Algebras and Representation Theory 5: 163–186, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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Algebras and Representation Theory 5: 163–186, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

163

∗-Representations of Twisted Generalized Weyl Constructions VOLODYMYR MAZORCHUK1 and LYUDMYLA TUROWSKA2

1 Department of Mathematics, University of Uppsala, Box 480, SE 75106, Sweden. e-mail: [email protected] 2 Department of Mathematics, Chalmers University of Technology and GU, SE 41296, Göteborg, Sweden. e-mail: [email protected]

(Received: July 1999) Presented by Y. Drozd Abstract. We study bounded and unbounded ∗-representations of Twisted Generalized Weyl Algebras and algebras similar to them for different choices of involutions. Mathematics Subject Classifications (2000): Primary 47C10; secondary: 47D40, 16W10. Key words: generalized Weyl algebra, twisted generalized Weyl construction, involution, ∗-representation, irreducible representation.

1. Introduction Generalized Weyl Algebras (GWA) were first introduced by Bavula as some natural generalization of the Weyl algebra A1 (see [B2] and references therein). Since then, GWA have become objects of much interest (see for example [B1, B2, KMP, Sm, Sk, DGO]). Many known algebras such as U(sl(2, C)), Uq (sl(2, C)), down-up algebras and others can be viewed as generalized Weyl algebras and thus can be studied from some unifying point of view. In [MT] we introduce a nontrivial higher rank generalization of GWA, which we call Twisted Generalized Weyl Algebras. We study simple weight modules over twisted GWA in a special (torsion-free) case and show that there arise new effects which might be of interest for further investigations. We also note, that Twisted GWA are not isomorphic to the higher rank GWA, considered in [B1] in general. For such algebras it is natural to study their unitarizable modules, i.e. ∗-representations in a Hilbert space. The purpose of this paper is to introduce natural ∗-structures over twisted generalized Weyl algebras and some of their noncommutative (‘quantum’) deformations and to study Hilbert space representations of the corresponding ∗-algebras (real forms) by bounded and unbounded operators. The class of ∗-algebras considered in the paper contains a number of known ∗-algebras such as U(su(2)), U(sl(2, R)), Uq (su(2)), Uq (su(1, 1)), SUq (2) as well as

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∗-algebras generated by Qij -CCR ([Jor]), twisted canonical (anti)-commutation relations ([Pus, PW]) and others. The technique of study of ∗-representations used in the paper is based on the study of structure and properties of some dynamical systems. This approach goes back to the classical papers [M1, M2, EH, Kir1, Ped] and turns out to be a useful tool for investigation of representations of many ∗-algebras ([V1, V2, OS1, OT]). Using this method we obtain a complete classification of irreducible representations of the introduced real forms of twisted GWA (the case of commutative ground ∗-algebra R, see Section 2 for the precise definitions) provided that the corresponding dynamical system is simple, i.e., it possesses a measurable section. Any such representation is related to an orbit of the dynamical system. Otherwise, the problem of unitary classification of their representations can be problematic. Namely, if the dynamical system does not have a measurable section there might exist nonatomic quasi-invariant measures which generate factor-representations which are not of type I ([MN]). For a noncommutative ∗-algebra R of type I, we show that there is a one-to-one correspondence between weight irreducible representations of the corresponding ∗-algebras and projective unitary irreducible representations of some groups isomorphic to Zl . We study bounded and unbounded ∗-representations of our algebras. Note that the first problem that arises when one deals with representations by unbounded operators is to select the ‘well-behaved’ representations like the integrable representations of Lie algebras. In the paper we define a class of ‘well-behaved’ representations for our ∗-algebras and study them up to unitary equivalence. The paper is organised in the following way: in Section 2 we introduce a deformation of twisted GWA and define ∗-structures on it. In Section 3 we study bounded representations of the corresponding ∗-algebras (real forms). After discussing some properties of representations, we describe irreducible ones in terms of two models. As a results of our classification all irreducible weight ∗-representations of real forms for twisted GWA are listed in Theorem 4. In Section 4 the results obtained in the previous section are generalized to a class of unbounded representations.

2. Twisted Generalized Weyl Construction and its ∗-Structures 2.1. DEFINITION OF THE ALGEBRAS AND ∗- STRUCTURES Throughout the paper C is the complex field, R is the field of real numbers, Z is the ring of integers, N is the set of all positive integers. Fix a positive integer n and set Nn = {1, 2, . . . , n}. Let R be a unital algebra over C, {σi | 1 i n} a set of pairwise commuting automorphisms of R and M

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a matrix (µij )i,j ∈Nn with complex nonzero entries µij ∈ C, i, j ∈ Nn . Fix central elements ti ∈ R, i ∈ Nn , satisfying the following relations: ti tj = µij µj i σi−1 (tj )σj−1 (ti ),

i, j ∈ Nn , i = j.

(1)

We define A to be an R-algebra generated over R by indeterminates Xi , Yi , i ∈ Nn , subject to the relations • • • • •

Xi r = σi (r)Xi for any r ∈ R, i ∈ Nn ; Yi r = σi−1 (r)Yi for any r ∈ R, i ∈ Nn ; Xi Yj = µij Yj Xi for any i, j ∈ Nn , i = j ; Yi Xi = ti , i ∈ Nn ; Xi Yi = σi (ti ), i ∈ Nn .

We will say that A is obtained from R, M, σi , ti , i ∈ Nn by twisted generalized Weyl construction. One can easily show that the elements Xi , Xj , Yi and Yj satisfy additionally the relations of the form: Xi Xj Yi Xi = µj i Xi Yi Xj Xi ⇔ Xi Xj ti = µj i Xj Xi σj−1 (ti ), (2) −1 Y X Y Y ⇔ Y Y σ (t ) = µ Y Y σ (σ (t )). Yi Yj Xi Yi = µ−1 i i j i i j i i j i i j i ij ij The algebra A possesses a natural structure of Zn -graded algebra by setting deg R = 0,

deg Xi = gi ,

deg Yi = −gi ,

i ∈ Nn ,

where gi , i ∈ Nn , are the standard generators of Z . n

Remark 1. If R is commutative and µij = 1, i, j ∈ Nn , then A coincides with the algebra A defined in [MT]. The twisted GWA A(R, σ1 , . . . , σn , t1 , . . . , tn ) of rank n can be obtained as the quotient ring A /I , where I is the maximal graded two-sided ideal of A intersecting R trivially. By [MT, Lemma 2], this ideal is unique. It is worth to point out that the requirement for µij to be equal 1 is not important. All the results obtained in [MT] can be reformulated easily for the case µij = 1. In particular, given a commutative ring R, its automorphisms σi , i ∈ Nn , a matrix M = (µij )i,j ∈Nn , µij = 0, and elements ti ∈ R, i ∈ Nn , satisfying the relations ti tj = µij µj i σi−1 (tj )σj−1 (ti ), i, j ∈ Nn , i = j , we can define a twisted GWA as follows: A(R, σ1 , . . . , σn , t1 , . . . , tn , M) = A /I , where I is the ideal defined above. This class of algebras contains beside the algebras U(sl(2)), Uq (sl(2)), the algebras of skew differential operators on the quantum n-space known as the quantized Weyl algebras [DJ, Jord] and some other coordinate rings of quantum symplectic and Euclidean spaces. Assume that µij = µj i ∈ R and R is a ∗-algebra satisfying the condition σi (r ∗ ) = (σi (r))∗ for any r ∈ R, i ∈ Nn . Then the algebra A possesses the following ∗-structures: Xi∗ = εi Yi ,

ti∗ = ti ,

where εi = ±1, i ∈ Nn .

We will denote the corresponding ∗-algebras by AεR1 ,...,εn .

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Remark 2. It is clear that any maximal graded two-sided ideal of the ∗-algebra is a ∗-ideal. Thus in the case of commutative R, the above ∗-structures generate ∗-structures on the corresponding twisted GWA. AεR1 ,...,εn

2.2. EXAMPLES 1. The universal enveloping algebra U(sl(2, C)). Let R = C[H, T ] be the polynomial ring in two variables, t = T , T ∗ = T , H ∗ = H , σ (H ) = H − 1, σ (T ) = T + H . Then A1R U(su(2)) and A−1 R U(sl(2, R)). 2. The quantum algebra Uq (sl(2, C)). Let R = C[T , k, k −1 ] (polynomials in T and Laurent polynomials in k), t = T,

T ∗ = T,

σ (k) = q −1 k,

σ (T ) = T +

k − k −1 . q − q −1

• If q ∈ R and k = k ∗ then A1R suq (2), A−1 R suq (1, 1). suq (2). • If |q| = 1 and k ∗ = k −1 then A1R A−1 R Irreducible representations of the ∗-algebras suq (2), suq (1, 1) were studied in [V3]. 3. Quantized Weyl algebras. Let = (λij ) be an n × n matrix with nonzero complex entries such that λij = λ−1 j i , let q = (q1 , . . . , qn ) be an n-tuple of elements q, from C\{0, 1}. The n-th quantized Weyl algebra An ([Jord]) is the C-algebra with generators xi , yi , 1 i n, and relations xi xj = qi λij xj xi , xi yj = λj i yj xi ,

yi yj = λij yj yi , xj yi = qi λij yi xj ,

xj yj − qj yj xj = 1 +

j −1

(3)

(qi − 1)yi xi ,

i=1

for 1 i < j n. Let R = C[t1 , . . . , tn ] be the polynomial ring in n variables, σi the automorphisms of R defined by i−1 (qj − 1)tj , qi ti+1 , . . . , qi tn ), σi : p(t1 , . . . , tn ) → p(t1 , . . . , ti−1 , 1 + qi ti + j =1 q,

and M = (µij )ni,j =1 , where µij = λj i and µj i = qi λij for i < j . Then An is isomorphic to a quotient of the algebra A which is obtained from R, M, σi , ti , i ∈ Nn by twisted generalized Weyl construction. It is easy to show that the maximal graded ideal of A intersecting R trivially is generated by the elements xi xj − qi λij xj xi , yi yj − λij yj yi ,

1 i < j n,

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167

q,

hence A(R, σ1 , . . . , σn , t1 , . . . , tn , M) An . Assume that qi , λij ∈ R \{0}, xi∗ = εi yi , i, j = 1, . . . , n. The involutions define q, ∗-structures in A and the quantum Weyl algebra An . Note that in the case λj i = qi λij = µ ∈ (0, 1) relations (3) are known as twisted canonical commutation relations ([PW]). ∗-Representations of the algebra which correspond to the involution xi∗ = yi , i = 1, . . . , n were classified in [PW]. 4. Qij -CCR. Let Ad be a ∗-algebra generated by elements ai∗ , ai , i = 1, . . . , d, satisfying the following Qij -commutation relations: ai∗ aj = Qij aj ai∗ , ai∗ ai − Qii ai ai∗ = 1, ai aj = Qj i aj ai , i = j,

i = j,

(4) (5)

where Qii ∈ (0, 1), |Qij | = 1 if i = j , Qij = Qj i , i, j = 1, . . . , d. The ∗algebra A1 is a real form of the generalized Weyl algebra with R = C[T ] and t = t ∗ = T , σ (T ) = Q−1 11 (T − 1). For d > 1 set R = Ad−1 ⊕ C[T ], where T = T ∗ and [T , a] = 0 for any a ∈ Ad−1 . Let σ (ai ) = Qid ai , i = 1, . . . d − 1, 1 σ (T ) = Q−1 dd (T − 1) and t = T . Then AR Ad . Representations of Qij -CCR were studied in [Pr] using a method different from the one presented in this paper. For other examples of twisted generalized Weyl constructions see also Remark 9. 3. ∗-Representations of Twisted Generalized Weyl Constructions 3.1. BOUNDED REPRESENTATIONS OF AεR1 ,...,εn Let H be a separable Hilbert space. Throughout this section L(H ) denotes the set of all bounded operators on H . Let B(R) be the class of Borel subsets of R. For a selfadjoint operator A we will denote by EA (·) the corresponding resolution of the identity. Let M be any subset of L(H ). We denote by M the commutant of M, i.e. the set of those elements of L(H ) that commute with all the elements of M. For a group G we will denote by G∗ the set of its characters. In this section we study bounded representations of AεR1 ,...,εn , i.e. ∗-homomorphisms π : AεR1 ,...,εn → L(H ) up to unitary equivalence. We recall that representations π in H and π˜ in H˜ of a ∗-algebra A are said to be unitarily equivalent if there exists a unitary operator U : H → H˜ such that U π(a) = π˜ (a)U for any a ∈ A. Throughout the paper we will use the notation π1 π2 for unitarily equivalent representations π1 , π2 . We will assume also that µij > 0 for i, j ∈ Nn . Let π be a representation of AεR1 ,...,εn . We will denote the operators π(x), x ∈ AεR1 ,...,εn simply by x if no confusion can arise. Let r = Ur |r| (r ∈ R), Xi = Ui |Xi | be the polar decomposition of operators r and Xi respectively, where |r| = (r ∗ r)1/2 , |Xi | = (Xi∗ Xi )1/2, Ur and Ui are phases of the operators r and Xi . We recall, that the phase of an operator B is a partial isometry with the initial

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space (ker B)⊥ = (ker B ∗ B)⊥ and the final space (ker B ∗ )⊥ = (ker BB ∗ )⊥ . Since Xi∗ Xi = εi ti and Xi Xi∗ = εi σi (ti ), we have εi ti 0, εi σi (ti ) 0, and |Xi | = (εi ti )1/2 , |Xi∗ | = (εi σi (ti ))1/2 , i ∈ Nn . PROPOSITION 1. For any bounded representation of the ∗-algebra AεR1 ,...,εn the following relations hold Ui Ur = Uσi (r)Ui , Ui E|r| (+) = E|σi (r)| (+)Ui , + ∈ B(R), r ∈ R, i ∈ Nn , [Ui , Uj∗ ]Qij = 0, i = j, [Ui , Uj ]Pij = 0,

(6) (7)

where Pij and Qij are projections onto (ker ti tj )⊥ and (ker σi (ti tj ))⊥ respectively. Moreover, each Ui is centered, i.e. [Uik (Ui∗ )k , Uil (Ui∗ )l ] = 0,

[Uik (Ui∗ )k , (Ui∗ )l Uil ] = 0,

[(Ui∗ )k Uik , (Ui∗ )l Uil ] = 0 for any k, l ∈ N.

(8)

Conversely, any family of operators r = Ur |r|, Xi = Ui (εi ti )1/2 , i ∈ Nn , determines a bounded representation of AεR1 ,...,εn if |r|, εi ti , εi σi (ti ), i ∈ Nn are bounded positive operators and Ur , Ui , i ∈ Nn are partial isometries satisfying (6)–(7) and the conditions ker Ur = ker |r|, ker Ui = ker ti . Proof. From the relations in the algebra AεR1 ,...,εn it follows that Xi |r|2 = |σi (r)|2 Xi . By [SST, Theorem 2.1] we have Xi E|r| (+) = E|σi (r)|(+)Xi for any + ∈ B(R). To obtain relations (6) we note that |Xi | = (εi ti )1/2 , ker Xi = ker Ui = ker ti and ti commutes with any projection E|r| (+) as a central element of the algebra R. By the definition of polar decomposition, Ui∗ Ui = Eεi ti (R \ {0}) and Ui Ui∗ = Eεi σi (ti ) (R \ {0}). From this and relations (6) it follows that Ui is centered. The relations connecting Ui and Uj follow from (2). Indeed, Xi Xj ti = µj i Xj Xi σj−1 (ti ) ⇔ Ui (εi ti )1/2 Uj (εj tj )1/2 ti = µj i Uj (εj tj )1/2Ui (εi ti )1/2 σj−1 (ti ). From the relation εi ti Uj = Uj εi σj−1 (ti ) and the fact that (ker Uj )⊥ = (ker tj )⊥ is invariant with respect to σj−1 (ti ) we can conclude that εi σj−1 (ti )|(ker Uj )⊥ is positive and (εi ti )1/2 Uj = Uj (εi σj−1 (ti ))1/2 . This gives Ui Uj (εi εj σj−1 (ti )tj )1/2 ti = µj i Uj Ui (εi εj σi−1 (tj )ti )1/2 σj−1 (ti ) ⇔ Ui Uj (εi εj σj−1 (ti )tj ti2 )1/2 = Uj Ui (µj i µij εi εj σi−1 (tj )ti (σj−1 (ti ))2 )1/2 ⇔ Ui Uj (εi εj σj−1 (ti )tj ti2 )1/2 = Uj Ui (εi εj σj−1 (ti )tj ti2 )1/2 . By (1) we can conclude now that [Ui , Uj ]Pij = 0. Note that another relation of (2) will give us the same result. Similar arguments applying to the equality Xi Yj = µij Yj Xi imply the relations connecting Ui and Uj∗ .

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169

The converse implication follows by standard arguments from spectral decomposition of the operators |r|. ✷ The question of classifying of ∗-representations up to unitary equivalence can be very difficult in general for arbitrary ∗-algebra. If the ∗-algebra is of type I one has a more satisfactory theory (see [Dix2]). From now on we will assume that R is an algebra of type I, i.e. for any representation π of R the W ∗ -algebra π(R) = {π(r), r ∈ R} is of type I (see [Dix2] for the precise definition). The algebra π(R) coincides with the W ∗ -algebra generated by π(R). Moreover, we will restrict ourselves to the case of countably generated ∗-algebras. Denote by Rˆ the set of equivalence classes of irreducible representations of R and by Hn the standard n-dimensional Hilbert space. THEOREM 1. Let H be a separable Hilbert space, π a representation of R. Then there exist a standard Borel space .π , mutually singular positive measures (µk )k∈K on .π , µk -measurable fields ξ → Hk (ξ ) of Hilbert spaces, µk -measurable fields ξ → πk (ξ ) of nontrivial unitarily nonequivalent ⊕ irreducible representations on Hk (ξ ) and an isomorphism of H onto k∈K nk .π Hk (ξ ) dµk (ξ ), where the nk ∈ N ∪ {∞} are mutually distinct, which transforms the representation π into ⊕ nk πk (ξ ) dµk (ξ ). (9) k∈K

.π

⊕ Moreover, the representations .π πk (ξ ) dµk (ξ ) are mutually disjoint and the set (nk )k∈K is unique up to a permutation of the set of indices. Proof. Let A be the closure of π(R) in the operator norm. Since R is of type I and countably generated, we have that A is a separable C ∗ -algebra of type I. The theorem now follows from the general result about the same decomposition of any representation of A applied to the identity representation 2(a) = a for any a ∈ A (see [Dix2, Theorem 8.6.6]). Here .π = Aˆ is the set of equivalence classes of irreducible representations of A. We also note that the measures µk on .π are defined uniquely up to equivalence. ✷ Using standard arguments one can show the uniqueness of the decomposition (9). Namely, if .π1 , (µ1k )k∈K , ξ → Hk1 (ξ ), ξ → πk1 (ξ ) have the same properties then there exists a µk -negligible set N ∈ .π and µ1k -negligible set N1 ∈ .π1 , k ∈ K, a Borel isomorphism ν of .π /N onto .π1 /N1 transforming µk into a measure µ˜ 1k equivalent to µ1k for any k ∈ K, an isomorphism ξ → V (ξ ) of the field ξ → Hk (ξ ), ξ ∈ .π /N onto the field ξ1 → Hk1 (ξ1 ), ξ1 ∈ .π1 /N1 such that V (ξ ) transforms πk (ξ ) into πk1 (ν(ξ )). Define µ = k∈K µk and π(ξ ) = πk (ξ ) for ξ ∈ .k . Since the πk are mutually disjoint, there exists a set M ⊂ .π , µ(M) = 0 such that π(ξ ) and π(ξ ) are unitarily nonequivalent for any ξ, ξ ∈ .π \ M, ξ = ξ . Let m(ξ ) = nk for

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ξ ∈ .k . The field .π ξ → m(ξ ) is µ-measurable, hence, ξ → π(ξ ) ⊗ Iξ is a µmeasurable field of representations of R on .π (here Iξ is the identity operator on the space K(ξ ) = Hm(ξ ) ) and π .π π(ξ ) ⊗ Iξ dµ(ξ ). This is the disintegration of π into primary components. Remark 3. The set .π depends on the representation π . If R is a C ∗ -algebra of type I then .π can be identified with Rˆ with the Borel structure defined by the topology, which coincides with the Mackey structure (see [Dix2]). Also, in this case one has π(ξ ) ∈ ξ for any ξ ∈ Rˆ and the equivalence class of µk is uniquely determined by πk . Our basic assumption is the following: there exist a Borel set . and a one-toone map ϕ: . → Rˆ such that for any representation πof R there exist mutually singular standard measures µk , k ∈ K, on . and a k∈K µk -measurable field ξ → π(ξ ) of unitarily nonequivalent . such irreducible representations of R on that π(ξ ) ∈ ϕ(ξ ) and π k∈K nk . π(ξ ) dµk (ξ ) or equivalently π . π(ξ )⊗ Iξ dµ(ξ ), where µ = k∈K µk and Iξ is the identity operator on K(ξ ) defined above. The set nk is defined uniquely up to permutation of indices. Clearly, π(σi ) is irreducible for any irreducible representation π of R and i ∈ Nn . Moreover, any two representations π1 and π2 of R are unitarily equivalent if and only if π1 (σi ) and π2 (σi ) are unitarily equivalent. Thus, we can define the action of σi on Rˆ as follows: for any ξ ∈ Rˆ and any π ∈ ξ set σi (ξ ) to be the equivalence class of the representation π(σi ). Since ϕ: . → Rˆ is one-to-one, we can define σi : . → . to be σi (ξ ) = ϕ −1 (σi (ϕ(ξ ))). In general σi : . → . is not necessarily a Borel isomorphism. If π is a representations of AεR1 ,...,εn then the restriction of π to R is a representation of R. The next theorem is a realization of π in the space H of disintegration of π(R) into primary components. THEOREM 2. Let π be a representation of AεR1 ,...,εn in a Hilbert space H = ⊕ ⊕ ˜ . H (ξ ) ⊗ K(ξ ) dµ(ξ ) such that π |R = . π(ξ ) ⊗ Iξ dµ(ξ ), where H (ξ ) = H (ξ ) ⊗ K(ξ ), π˜ (ξ ) = π(ξ ) ⊗ Iξ and (., µ) satisfy the basic assumption. Let +1i = {ξ ∈ . | π(ξ )(ti ) = 0}, +2i = {ξ ∈ . | π(ξ )(σi (ti )) = 0}. Then there exist µ-negligible Borel sets N1 , N2 ⊂ . and Borel maps 8i , i = 1, . . . , n, such that 8i is an isomorphism of +2i \ N2 onto +1i \ N1 and (π(r)f )(ξ ) = π(ξ ˜ )(r)f (ξ ), (π(Xi )f )(ξ ) d(8i (µ)) (8i (ξ )) εi π˜ (8i (ξ ))(ti )f (8i (ξ )), Ui (8i (ξ )) dµ = 2 ξ ∈ +i \ N2 0, otherwise.

(10)

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Here the measure χ+1i (ξ ) d8i (µ)(ξ ) is absolutely continuous with respect to dµ(ξ ), +1i \ N1 ξ → Ui (ξ ) is a measurable field of unitary operators from H˜ (ξ ) into H˜ (8−1 i (ξ )) satisfying the relations ˜ )Ui∗ (ξ ) = π˜ (8−1 Ui (ξ )π(ξ i (ξ ))(σi ), −1 Uj (8−1 i (ξ ))Ui (ξ ) = Ui (8j (ξ ))Uj (ξ )

(11)

µ-almost everywhere on +1i ∪ +1j . Moreover, σi (ξ ) = 8i (ξ ), ξ ∈ +2i µ-a.e. Proof. Here we retain the notations from Theorem 1. Denote by P1i , P2i , i = 1, . . . , n, the projections onto (ker π(ti ))⊥ and (ker π(σi (ti )))⊥ respectively, Rπ the von Neumann algebra generated by π(R) ⊕ and Z the center of Rπ . Fix i ∈ Nn . It is clear that P1i , P2i ∈ Z ⊂ Rπ and Pji = . χ+j (ξ ) dµ(ξ ) for j = 1, 2. Moreover, i

the subspace Pji H is invariant with respect to π(r), r ∈ R which implies that the operators {π1i (r) ≡ π(r)P1i | r ∈ R} and {π2i (r) ≡ π(σi (r))P1i | r ∈ R} define representations of R on Pi1 H and Pi2 H respectively and ⊕ ⊕ i i π˜ (ξ )(r) dµ(ξ ), π2 (r) = π(ξ ˜ )(σi (r)) dµ(ξ ). π1 (r) = +1i

+2i

Let Ui be the phase of π(Xi ). Ui is a partial isometry with the initial and final spaces (ker π(ti ))⊥ and (ker π(σi (ti )))⊥ respectively and hence it is a unitary operator from P1i H onto P2i H . Moreover, by Proposition 1 we have Ui E|π(r)| (+)P1i Ui∗ = Ui E|π(r)| (+)Ui∗ = E|π(σi (r))| (+)Ui Ui∗ = E|π(σi (r))|(+)P2i and Ui Uπ(r)P1i Ui∗ = Ui Uπ(r)Ui∗ = Uπ(σi (r))Ui Ui∗ = Uπ(σi (r)) P2i which implies Ui π1i (r)Ui∗ = π2i (r), Ui Z1i Ui∗ = Z2i .

r ∈ R,

Ui π1i (R) Ui∗ = π2i (R)

and

⊕ Clearly, the center Zji is the algebra of diagonalizable operators in +j H (ξ ) ⊗ i K(ξ ) dµ(ξ ). Now from [Dix1, Theorem 4, p. 238] or [Ta, Theorem 8.23] we have that there exist µ-negligible Borel sets N1 , N2 ⊂ . and Borel isomorphisms 8i of +2i \ N2 onto +1i \ N1 such that the measures µ1 and 8i (µ2 ) are equivalent (i.e. µ1 (+) = 0 1 if and only if µ2 (8−1 i (+)) = 0 for any Borel set + ⊂ +i ), where µ1 and µ2 are 1 2 the restrictions of the measure µ onto +i and +i , respectively. Further, there exists a µ-measurable field ξ → Ui (ξ ), ξ ∈ +1i \ N1 , of unitary operators from H˜ (ξ ) ˜ )Ui∗ (ξ ) = π(8 ˜ −1 onto H˜ (8−1 i (ξ )) such that Ui (ξ )π(ξ i (ξ ))(σi ) and

⊕ d(8i (µ)) (ξ ) dµ(ξ ). Ui (ξ ) Ui = dµ +1i

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Let π1i (a) be a central element from π1i (R). Then π2i (a) is central in π2i (R). Hence π˜ (ξ )(a) = λ1 (ξ )IH˜ (ξ ) , π˜ (ξ )(σi (a)) = λ2 (ξ )IH˜ (ξ ) , where λj (·), j = 1, 2 are Borel complex functions. From Ui π1i (a)Ui∗ = π2i (a) we obtain λ1 (8i (ξ )) = λ2 (ξ ) µ-a.e. +2i . Moreover, it follows from the definition of the map σi : . → . that λ2 (ξ ) = λ1 (σi (ξ )) µ-a.e. Thus λ1 (8i (ξ )) = λ1 (σi (ξ )) for almost all ξ ∈ +2i \ N2 . Since π1i (R) ∩ π1i (R) is dense in Z we get f (8i (ξ )) = f (σi (ξ )) for any Borel function and hence σi (ξ ) = 8i (ξ ) µ-a.e. on +2i . Finally, (11) follows from Proposition 1 ([Ui , Uj ]Pij = 0). Conversely, one can easily check that any family of operators defined by (10) ✷ determines a representation of AεR1 ,...,εn . It follows from the above theorem that σi : . → . is equal to a Borel function µ-almost everywhere if ker π(σi (ti )) = {0}. From now on we will assume that σi , i = 1, . . . , n are Borel. The mappings σi , i ∈ Nn determine an action of the group Zn on . by (i1 , . . . , in )ξ = σ1i1 (σ2i2 (. . . σnin (ξ ) . . .)) and generate the dynamical system (., (σi )ni=1 ). Let 0. The sets ˜ = + ˜ ∪+ ˜ + sets of positive measure: + ˜ i }, i = 1, 2 are invariant with respect to ˜ i ) = {σ1i1 (. . . (σnin (ξ ) . . .) | ξ ∈ +

163

∗-Representations of Twisted Generalized Weyl Constructions VOLODYMYR MAZORCHUK1 and LYUDMYLA TUROWSKA2

1 Department of Mathematics, University of Uppsala, Box 480, SE 75106, Sweden. e-mail: [email protected] 2 Department of Mathematics, Chalmers University of Technology and GU, SE 41296, Göteborg, Sweden. e-mail: [email protected]

(Received: July 1999) Presented by Y. Drozd Abstract. We study bounded and unbounded ∗-representations of Twisted Generalized Weyl Algebras and algebras similar to them for different choices of involutions. Mathematics Subject Classifications (2000): Primary 47C10; secondary: 47D40, 16W10. Key words: generalized Weyl algebra, twisted generalized Weyl construction, involution, ∗-representation, irreducible representation.

1. Introduction Generalized Weyl Algebras (GWA) were first introduced by Bavula as some natural generalization of the Weyl algebra A1 (see [B2] and references therein). Since then, GWA have become objects of much interest (see for example [B1, B2, KMP, Sm, Sk, DGO]). Many known algebras such as U(sl(2, C)), Uq (sl(2, C)), down-up algebras and others can be viewed as generalized Weyl algebras and thus can be studied from some unifying point of view. In [MT] we introduce a nontrivial higher rank generalization of GWA, which we call Twisted Generalized Weyl Algebras. We study simple weight modules over twisted GWA in a special (torsion-free) case and show that there arise new effects which might be of interest for further investigations. We also note, that Twisted GWA are not isomorphic to the higher rank GWA, considered in [B1] in general. For such algebras it is natural to study their unitarizable modules, i.e. ∗-representations in a Hilbert space. The purpose of this paper is to introduce natural ∗-structures over twisted generalized Weyl algebras and some of their noncommutative (‘quantum’) deformations and to study Hilbert space representations of the corresponding ∗-algebras (real forms) by bounded and unbounded operators. The class of ∗-algebras considered in the paper contains a number of known ∗-algebras such as U(su(2)), U(sl(2, R)), Uq (su(2)), Uq (su(1, 1)), SUq (2) as well as

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∗-algebras generated by Qij -CCR ([Jor]), twisted canonical (anti)-commutation relations ([Pus, PW]) and others. The technique of study of ∗-representations used in the paper is based on the study of structure and properties of some dynamical systems. This approach goes back to the classical papers [M1, M2, EH, Kir1, Ped] and turns out to be a useful tool for investigation of representations of many ∗-algebras ([V1, V2, OS1, OT]). Using this method we obtain a complete classification of irreducible representations of the introduced real forms of twisted GWA (the case of commutative ground ∗-algebra R, see Section 2 for the precise definitions) provided that the corresponding dynamical system is simple, i.e., it possesses a measurable section. Any such representation is related to an orbit of the dynamical system. Otherwise, the problem of unitary classification of their representations can be problematic. Namely, if the dynamical system does not have a measurable section there might exist nonatomic quasi-invariant measures which generate factor-representations which are not of type I ([MN]). For a noncommutative ∗-algebra R of type I, we show that there is a one-to-one correspondence between weight irreducible representations of the corresponding ∗-algebras and projective unitary irreducible representations of some groups isomorphic to Zl . We study bounded and unbounded ∗-representations of our algebras. Note that the first problem that arises when one deals with representations by unbounded operators is to select the ‘well-behaved’ representations like the integrable representations of Lie algebras. In the paper we define a class of ‘well-behaved’ representations for our ∗-algebras and study them up to unitary equivalence. The paper is organised in the following way: in Section 2 we introduce a deformation of twisted GWA and define ∗-structures on it. In Section 3 we study bounded representations of the corresponding ∗-algebras (real forms). After discussing some properties of representations, we describe irreducible ones in terms of two models. As a results of our classification all irreducible weight ∗-representations of real forms for twisted GWA are listed in Theorem 4. In Section 4 the results obtained in the previous section are generalized to a class of unbounded representations.

2. Twisted Generalized Weyl Construction and its ∗-Structures 2.1. DEFINITION OF THE ALGEBRAS AND ∗- STRUCTURES Throughout the paper C is the complex field, R is the field of real numbers, Z is the ring of integers, N is the set of all positive integers. Fix a positive integer n and set Nn = {1, 2, . . . , n}. Let R be a unital algebra over C, {σi | 1 i n} a set of pairwise commuting automorphisms of R and M

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165

a matrix (µij )i,j ∈Nn with complex nonzero entries µij ∈ C, i, j ∈ Nn . Fix central elements ti ∈ R, i ∈ Nn , satisfying the following relations: ti tj = µij µj i σi−1 (tj )σj−1 (ti ),

i, j ∈ Nn , i = j.

(1)

We define A to be an R-algebra generated over R by indeterminates Xi , Yi , i ∈ Nn , subject to the relations • • • • •

Xi r = σi (r)Xi for any r ∈ R, i ∈ Nn ; Yi r = σi−1 (r)Yi for any r ∈ R, i ∈ Nn ; Xi Yj = µij Yj Xi for any i, j ∈ Nn , i = j ; Yi Xi = ti , i ∈ Nn ; Xi Yi = σi (ti ), i ∈ Nn .

We will say that A is obtained from R, M, σi , ti , i ∈ Nn by twisted generalized Weyl construction. One can easily show that the elements Xi , Xj , Yi and Yj satisfy additionally the relations of the form: Xi Xj Yi Xi = µj i Xi Yi Xj Xi ⇔ Xi Xj ti = µj i Xj Xi σj−1 (ti ), (2) −1 Y X Y Y ⇔ Y Y σ (t ) = µ Y Y σ (σ (t )). Yi Yj Xi Yi = µ−1 i i j i i j i i j i i j i ij ij The algebra A possesses a natural structure of Zn -graded algebra by setting deg R = 0,

deg Xi = gi ,

deg Yi = −gi ,

i ∈ Nn ,

where gi , i ∈ Nn , are the standard generators of Z . n

Remark 1. If R is commutative and µij = 1, i, j ∈ Nn , then A coincides with the algebra A defined in [MT]. The twisted GWA A(R, σ1 , . . . , σn , t1 , . . . , tn ) of rank n can be obtained as the quotient ring A /I , where I is the maximal graded two-sided ideal of A intersecting R trivially. By [MT, Lemma 2], this ideal is unique. It is worth to point out that the requirement for µij to be equal 1 is not important. All the results obtained in [MT] can be reformulated easily for the case µij = 1. In particular, given a commutative ring R, its automorphisms σi , i ∈ Nn , a matrix M = (µij )i,j ∈Nn , µij = 0, and elements ti ∈ R, i ∈ Nn , satisfying the relations ti tj = µij µj i σi−1 (tj )σj−1 (ti ), i, j ∈ Nn , i = j , we can define a twisted GWA as follows: A(R, σ1 , . . . , σn , t1 , . . . , tn , M) = A /I , where I is the ideal defined above. This class of algebras contains beside the algebras U(sl(2)), Uq (sl(2)), the algebras of skew differential operators on the quantum n-space known as the quantized Weyl algebras [DJ, Jord] and some other coordinate rings of quantum symplectic and Euclidean spaces. Assume that µij = µj i ∈ R and R is a ∗-algebra satisfying the condition σi (r ∗ ) = (σi (r))∗ for any r ∈ R, i ∈ Nn . Then the algebra A possesses the following ∗-structures: Xi∗ = εi Yi ,

ti∗ = ti ,

where εi = ±1, i ∈ Nn .

We will denote the corresponding ∗-algebras by AεR1 ,...,εn .

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Remark 2. It is clear that any maximal graded two-sided ideal of the ∗-algebra is a ∗-ideal. Thus in the case of commutative R, the above ∗-structures generate ∗-structures on the corresponding twisted GWA. AεR1 ,...,εn

2.2. EXAMPLES 1. The universal enveloping algebra U(sl(2, C)). Let R = C[H, T ] be the polynomial ring in two variables, t = T , T ∗ = T , H ∗ = H , σ (H ) = H − 1, σ (T ) = T + H . Then A1R U(su(2)) and A−1 R U(sl(2, R)). 2. The quantum algebra Uq (sl(2, C)). Let R = C[T , k, k −1 ] (polynomials in T and Laurent polynomials in k), t = T,

T ∗ = T,

σ (k) = q −1 k,

σ (T ) = T +

k − k −1 . q − q −1

• If q ∈ R and k = k ∗ then A1R suq (2), A−1 R suq (1, 1). suq (2). • If |q| = 1 and k ∗ = k −1 then A1R A−1 R Irreducible representations of the ∗-algebras suq (2), suq (1, 1) were studied in [V3]. 3. Quantized Weyl algebras. Let = (λij ) be an n × n matrix with nonzero complex entries such that λij = λ−1 j i , let q = (q1 , . . . , qn ) be an n-tuple of elements q, from C\{0, 1}. The n-th quantized Weyl algebra An ([Jord]) is the C-algebra with generators xi , yi , 1 i n, and relations xi xj = qi λij xj xi , xi yj = λj i yj xi ,

yi yj = λij yj yi , xj yi = qi λij yi xj ,

xj yj − qj yj xj = 1 +

j −1

(3)

(qi − 1)yi xi ,

i=1

for 1 i < j n. Let R = C[t1 , . . . , tn ] be the polynomial ring in n variables, σi the automorphisms of R defined by i−1 (qj − 1)tj , qi ti+1 , . . . , qi tn ), σi : p(t1 , . . . , tn ) → p(t1 , . . . , ti−1 , 1 + qi ti + j =1 q,

and M = (µij )ni,j =1 , where µij = λj i and µj i = qi λij for i < j . Then An is isomorphic to a quotient of the algebra A which is obtained from R, M, σi , ti , i ∈ Nn by twisted generalized Weyl construction. It is easy to show that the maximal graded ideal of A intersecting R trivially is generated by the elements xi xj − qi λij xj xi , yi yj − λij yj yi ,

1 i < j n,

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167

q,

hence A(R, σ1 , . . . , σn , t1 , . . . , tn , M) An . Assume that qi , λij ∈ R \{0}, xi∗ = εi yi , i, j = 1, . . . , n. The involutions define q, ∗-structures in A and the quantum Weyl algebra An . Note that in the case λj i = qi λij = µ ∈ (0, 1) relations (3) are known as twisted canonical commutation relations ([PW]). ∗-Representations of the algebra which correspond to the involution xi∗ = yi , i = 1, . . . , n were classified in [PW]. 4. Qij -CCR. Let Ad be a ∗-algebra generated by elements ai∗ , ai , i = 1, . . . , d, satisfying the following Qij -commutation relations: ai∗ aj = Qij aj ai∗ , ai∗ ai − Qii ai ai∗ = 1, ai aj = Qj i aj ai , i = j,

i = j,

(4) (5)

where Qii ∈ (0, 1), |Qij | = 1 if i = j , Qij = Qj i , i, j = 1, . . . , d. The ∗algebra A1 is a real form of the generalized Weyl algebra with R = C[T ] and t = t ∗ = T , σ (T ) = Q−1 11 (T − 1). For d > 1 set R = Ad−1 ⊕ C[T ], where T = T ∗ and [T , a] = 0 for any a ∈ Ad−1 . Let σ (ai ) = Qid ai , i = 1, . . . d − 1, 1 σ (T ) = Q−1 dd (T − 1) and t = T . Then AR Ad . Representations of Qij -CCR were studied in [Pr] using a method different from the one presented in this paper. For other examples of twisted generalized Weyl constructions see also Remark 9. 3. ∗-Representations of Twisted Generalized Weyl Constructions 3.1. BOUNDED REPRESENTATIONS OF AεR1 ,...,εn Let H be a separable Hilbert space. Throughout this section L(H ) denotes the set of all bounded operators on H . Let B(R) be the class of Borel subsets of R. For a selfadjoint operator A we will denote by EA (·) the corresponding resolution of the identity. Let M be any subset of L(H ). We denote by M the commutant of M, i.e. the set of those elements of L(H ) that commute with all the elements of M. For a group G we will denote by G∗ the set of its characters. In this section we study bounded representations of AεR1 ,...,εn , i.e. ∗-homomorphisms π : AεR1 ,...,εn → L(H ) up to unitary equivalence. We recall that representations π in H and π˜ in H˜ of a ∗-algebra A are said to be unitarily equivalent if there exists a unitary operator U : H → H˜ such that U π(a) = π˜ (a)U for any a ∈ A. Throughout the paper we will use the notation π1 π2 for unitarily equivalent representations π1 , π2 . We will assume also that µij > 0 for i, j ∈ Nn . Let π be a representation of AεR1 ,...,εn . We will denote the operators π(x), x ∈ AεR1 ,...,εn simply by x if no confusion can arise. Let r = Ur |r| (r ∈ R), Xi = Ui |Xi | be the polar decomposition of operators r and Xi respectively, where |r| = (r ∗ r)1/2 , |Xi | = (Xi∗ Xi )1/2, Ur and Ui are phases of the operators r and Xi . We recall, that the phase of an operator B is a partial isometry with the initial

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space (ker B)⊥ = (ker B ∗ B)⊥ and the final space (ker B ∗ )⊥ = (ker BB ∗ )⊥ . Since Xi∗ Xi = εi ti and Xi Xi∗ = εi σi (ti ), we have εi ti 0, εi σi (ti ) 0, and |Xi | = (εi ti )1/2 , |Xi∗ | = (εi σi (ti ))1/2 , i ∈ Nn . PROPOSITION 1. For any bounded representation of the ∗-algebra AεR1 ,...,εn the following relations hold Ui Ur = Uσi (r)Ui , Ui E|r| (+) = E|σi (r)| (+)Ui , + ∈ B(R), r ∈ R, i ∈ Nn , [Ui , Uj∗ ]Qij = 0, i = j, [Ui , Uj ]Pij = 0,

(6) (7)

where Pij and Qij are projections onto (ker ti tj )⊥ and (ker σi (ti tj ))⊥ respectively. Moreover, each Ui is centered, i.e. [Uik (Ui∗ )k , Uil (Ui∗ )l ] = 0,

[Uik (Ui∗ )k , (Ui∗ )l Uil ] = 0,

[(Ui∗ )k Uik , (Ui∗ )l Uil ] = 0 for any k, l ∈ N.

(8)

Conversely, any family of operators r = Ur |r|, Xi = Ui (εi ti )1/2 , i ∈ Nn , determines a bounded representation of AεR1 ,...,εn if |r|, εi ti , εi σi (ti ), i ∈ Nn are bounded positive operators and Ur , Ui , i ∈ Nn are partial isometries satisfying (6)–(7) and the conditions ker Ur = ker |r|, ker Ui = ker ti . Proof. From the relations in the algebra AεR1 ,...,εn it follows that Xi |r|2 = |σi (r)|2 Xi . By [SST, Theorem 2.1] we have Xi E|r| (+) = E|σi (r)|(+)Xi for any + ∈ B(R). To obtain relations (6) we note that |Xi | = (εi ti )1/2 , ker Xi = ker Ui = ker ti and ti commutes with any projection E|r| (+) as a central element of the algebra R. By the definition of polar decomposition, Ui∗ Ui = Eεi ti (R \ {0}) and Ui Ui∗ = Eεi σi (ti ) (R \ {0}). From this and relations (6) it follows that Ui is centered. The relations connecting Ui and Uj follow from (2). Indeed, Xi Xj ti = µj i Xj Xi σj−1 (ti ) ⇔ Ui (εi ti )1/2 Uj (εj tj )1/2 ti = µj i Uj (εj tj )1/2Ui (εi ti )1/2 σj−1 (ti ). From the relation εi ti Uj = Uj εi σj−1 (ti ) and the fact that (ker Uj )⊥ = (ker tj )⊥ is invariant with respect to σj−1 (ti ) we can conclude that εi σj−1 (ti )|(ker Uj )⊥ is positive and (εi ti )1/2 Uj = Uj (εi σj−1 (ti ))1/2 . This gives Ui Uj (εi εj σj−1 (ti )tj )1/2 ti = µj i Uj Ui (εi εj σi−1 (tj )ti )1/2 σj−1 (ti ) ⇔ Ui Uj (εi εj σj−1 (ti )tj ti2 )1/2 = Uj Ui (µj i µij εi εj σi−1 (tj )ti (σj−1 (ti ))2 )1/2 ⇔ Ui Uj (εi εj σj−1 (ti )tj ti2 )1/2 = Uj Ui (εi εj σj−1 (ti )tj ti2 )1/2 . By (1) we can conclude now that [Ui , Uj ]Pij = 0. Note that another relation of (2) will give us the same result. Similar arguments applying to the equality Xi Yj = µij Yj Xi imply the relations connecting Ui and Uj∗ .

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169

The converse implication follows by standard arguments from spectral decomposition of the operators |r|. ✷ The question of classifying of ∗-representations up to unitary equivalence can be very difficult in general for arbitrary ∗-algebra. If the ∗-algebra is of type I one has a more satisfactory theory (see [Dix2]). From now on we will assume that R is an algebra of type I, i.e. for any representation π of R the W ∗ -algebra π(R) = {π(r), r ∈ R} is of type I (see [Dix2] for the precise definition). The algebra π(R) coincides with the W ∗ -algebra generated by π(R). Moreover, we will restrict ourselves to the case of countably generated ∗-algebras. Denote by Rˆ the set of equivalence classes of irreducible representations of R and by Hn the standard n-dimensional Hilbert space. THEOREM 1. Let H be a separable Hilbert space, π a representation of R. Then there exist a standard Borel space .π , mutually singular positive measures (µk )k∈K on .π , µk -measurable fields ξ → Hk (ξ ) of Hilbert spaces, µk -measurable fields ξ → πk (ξ ) of nontrivial unitarily nonequivalent ⊕ irreducible representations on Hk (ξ ) and an isomorphism of H onto k∈K nk .π Hk (ξ ) dµk (ξ ), where the nk ∈ N ∪ {∞} are mutually distinct, which transforms the representation π into ⊕ nk πk (ξ ) dµk (ξ ). (9) k∈K

.π

⊕ Moreover, the representations .π πk (ξ ) dµk (ξ ) are mutually disjoint and the set (nk )k∈K is unique up to a permutation of the set of indices. Proof. Let A be the closure of π(R) in the operator norm. Since R is of type I and countably generated, we have that A is a separable C ∗ -algebra of type I. The theorem now follows from the general result about the same decomposition of any representation of A applied to the identity representation 2(a) = a for any a ∈ A (see [Dix2, Theorem 8.6.6]). Here .π = Aˆ is the set of equivalence classes of irreducible representations of A. We also note that the measures µk on .π are defined uniquely up to equivalence. ✷ Using standard arguments one can show the uniqueness of the decomposition (9). Namely, if .π1 , (µ1k )k∈K , ξ → Hk1 (ξ ), ξ → πk1 (ξ ) have the same properties then there exists a µk -negligible set N ∈ .π and µ1k -negligible set N1 ∈ .π1 , k ∈ K, a Borel isomorphism ν of .π /N onto .π1 /N1 transforming µk into a measure µ˜ 1k equivalent to µ1k for any k ∈ K, an isomorphism ξ → V (ξ ) of the field ξ → Hk (ξ ), ξ ∈ .π /N onto the field ξ1 → Hk1 (ξ1 ), ξ1 ∈ .π1 /N1 such that V (ξ ) transforms πk (ξ ) into πk1 (ν(ξ )). Define µ = k∈K µk and π(ξ ) = πk (ξ ) for ξ ∈ .k . Since the πk are mutually disjoint, there exists a set M ⊂ .π , µ(M) = 0 such that π(ξ ) and π(ξ ) are unitarily nonequivalent for any ξ, ξ ∈ .π \ M, ξ = ξ . Let m(ξ ) = nk for

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ξ ∈ .k . The field .π ξ → m(ξ ) is µ-measurable, hence, ξ → π(ξ ) ⊗ Iξ is a µmeasurable field of representations of R on .π (here Iξ is the identity operator on the space K(ξ ) = Hm(ξ ) ) and π .π π(ξ ) ⊗ Iξ dµ(ξ ). This is the disintegration of π into primary components. Remark 3. The set .π depends on the representation π . If R is a C ∗ -algebra of type I then .π can be identified with Rˆ with the Borel structure defined by the topology, which coincides with the Mackey structure (see [Dix2]). Also, in this case one has π(ξ ) ∈ ξ for any ξ ∈ Rˆ and the equivalence class of µk is uniquely determined by πk . Our basic assumption is the following: there exist a Borel set . and a one-toone map ϕ: . → Rˆ such that for any representation πof R there exist mutually singular standard measures µk , k ∈ K, on . and a k∈K µk -measurable field ξ → π(ξ ) of unitarily nonequivalent . such irreducible representations of R on that π(ξ ) ∈ ϕ(ξ ) and π k∈K nk . π(ξ ) dµk (ξ ) or equivalently π . π(ξ )⊗ Iξ dµ(ξ ), where µ = k∈K µk and Iξ is the identity operator on K(ξ ) defined above. The set nk is defined uniquely up to permutation of indices. Clearly, π(σi ) is irreducible for any irreducible representation π of R and i ∈ Nn . Moreover, any two representations π1 and π2 of R are unitarily equivalent if and only if π1 (σi ) and π2 (σi ) are unitarily equivalent. Thus, we can define the action of σi on Rˆ as follows: for any ξ ∈ Rˆ and any π ∈ ξ set σi (ξ ) to be the equivalence class of the representation π(σi ). Since ϕ: . → Rˆ is one-to-one, we can define σi : . → . to be σi (ξ ) = ϕ −1 (σi (ϕ(ξ ))). In general σi : . → . is not necessarily a Borel isomorphism. If π is a representations of AεR1 ,...,εn then the restriction of π to R is a representation of R. The next theorem is a realization of π in the space H of disintegration of π(R) into primary components. THEOREM 2. Let π be a representation of AεR1 ,...,εn in a Hilbert space H = ⊕ ⊕ ˜ . H (ξ ) ⊗ K(ξ ) dµ(ξ ) such that π |R = . π(ξ ) ⊗ Iξ dµ(ξ ), where H (ξ ) = H (ξ ) ⊗ K(ξ ), π˜ (ξ ) = π(ξ ) ⊗ Iξ and (., µ) satisfy the basic assumption. Let +1i = {ξ ∈ . | π(ξ )(ti ) = 0}, +2i = {ξ ∈ . | π(ξ )(σi (ti )) = 0}. Then there exist µ-negligible Borel sets N1 , N2 ⊂ . and Borel maps 8i , i = 1, . . . , n, such that 8i is an isomorphism of +2i \ N2 onto +1i \ N1 and (π(r)f )(ξ ) = π(ξ ˜ )(r)f (ξ ), (π(Xi )f )(ξ ) d(8i (µ)) (8i (ξ )) εi π˜ (8i (ξ ))(ti )f (8i (ξ )), Ui (8i (ξ )) dµ = 2 ξ ∈ +i \ N2 0, otherwise.

(10)

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Here the measure χ+1i (ξ ) d8i (µ)(ξ ) is absolutely continuous with respect to dµ(ξ ), +1i \ N1 ξ → Ui (ξ ) is a measurable field of unitary operators from H˜ (ξ ) into H˜ (8−1 i (ξ )) satisfying the relations ˜ )Ui∗ (ξ ) = π˜ (8−1 Ui (ξ )π(ξ i (ξ ))(σi ), −1 Uj (8−1 i (ξ ))Ui (ξ ) = Ui (8j (ξ ))Uj (ξ )

(11)

µ-almost everywhere on +1i ∪ +1j . Moreover, σi (ξ ) = 8i (ξ ), ξ ∈ +2i µ-a.e. Proof. Here we retain the notations from Theorem 1. Denote by P1i , P2i , i = 1, . . . , n, the projections onto (ker π(ti ))⊥ and (ker π(σi (ti )))⊥ respectively, Rπ the von Neumann algebra generated by π(R) ⊕ and Z the center of Rπ . Fix i ∈ Nn . It is clear that P1i , P2i ∈ Z ⊂ Rπ and Pji = . χ+j (ξ ) dµ(ξ ) for j = 1, 2. Moreover, i

the subspace Pji H is invariant with respect to π(r), r ∈ R which implies that the operators {π1i (r) ≡ π(r)P1i | r ∈ R} and {π2i (r) ≡ π(σi (r))P1i | r ∈ R} define representations of R on Pi1 H and Pi2 H respectively and ⊕ ⊕ i i π˜ (ξ )(r) dµ(ξ ), π2 (r) = π(ξ ˜ )(σi (r)) dµ(ξ ). π1 (r) = +1i

+2i

Let Ui be the phase of π(Xi ). Ui is a partial isometry with the initial and final spaces (ker π(ti ))⊥ and (ker π(σi (ti )))⊥ respectively and hence it is a unitary operator from P1i H onto P2i H . Moreover, by Proposition 1 we have Ui E|π(r)| (+)P1i Ui∗ = Ui E|π(r)| (+)Ui∗ = E|π(σi (r))| (+)Ui Ui∗ = E|π(σi (r))|(+)P2i and Ui Uπ(r)P1i Ui∗ = Ui Uπ(r)Ui∗ = Uπ(σi (r))Ui Ui∗ = Uπ(σi (r)) P2i which implies Ui π1i (r)Ui∗ = π2i (r), Ui Z1i Ui∗ = Z2i .

r ∈ R,

Ui π1i (R) Ui∗ = π2i (R)

and

⊕ Clearly, the center Zji is the algebra of diagonalizable operators in +j H (ξ ) ⊗ i K(ξ ) dµ(ξ ). Now from [Dix1, Theorem 4, p. 238] or [Ta, Theorem 8.23] we have that there exist µ-negligible Borel sets N1 , N2 ⊂ . and Borel isomorphisms 8i of +2i \ N2 onto +1i \ N1 such that the measures µ1 and 8i (µ2 ) are equivalent (i.e. µ1 (+) = 0 1 if and only if µ2 (8−1 i (+)) = 0 for any Borel set + ⊂ +i ), where µ1 and µ2 are 1 2 the restrictions of the measure µ onto +i and +i , respectively. Further, there exists a µ-measurable field ξ → Ui (ξ ), ξ ∈ +1i \ N1 , of unitary operators from H˜ (ξ ) ˜ )Ui∗ (ξ ) = π(8 ˜ −1 onto H˜ (8−1 i (ξ )) such that Ui (ξ )π(ξ i (ξ ))(σi ) and

⊕ d(8i (µ)) (ξ ) dµ(ξ ). Ui (ξ ) Ui = dµ +1i

172

VOLODYMYR MAZORCHUK AND LYUDMYLA TUROWSKA

Let π1i (a) be a central element from π1i (R). Then π2i (a) is central in π2i (R). Hence π˜ (ξ )(a) = λ1 (ξ )IH˜ (ξ ) , π˜ (ξ )(σi (a)) = λ2 (ξ )IH˜ (ξ ) , where λj (·), j = 1, 2 are Borel complex functions. From Ui π1i (a)Ui∗ = π2i (a) we obtain λ1 (8i (ξ )) = λ2 (ξ ) µ-a.e. +2i . Moreover, it follows from the definition of the map σi : . → . that λ2 (ξ ) = λ1 (σi (ξ )) µ-a.e. Thus λ1 (8i (ξ )) = λ1 (σi (ξ )) for almost all ξ ∈ +2i \ N2 . Since π1i (R) ∩ π1i (R) is dense in Z we get f (8i (ξ )) = f (σi (ξ )) for any Borel function and hence σi (ξ ) = 8i (ξ ) µ-a.e. on +2i . Finally, (11) follows from Proposition 1 ([Ui , Uj ]Pij = 0). Conversely, one can easily check that any family of operators defined by (10) ✷ determines a representation of AεR1 ,...,εn . It follows from the above theorem that σi : . → . is equal to a Borel function µ-almost everywhere if ker π(σi (ti )) = {0}. From now on we will assume that σi , i = 1, . . . , n are Borel. The mappings σi , i ∈ Nn determine an action of the group Zn on . by (i1 , . . . , in )ξ = σ1i1 (σ2i2 (. . . σnin (ξ ) . . .)) and generate the dynamical system (., (σi )ni=1 ). Let 0. The sets ˜ = + ˜ ∪+ ˜ + sets of positive measure: + ˜ i }, i = 1, 2 are invariant with respect to ˜ i ) = {σ1i1 (. . . (σnin (ξ ) . . .) | ξ ∈ +

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