1, say, and

thus have &b 5 /I'

with ak = alkbn + &p&

a s k is a bounded function of a single element, X,, we k

a s k 5 /I1 for some /I < 00. I k

We now define a linear flubspace (not a subalgebra), % c

% := linear span (pbq I b E b and p , q E $}

and a convex cone %+ c m:

We denote the hermitean part of % by %b. The cone d on %h.

there is a p E $h swh that --p 5 m 5 p .

defines an order relation

2.6 Proposition. $A is a cofinal subspace of %h with r q e c t to %+, i.e. for any m E !!&

Proof. It suffices to consider m = pbq + q*b*p* with b E b, p , q E b. Write

4

k= 1 By Lemma 2.4 we can write this as

By proposition 2.3 it then follows that f m 5 /I

one obtains

r:bkrk with bk E bh and rk = a@* + a z k q E $.

rirk for Some /I E R+. I Combining proposition 2.5 and a standard extension theorem [7, Corollary 2.8, p. 821

2.6 Proposition. A linear functional w on '$ ?w.s an extension b to a linear functional

As a corollary we obtain

2.7 Theorem. Let w be a positive functional on 8 and let nw be the corrtwponding cyclic representation of 8 with cyclic vector Q, in the -BERT &pace X m . The fdlm'ng are equivalent:

on YR that is positive on %+ i f and only i f o is p M v e on $ n %+.

(1) o is positive on $ n %+. (2) The olperators z,(X), X E Q have self adjoint exte- 5t,(X) with dense d m i m

in a E ~ B E R T space S? 3 X,, such that all bounded functions of &,(X) and a,( Y) cmmute if ( X , Y ) E e. The extension cun be c h e n such that Z is cyclic in 3i? for the algebra generated by bounded functions of the operators It,(X), X E Q.

8'

116 Math. Nachr. 146 (1990)

Proof. If w is positive on 8 n 'B+ it has by proposition 2.6 a positive extension b to 'B. We we b to define a scalar product on the linear space '2 := linear span 8g c '331: (bp, cq) := b(p*b*cq) for b, c E b, p , q E '$3. This is well defined and positive semidefinite because b is positive on fm+. After dividing out the null space we obtain a HILBERT space 9; if bp E '92 we denote the corresponding element in S?' by [bp]. Since b is an extension of w, Sf is embedded into &' in a natural way as the closure of .B := ( [ p ] I p E '@I ; Q, corresponds to [ l ] and n,(p) [q] = b] for p , q E '$3. On &' there is also defined a representation of B which we call It, by slight abuse of notation: &(b) [cp] := [bcp] for b, c E b, p E '$. If X E B, the self adjoint operator h ( X ) is defined on the domain &X) = It(R+(X)) 3t' as

.-

a(x) It(R+(X)) y := y - iR+(X) y , y E 9. Uaing the relation8 (1)-(3) for R J X ) and X one checks easily that h ( X ) is well defined and has the required properties. Conversely, if the self adjoint extensions It,(X) are given, we obtain a representation, a h denoted by h,, of b by defining

m J l . . * b " ) = bl(a,<x,,) . -b "(%(XI)) if bk = a,(&) is a function of a single variable for k = 1 , .. ., n. The extension b is then defined by

b(Pbc7) = (n&*), %(b) %(P)) for p, q E 9, b E b (and linear extension to all of W). I

3. Concluding Remarks

In this note we have predented a second approach (different from the method of DUBOIS- VIOLETTE) to the noncommutative moment problem. The main advantage seems to 11s that only algebraic and order structures are used. In this context it is worth recalling that the topology of a C*-algebra % with unit i0 in fact determined by the order struc- ture; one haa llall = sup Iln,(a)ll for all a E %, where the supremum ki taken over all positive, linear functionals w on % with w(1) = 1. PropoElifion 2.3 establishes the result that also our *-algebra '23 of bounded elements has 1 as an order unit, and one could in the same way define a C*-topology on this algebra using its positive linear func- tionab. Since a topological completion i0 not needed in our approach, however, it is not necessary to introduce this topology.

As demonstrated by example 1.1 there are cases where the DUBOIS-VIOLE~E C*- algebra is trivial, while our algebra 8 is not. Although theorem 2.7 gives in principle a complete answer to the question posed in all cases, the criterium (1) is not necessarily a very practical one. In fact we only expect it to be manageable for ccgood" relations e such that the quotient algebra Lgp(E) = T(E) /3 , possesses sufficiently many C*-semi- norms. As remarked in the introduction, simple relations, i.e. such that are defined on a basis Q for 1, belong to this category, but an abstract definition of "good" relations Reems to be desirable in order to have a better understanding of the difference between the approach of DUBOIS-VIOLETTE and OUW.

Acknowledgement. I. Y. gratefully acknowledges a DAAD grant.

Borchers/Yngvason, Partially Commutative Moment Problems 117

References

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[8] R. HAAQ, D. KASTLER, J. Math. Phys. 6 (1964) 848 [a] R. T. POWERS, 1974, Trans. Am. Math. SOC. 187,281-293 [7] A. L. PEREYSINI, Ordered Topological Vector Spaces, Harper t Row, New York 1987 [8] A. UHLMANN, Wiss. Z. Karl-Marx-Univ. (1982) 213-217 [9] J. YNGVASON, Publ. RIMS Kyoto 80 (1982) 1083-1081

(1977) 161-172

[lo] -, in: Proceedings of the 19th Nordic Congress of Mathematicians, ed. by J. R. Steftinseon,

[ll] -, Ann. Inst. Henri Poincarb 48 (1988) 181-173 Icelandic Mathematical SOC., Reykjavik, 1986, pp. 229-260

Inatitut fiir Theoretiach Physik Univeraikit W i n g e n Univeraity of Iceland D - 3400 Gottingen Iceland

Department of Physics