*-Representations of Twisted Generalized Weyl Constructions

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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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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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 <ξ = {σ1i1 (σ2i2 (. . . (σnin (ξ ) . . .)) | (i1 , . . . , in ) ∈ Zn } denote the orbit of ξ ∈ .. We will say that (., (σi )ni=1 ) is simple if there exists a Borel set τ ⊂ . intersecting any orbit of the dynamical system exactly in one point. This set is called a measurable section of the dynamical system. PROPOSITION 2. Assume that (., (σi )ni=1 ) is simple. Then for any irreducible representation of AεR1 ,...,εn the corresponding measure µ is concentrated on a single orbit of the dynamical system. Proof. Let π be an irreducible representation of AεR1 ,...,εn . Then the corresponding measure µ is ergodic with respect to (σi )ni=1 , i.e. given a Borel set + such that σi (+) = + for any i = 1, . . . , n, we have eitherµ(+) = 0 or µ(. \ +) = 0. In ⊕ order to prove this, consider the projection P = . χ+ (ξ ) dµ(ξ ). By Theorem 2, P commutes with any operator from Rπ and Xi , Xi∗ , i ∈ Nn . Thus, P = λI for some λ ∈ R. Since P is a projection, we obtain that either λ = 0 or λ = 1 and the statement follows. Now we show that the existence of a measurable section is a sufficient condition for µ to be concentrated on an orbit. Indeed, suppose that this is not the case and denote by + the support of µ. Then there exists (i1 , . . . , in ) ∈ Zn such that µ(σ1i1 (. . . σnin (τ ) . . .) ∩ +) = 0. Let µ˜ be the restriction of µ to the set ˜ By the assumption, µ˜ is σ1i1 (. . . σnin (τ ) . . .) ∩ + which will be denoted by +. ˜ into two not concentrated at one point x. Hence, there exists a partition of + 1 2 1 2 i ˜ ,+ ˜ ∩+ ˜ = ∅, µ( ˜ ) > 0. The sets ˜ = + ˜ ∪+ ˜ + sets of positive measure: + ˜ i }, i = 1, 2 are invariant with respect to ˜ i ) = {σ1i1 (. . . (σnin (ξ ) . . .) | ξ ∈ + <(+ ˜ 1 ) ∩ <(+ ˜ 2 ) = ∅. This contradicts the ergodicity of the each σi . Moreover, <(+ measure µ. Thus µ˜ is concentrated at one point. This completes the proof. ✷

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Remark 4. If the dynamical system does not have a measurable section the problem of unitary classification of all irreducible representation of the ∗-algebra can be very difficult. In this case there might exist ergodic quasi-invariant measures which are not concentrated on a single orbit. Such measures generate a wide class of representations which are not of type I and the description of which is problematic (see, for example, [MN, OS1, OS2, V2]). For the rest of the paper we will assume that the dynamical system defined above is simple. The investigation of irreducible representations can be restricted to that of irreducible representations with supp µ ⊂ γ for a fixed orbit γ ⊂ .. For i ∈ Nn we will call an element λ ∈ γ a forward (a backward) i-break if π(λ)(ti ) = 0 (π(λ)(σi (ti )) = 0), where π(λ) is an irreducible representation in H (λ). For λ ∈ γ denote by Pλ the set of all ξ ∈ γ with the following properties: εi π(ξ )(ti )  0, εi π(ξ )(σi (ti ))  0, there exists (i1 , . . . , ik ) ∈ Zn such that ξ = −δ k σiδkk (. . . (σiδ11 (λ) . . .)), δl = ±1, 1
l < k and each σil+1l+1 (. . . σi−δ (ξ ) . . .), 1
k l
k is not forward il -break if δl = 1 and is not backward il -break if δl = −1. Let gi denote the canonical generator (0, . . . , 0, 1, 0, . . . , 0) of Zn . We define    i

P˜λ to be the set of all elements g ∈ Zn such that there exists a decomposition g = giδss . . . giδ11 , where giδkk . . . giδ11 λ ∈ Pλ for any k
s. Below we present some constructions of irreducible representations of AεR1 ,...,εn . For a representation π with supp π = Pλ consider the stabilizer K λ of λ ∈ γ , i.e. K λ = {g ∈ Zn | gλ = λ}, and put K˜ λ = K λ ∩ P˜λ . Note that if λ1 , λ2 ∈ Pλ then K˜ λ1 = K˜ λ2 . ε1 ,...,εn . Consider the following two models of representations of A R λ (M1 ): Assume that K is trivial and Pλ = ∅. Let H = g∈P˜λ H (gλ) where H (gλ) = H (λ) for any g ∈ P˜λ . We define (π(r)f )(gλ) = π(λ)(σ1i1 . . . σnin (r))f (gλ) for g = (i1 , . . . , in ) ∈ Zn , and partial isometries Ui , i = 1, . . . , n, as follows:  0, π(gλ)(σi (ti )) = 0, (Ui f )(gλ) = f (gi gλ), otherwise.  (M2 ): Assume that K λ is nontrivial, Pλ = ∅. Put H = g∈P˜λ H˜ (gλ), where H˜ (gλ) = H (gλ) ⊗ K(gλ), H (gλ) = H (λ) and K(gλ) = K(λ), g ∈ P˜λ , are separable Hilbert spaces. We define (π(r)(f ⊗ h))(gλ) = π(λ)(σ1i1 . . . σnin (r))f (gλ) ⊗ h(gλ) for g = (i1 , . . . , in ) ∈ Zn . The operators Ui , i ∈ Nn , are defined by imposing the following conditions: Ui∗ H˜ (gλ) ⊂ H˜ (gi gλ), g ∈ P˜λ , Ui H˜ (gλ) ⊂ H˜ (gi−1 gλ), (Uiδkk . . . Uiδ11 (f ⊗ h))(gλ) = 0,

k 1 g = gi−δ . . . gi−δ k 1

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VOLODYMYR MAZORCHUK AND LYUDMYLA TUROWSKA −δ

1 provided that there exists 1
s
k such that for g s = gis−1s−1 . . . gi−δ we have that 1 either π(g s λ)(tis ) = 0 and δs = 1, or π(gis g s λ)(tis ) = 0 and δs = −1, (by Uiδi , where δi = −1 we mean Ui∗ ), otherwise

(Uiδkk . . . Uiδ11 (f ⊗ h))(gλ)  f (λ) ⊗ h(λ), = k k 1 1 . . . gi−δ )f (λ) ⊗ S(gi−δ . . . gi−δ )h(λ), W (gi−δ k k 1 1

g∈ / K˜ λ , g ∈ K˜ λ .

k 1 . . . gi−δ ) is a unitary operator acting in H (λ) and such that Here W (gi−δ k 1 k k 1 1 . . . gi−δ )π(λ)(r)W −1 (gi−δ . . . gi−δ ) = π(λ)(σiδkk . . . σiδ11 (r)), W (gi−δ k k 1 1

(12)

S(·) is a unitary irreducible projective representation of K˜ λ on the space K(λ) with multiplier c(k1 , k2 ) defined by c(k1 , k2 )I = W (k1 )W (k2 )W −1 (k1 k2 ), i.e. S(k1 k2 ) = c(k1 , k2 )S(k1 )S(k2 )

for any k1 , k2 ∈ K˜ λ

(see, for example, [Kir2] for the definition). The existence of such operators W (k), k ∈ K˜ λ , follows from the fact that π(λ) and π(λ)(σiδkk . . . σiδ11 ) are unitarily equivalent irreducible representations for k = k 1 . . . gi−δ ∈ K˜ λ . One can show also that W (k1 )W (k2 )W −1 (k1 k2 ), k1 , k2 ∈ K˜ λ gi−δ k 1 commutes with π(λ)(r), r ∈ R, and hence is a multiple of the identity operator. Remark 5. If R is commutative then any irreducible representation of R is onedimensional. Hence dim H (gλ) = 1. Moreover, since any W (k) = λ(k) ∈ C for some |λ(k)| = 1, we have [S(k1 ), S(k2 )] = 0 and hence any irreducible representation s(·) is one-dimensional. This gives us dim H˜ (gλ) = 1. THEOREM 3. Any irreducible representation of AεR1 ,...,εn is unitarily equivalent to one described in the models Mi , i = 1, 2. Proof. Let π be an irreducible representation of AεR1 ,...,εn in a Hilbert space H . By Theorem 2 and Proposition 2, there exists an orbit γ of the dynamical system generated by σi , i ∈ Nn , such that H = λ∈supp π⊆γ H˜ (λ), where H˜ (λ) = H (λ) ⊗ K(λ), (π(r)(f ⊗ h))(λ) = π(λ)(r)f (λ) ⊗ h(λ), and Ui H˜ (λ) ⊂ H˜ (gi−1 λ),

Ui∗ H˜ (λ) ⊂ H˜ (gi λ).

(13)

Let Z be the center of π(R) . Denote by A0 the W ∗ -algebra generated by π(R) and polynomials in Ui , Ui∗ that commute with any operator of Z. Then A0 =  ˜ λ∈supp π A0,λ , where A0,λ is the subalgebra of operators acting in H (λ).

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We claim that A0,λ = L(H˜ (λ)), λ ∈ supp π . Suppose, that the statement is false. Then A0,λ = A10,λ ⊕ A20,λ which implies A = A1 ⊕ A2 , where A is the W ∗ -algebra generated by π(AεR1 ,...,εn ) and Ai = AAi0,λ, i = 1, 2. Indeed, given X = X1 X2 . . . Xn ∈ A, where Xi ∈ {r ∈ R, Ui , Ui∗ , i = 1, . . . , n}, we have either that X H˜ (λ) is orthogonal to H˜ (λ) or X H˜ (λ) ⊆ H˜ (λ). If X H˜ (λ) ⊆ H˜ (λ) and X H˜ (λ) = 0 then one can easily see that XPH˜ (λ) commutes with any operator from Z, where PH˜ (λ) is the projection onto H˜ (λ). Hence XA10,λ ⊆ A10,λ and XA10,λ ⊥ Y A20,λ for any X, Y ∈ A which forces immediately A1 ⊥ A2 . The decomposition A = A1 ⊕ A2 is impossible due to irreducibility. Fix now λ ∈ supp π . It is clear that the standard gradation of AεR1 ,...,εn induces the gradation on the algebra A. From (13) and the equalities ker Ui = ker ti , ker Ui∗ = ker σi (ti ) it follows that Uiδkk . . . Uiδ11 H˜ (λ) = 0 if there exists 1
s
k such that either π(g s λ)(tis ) = 0 and δs = 1 or π(gis g s λ)(tis ) = 0 and δs = −1, where −δ 1 (for δl = −1 we mean by Uiδl l the operator Ui∗l ). In the other g s = gis−1s−1 . . . gi−δ 1 case, U ∗ U |H˜ (λ) = (Uiδkk . . . Uiδ11 )∗ Uiδkk . . . Uiδ11 |H˜ (λ) = I which implies U ∗ U ∈ Z. Let S1 , S2 be two products of some operators Ui , Ui∗ , i = 1, . . . , n, such that deg S1 = deg S2 . We claim that S1 |H˜ (λ) = S2 |H˜ (λ) if both S1 |H˜ (λ) and S2 |H˜ (λ) are nonzero. This follows from the fact that δ

δ

Uiδi Uj j f = Uj j Uiδi f, δ

f ∈ H˜ (ξ ), ξ ∈ supp π, δ

implies either Uiδi Uj j f = 0 or Uj j Uiδi f = 0. Indeed, suppose Ui Uj f = Uj Ui f then Pij |H˜ (ξ ) = Uj∗ Uj Ui∗ Ui |H˜ (ξ ) = Ui∗ Ui Uj∗ Uj |H˜ (ξ ) = 0 which yield that either Ui f = 0 or Uj f = 0. The same conclusion can be drawn for other cases. Thus for any g ∈ P˜λ we can define an operator U (g −1 )PH (λ) to s 1 . . . Ui−δ PH (λ) for some decomposition be the nonzero operator of the form Ui−δ s 1  δ1 δs −1 ˜ g = gi1 . . . gis . It is obvious that the subspace g∈P˜λ U (g )H (λ) is invariant ε1 ,...,εn ), hence it coincides with H . Clearly, operators U (g −1 ) with respect to π(AR are unitary operators from H˜ (λ) to U (g −1 )H˜ (λ). Moreover, Ui U (g −1 )H˜ (λ) = 0 if π(gλ)(ti ) = 0 and Ui U (g −1 )H˜ (λ) = U (gi g −1 )H˜ (λ) if gi−1 g ∈ P˜λ . Analogously we have that Ui∗ U (g −1 )H˜ (λ) = 0

if π(gi gλ)(ti ) = 0

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and Ui∗ U (g −1 )H˜ (λ) = U ((gi g)−1 )H˜ (λ)

if gi g ∈ P˜λ .

Assume now that K λ is trivial. Then U H˜ (λ) := Uiδkk . . . Uiδ11 H˜ (λ) ⊆ H˜ (λ) if and only if either U |H˜ (λ) = 0, or giδkk . . . giδ11 = e and in the last case U |H˜ (λ) = U (e)|H˜ (λ) = I . From this one can conclude that A0 = (π(R)),

A0,λ = (π(λ)(R)),

and

H˜ (λ) = H (λ).

Since U ∗ (g −1 )π(r)U (g −1 )|H (λ) = π(λ)(σiδss . . . σiδ11 )(r), one can easily get that the representation π is unitarily equivalent to one of the representations from M1 . If the dynamical system is not free then A0 = (π(R)) in general. Fix λ ∈ γ . By the same arguments as given above it follows that the representation of our algebra is irreducible if and only if the family of operators A0,λ is irreducible or, equivalently, the family {π(r), Uiδkk . . . Uiδ11 | r ∈ R, giδkk . . . giδ11 ∈ K λ } is irreducible as operators on H˜ (λ). As before, for giδ11 . . . giδkk ∈ K λ we have that either the k 1 . . . Ui−δ onto H˜ (λ) is zero or it is unitary as an restriction of the operator Ui−δ k 1 operator from H˜ (λ) to H˜ (λ). Thus for any k ∈ K˜ λ we can define the operators U (k −1 )PH˜ (λ) as it was done above. It is clear that U (k −1 )PH˜ (λ) transforms any irreducible representation into a unitarily equivalent one. Let W (k) be a unitary operator that transforms π(λ) into π(λ)(σl−il . . . σ1−i1 ) for k = (i1 , . . . , il ) ∈ K˜ λ , i.e. W (k)π(λ)(r)W −1 (k) = π(λ)(σl−il . . . σ1−i1 )(r),

r ∈ R.

(14)

Then U (k −1 )(W −1 (k) ⊗ IK(λ)) commutes with any operator π(λ)(r) ⊗ IK(λ), r ∈ R which implies U (k −1 )(W −1 (k) ⊗ Iλ ) = IH (λ) ⊗ S(k), where S(k) is an operator on K(λ). Since U (k1 ) and U (k2 ) commute for any k1 , k2 ∈ K˜ λ , [W (k1 ) ⊗ S(k1 ), W (k2 ) ⊗ S(k2 )] = 0. From this it follows easily that there exists s(k1 , k2 ) ∈ C such that W (k2 )W (k1 ) = s(k2 , k1 )W (k1 )W (k2 ) and S(k2 )S(k1 ) = s(k1 , k2 )−1 S(k1 )S(k2 ). Moreover, the corresponding representation of our algebra is irreducible if and only if the family {S(k), k ∈ K˜ λ } is irreducible. In the same way we can see that there exist c(k1 , k2 ), k1 , k2 ∈ K˜ λ , such that c(k1 , k2 ) = W (k1 )W (k2 )W −1 (k1 k2 ) and S(k1 k2 ) = c(k1 , k2 )S(k1 )S(k2 ) for any ˜ In particular, c(k2 , k1 )c(k1 , k2 )−1 = s(k1 , k2 ). The representation π is k1 , k2 ∈ K. ✷ unitarily equivalent to a representation from the model M2 . COROLLARY 1. Let π be an irreducible representation of the ∗-algebra AεR1 ,...,εn , λ ∈ supp π . Then supp π = Pλ .

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Remark 6. Theorem 3 shows that there is a one-to-one correspondence between irreducible representations π of the ∗-algebra AεR1 ,...,εn and irreducible projective unitary representations of K˜ λ , λ ∈ supp π . It is clear that K˜ λ is a subgroup of Zn . Hence, K˜ λ  Zl for some l
n. Let k1 , . . . , kl be the generators of K˜ λ and fix unitary operators W (k1 ), . . . , W (kl ) satisfying (14). Defining W (k) = W (kl )δl . . . W (k1 )δ1 for k = klδl . . . k1δ1 we get S(k) = S(kl )δl . . . S(k1 )δ1 , where S(ki ) are unitary operators satisfying the relations S(ki )S(kj ) = s(kj , ki )−1 S(kj )S(ki )

(15)

with s(kj , ki ) described in the proof of the last theorem. The numbers s(ki , kj ), i, j = 1, . . . , l, are uniquely defined by the representation. Moreover, any representation can be obtained in this way. Note that any family of unitary operators S(ki ) satisfying (15) defines a representation of C ∗ -algebra known as the noncommutative tori AD , where D = (θij ), e2πθij = s(kj , ki ) (see [R]). In particular, if l = 2, AD is a rotational algebra. The problem of unitary classification of all representations of such C ∗ -algebras might be very difficult. It is known, for example, that a 2-tori AD is of type I if and only if θ12 = −θ21 ∈ Q. For details we refer the reader to [R]. The next theorem provides a description of irreducible ∗-representations of twisted GWA. Suppose that R is commutative. Then any irreducible representation R is one-dimensional. For π(x) ∈ ϕ(x) ∈ Rˆ and r ∈ R we will denote by r(x) the operator π(x)(r). THEOREM 4. Any irreducible representations of the real form AεR1 ,...,εn of twisted GWA is unitarily equivalent to one of the following: (1) Let the orbit γ and λ ∈ γ√be such that K˜ λ is trivial, Pλ = ∅. Then H = l2 (Pλ ), rex = r(x)ex , Xi ex = εi ti (x)ui (x)eσ −1 (x) where x ∈ Pλ , σi (r)(x) = i r(σi (x)), i = 1, . . . , n and  1, σi−1 (x) ∈ Pλ , ui (x) = 0, σi−1 (x) ∈ Pλ , (2) Let the orbit γ and λ ∈ γ be such that K˜ λ is nontrivial, Pλ = ∅ and let s(·) ∈ (K˜ λ )∗ . Then H = l2 (Pλ ), Xi ex = εi ti (x)Ui ex , rex = r(x)ex , where x ∈ Pλ ,

σi (r)(x) = r(σi (x))

and

if there exists 1
s
k such that either π(g s λ)(ti ) = 0 and

δs = 1

Uiδkk . . . Uiδ11 eλ = 0

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or π(gi g s λ)(ti ) = 0 and

δs = −1

−δ

1 ). Otherwise (here g s = gis−1s−1 . . . gi−δ 1

Uiδkk . . . Uiδ11 eλ = eg −δk ...g −δ1 λ , ik

i1

k 1 if gi−δ . . . gi−δ ∈ / K˜ λ k 1

or k 1 . . . gi−δ )eλ , Uiδkk . . . Uiδ11 eλ = s(gi−δ k 1

k 1 if gi−δ . . . gi−δ ∈ K˜ λ . k 1

Remark 7. It is clear that the formulae above define an action of all operators Ui on any vector ex ∈ l2 (Pλ ). Proof. It follows from Remark 5 that dim H˜ (λ) = 1. For representations from the model M2 we have W (g) = w(g) ∈ C, |w(g)| = 1. If we take w(g) = 1, g ∈ K˜ λ we get S(g) = S( gl )δl . . . S(g1 )δ1 ∈ C

for g = glδl . . . g1δ1 .

Setting s(g) = W (g) ⊗ S(g) we obtain s(·) ∈ (K˜ λ )∗ . The rest follows from Theorem 3. ✷ Remark 8. It was shown in [MT] that the supports of finite-dimensional modules of twisted GWA might have more interesting geometrical structure than in the case of classical GWA. We constructed an example which provides some analogue between the structure of supports of finite-dimensional modules over classical simple Lie algebras and supports of finite-dimensional modules of some twisted GWA. The same analogue can be obtained for the ∗-algebras AεR1 ,...,εn and some real forms of simple Lie algebras. Remark 9. Using the above results one can obtain a complete classification of irreducible representations of the ∗-algebras from Examples 1–4. This technique was also applied in [STP] to study collections (u, v, j1 , . . . , jn ) of selfadjoint unitary operators satisfying the following commutation relations: ji jk = (−1)g(i,k) jk ji , ujk = (−1)h(k) jk u,

vjk = (−1)w(k) jk v,

here g(i, k) = g(k, i) ∈ {0, 1}, g(i, i) = 0, h(k), w(k) ∈ {0, 1} for any i, k = 1, . . . , n. The corresponding ∗-algebra can be treated as a ∗-algebra obtained by twisted generalized Weyl construction and has applications, for example, to a study of operator Banach algebras containing a dense ∗-subalgebra, and construction of invertibility symbols for operators in algebra. For details we refer the reader to [STP].

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3.2. UNBOUNDED REPRESENTATIONS OF AεR1 ,...,εn Often ∗-algebras do not have any bounded representations or their structure is not interesting. In general we have to deal with representations by unbounded operators. The problem of describing all such representations might be very difficult. Usually one studies some classes of ‘well-behaved’ representations. In the case of Lie algebras there were considered the so-called integrable representations which can be extended to unitary representations of the corresponding Lie group. For arbitrary ∗-algebras there is no canonical way how to select good unbounded representations. The problem of defining integrability for some ∗-algebras and operator relations was studied by many authors (see [S1, OS1, OT, S2, V2, Wor] and the bibliography therein). In this section we will try to define unbounded representations for the ∗-algebra AεR1 ,...,εn . Throughout this section we will use some notions and facts from the theory of unbounded operator algebras that can be found in [S1]. Let A be a ∗-algebra with a unit element. A ∗-representation π of A in a Hilbert space H (π ) is a homomorphism from A into the family of closable operators defined on a dense domain D(π ) that is invariant with respect to π(a), a ∈ A, and π(a ∗ ) ⊆ (π(a))∗ for any a ∈ A. For a closable operator A in a Hilbert space H we denote by D(A) the domain of A and (A)s the set {c ∈ L(H ) | cA ⊆ Ac} which is called the strong commutator of A. The  strong commutator of a representation π of a ∗-algebra A is defined by  π(A)s = a∈A (π(a))s . We will say that the set D ⊆ H is a core for a closed operator A if A|D = A. Here A denotes the closure of the operator A. Let π be a representation of AεR1 ,...,εn . For simplicity we shall write r instead of π(r) for r ∈ R if this does not lead to any confusion. Here r is always supposed to be closed. Consider the polar decompositions of the operators r = Ur |r|, where |r| = (r ∗ r)1/2 . Let Z denote the center of AεR1 ,...,εn . We will say that closed operators r ∈ R, Xi , Xi∗ i = 1, . . . , n, acting on a Hilbert space H define a representation of AεR1 ,...,εn if (1) the operators r ∈ R generate a closed representation of R on a dense domain D ⊆ H, (2) the operators t ∈ Z are normal and Et (+) ∈ (r)s for any + ∈ B(C), r ∈ R, (3) D is a core for any r ∈ R, (4) Xi∗ Xi = εi ti , Xi Xi∗ = εi σi (ti ) for all i = 1, . . . , n, (5) relations (6)–(7) hold on H , where Ui is the phase of the operator Xi . PROPOSITION 3. Assume that conditions (1)–(5) hold. Then the operators r ∈ R, Xi , Xi∗ i ∈ Nn define a closed representation π of AεR1 ,...,εn on D. Moreover, D is a core for r ∈ R, Xi , Xi∗ , i = 1, . . . , n. Proof. We first prove that D is invariant with respect to Ui , Ui∗ , i ∈ Nn . Indeed, since Ui E|r| (+) = E|σi (r)| (+)Ui for any + ∈ B(R), (Ui |r|ϕ, ψ) = (Ui ϕ, |σi (r)|ψ) for any ϕ ∈ D(|r|), ψ ∈ D(|σi (r)|), which gives Ui ϕ ∈ D(|σi (r)|) and Ui |r|ϕ = |σi (r)|Ui ϕ for ϕ ∈ D(|r|). From this and relations (6)–(7) it follows that Ui rϕ =

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σi (r)Ui ϕ for any ϕ ∈ D(r). By condition (1), the family {r | r ∈ R}  defines a closed representation π of R on D and hence D = r∈R D(π(r)) = r∈R D(r), the last equality is true due to condition (3). From this we have Ui D ⊆ D. The same holds for Ui∗ , i ∈ Nn . Since Xi∗ Xi = εi ti , εi ti  0. By condition (2), Eεi ti (+)D(r) ⊆ D(r) and Eεi ti (+)rϕ = rEεi ti (+)ϕ for any ϕ ∈ D(r), + ∈ B(R). In the same manner we get (εi ti )1/2 D ⊆ D, and hence Xi D ⊆ D. Analogously we obtain that D is invariant with respect to Xi∗ . From this we can conclude that r ∈ R, Xi , Xi∗ , i ∈ Nn , determine a representation π of AεR1 ,...,εn on the domain D. Since D is a (εi σi (ti ))1/2 and hence for Xi , Xi∗ . Finally core for ti , σi (ti ), D is a core for (εi ti )1/2 , π is closed since π |R is closed and D = r∈R D(r) is invariant with respect to Xi , ✷ Xi∗ , i = 1, . . . , n. In the rest of the paper we will consider only representations satisfying (1)– (5). Denote by Rπ the von Neumann algebra generated by E|r| (+) and Ur , where r ∈ R and + ∈ B(R). We will need the following auxiliary lemma. LEMMA 1. Let π be a representation of AεR1 ,...,εn as defined above. Then Rπ = π(R)s ∩ (π(R)s )∗ . Proof. By the definition, the restriction of π to R is closed and D is a core for all operators r. Hence, π(R)s ∩ (π(R)s )∗ = {c ∈ L(H ) | cr ⊆ rc, c∗ r ⊆ rc∗ , r ∈ R} (see [S1, Proposition 7.2.10]). We denote this set by π(R)ss . Let c ∈ π(R)ss . Then cD(r) ⊆ D(r) and c∗ D(r) ⊆ D(r) for every r ∈ R. Given ϕ ∈ D(r ∗ r), ψ ∈ D(r), we obtain (cr ∗ rϕ, ψ) = (r ∗ rϕ, c∗ ψ) = (rϕ, rc∗ ψ) = (rϕ, c∗ rψ) = (crϕ, rψ) = (rcϕ, rψ), which implies rcϕ ∈ D(r ∗ ) and cr ∗ rϕ = r ∗ rcϕ, ϕ ∈ D(r ∗ r). From this [c, E|r|2 (+)] = 0 and [c, E|r| (+)] = 0 for any + ∈ B(R). Using spectral properties of selfadjoint operators one obtains cD(|r|) ⊆ D(|r|) and c|r|ϕ = |r|cϕ for any ϕ ∈ D(|r|). Given ϕ ∈ D(r) it follows that (cUr |r|ϕ, ψ) = (Ur |r|cϕ, ψ) = (|r|cϕ, Ur∗ ψ) = (c|r|ϕ, Ur∗ ψ) = (|r|ϕ, c∗ Ur∗ ψ). On the other hand, (cUr |r|ϕ, ψ) = (|r|ϕ, Ur∗ c∗ ψ). Therefore, cUr ϕ = Ur cϕ for any ϕ from the image R(|r|) of the operator |r|. This implies cUr P = Ur cP where P is the projection onto the space (ker |r|)⊥ . Since P = E|r| (R \ {0}) = Ur∗ Ur , Ur P = Ur and c, P commute. This gives cUr = Ur c and consequently c ∈ Rπ .

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Let c ∈ Rπ . One has c|r| ⊆ |r|c and c∗ |r| ⊆ |r|c∗ which implies cr = cUr |r| = Ur c|r| ⊆ Ur |r|c = rc. By the same arguments we obtain c∗ r ⊆ rc∗ . ✷ This easily forces Rπ = π(R)ss . A representation π of an algebra A on a Hilbert space H is said to be irreducible if there is no nontrivial closed subspace K of H such that every operator π(a), a ∈ A can be written as a direct sum π(a) = a1 ⊕ a2 where a1 and a2 are linear operators on K and H ∩ K ⊥ respectively. A nontrivial closed subspace K ⊆ H possessing the above properties is called reducing for π . LEMMA 2. Let π be a ∗-representation of AεR1 ,...,εn by unbounded operators r ∈ R, Xi , Xi∗ , i = 1, . . . , n defined on a domain D of a Hilbert space H . For each closed linear subspace K of H the following conditions are equivalent: (1) K is reducing for π ; (2) The projection PK onto K belongs to the set Tπ =

Rπ

n  ∩ (Ui )s . i=1

Proof. By [S1, Lemma 8.3.3] K is reducing for π if and only if PK ∈ π(AεR1 ,...,εn )s . To obtain the statement it is sufficient to prove that {c ∈ Tπ | c = c∗ } = {c ∈ π(AεR1,...,εn )s | c = c∗ }. Let c = c∗ ∈ π(AεR1,...,εn )s . It follows that cD ⊆ D and c ∈ π(R)s ∩ (π(R)s )∗ = Rπ . Hence the operators c and ti (respectively c and (εi ti )1/2 ) strongly commute, i.e. their spectral projections commute. Applying the same argument as in Lemma 1 we get cUi = Ui c and, finally, c ∈ Tπ . Let c = c∗ ∈ Tπ . Thus c ∈ π(R)s and hence cD ⊆ D. Since Ui D ⊆ D, Ui∗ D ⊆ D and (εti )1/2D ⊆ D (see the proof of Proposition 3), we get cXi ϕ = cUi (εti )1/2 ϕ = Ui c(εti )1/2ϕ = Ui (εti )1/2 cϕ = Xi cϕ ∈ D. This implies c ∈ ✷ π(AεR1 ,...,εn )s . COROLLARY 2. A representation π of AεR1 ,...,εn is irreducible if and only if the family {x, Ui , Ui∗ | x ∈ Rπ , i = 1, . . . , n} is irreducible. Now we will make the following assumption. Let Rˆ be the set of equivalence classes of irreducible representations of R. As before, the set {σi }i∈Nn defines the ˆ We will assume that there exist a Borel set . and an injective map maps σi on R. ϕ: . → Rˆ such that ϕ(.) is invariant with respect to σi , i = 1, . . . , n and any representation of AεR1 ,...,εn restricted to R can be decomposed into a direct integral of primary representations of R which are multiples of irreducible representations

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from ϕ(.). Namely, given a separable Hilbert space H and a ∗-representation π of AεR1 ,...,εn with a domain D(π ), there exist a standard measure µ on ., µ-measurable fields ξ → H (ξ ), ξ → K(ξ ) of Hilbert spaces on ., a µ-measurable field ξ → π  (ξ ) of nontrivial irreducible representations on D(π  (ξ )) ⊆ H (ξ ) such ⊕ that π  (ξ ) ∈ ϕ(ξ ) and an isomorphism U of H onto . H (ξ ) ⊗ K(ξ ) dµ(ξ ) that  transforms the representation π into π such that  ⊕ D(π  (ξ )) ⊗ K(ξ ) dµ(ξ ) D(π  ) = .

and π  (r) =



⊕

π  (ξ )(r) ⊗ Iξ dµ(ξ ),

.

where Iξ is the identity operator on K(ξ ). LEMMA 3. Under the above assumptions we have  ⊕  π (r) = π  (ξ )(r) ⊗ Iξ dµ(ξ ). .

The operator U transforms the center Z of Rπ into the algebra of diagonalizable operators with respect to the decomposition. Moreover, if  ⊕ Rπ  (ξ ) ⊗ Iξ dµ(ξ ) Rπ  := U −1 Rπ U = .

then Rπ  (ξ ) = Rπ  (ξ ) almost everywhere with respect to µ. Proof. It follows from general properties of decomposable operators (see, for example, [S1, Section 12]). ✷ In a natural way we can define maps σi : . → ., i = 1, . . . , n. We will assume that these maps are Borel. The next theorem is an unbounded analogue of Theorem 2. ⊕ THEOREM 5. Let Rπ  = . Rπ  (ξ ) ⊗ Iξ dµ(ξ ) be a direct integral of von Neu⊕ mann algebras on . H˜ (ξ ) dµ(ξ ), H˜ (ξ ) = H (ξ ) ⊗ K(ξ ) from Lemma 3 and Ui be the phase of the operator π  (Xi ) (i = 1, . . . , n). Let +1i = {ξ ∈ . | π  (ξ )(ti ) = 0},

+2i = {ξ ∈ . | π  (ξ )(σi (ti )) = 0}.

Then (xf )(ξ ) = x(ξ )f (ξ ), x ∈ Rπ  ,   dσi (µ)  (σi (ξ ))f (σi (ξ )), Ui (σi (ξ )) (Ui f )(ξ ) = dµ  0, otherwise

ξ ∈ +2i ,

(16)

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almost everywhere with respect to µ. Here the measure χ+1i (ξ ) dσi (µ)(ξ ) is absolutely continuous with respect to dµ(ξ ), ξ → Ui (ξ ) is a measurable field of operators on H˜ (ξ ) into H˜ (σi−1 (ξ )) which are unitary µ-almost everywhere on +1i and Ui (ξ )x(ξ )U∗i (ξ ) = σi (x)(σi−1 (ξ )), Uj (σi−1 (ξ ))Ui (ξ ) = Ui (σj−1 (ξ ))Uj (ξ )

(17)

almost everywhere on +1i ∪ +1j with respect to µ. Here σi (x) is defined by σi (x) = Ui xUi∗ , x ∈ Rπ . Proof. Is similar to that of Theorem 2. ✷ To describe the structure of unbounded representations of AεR1 ,...,εn , we use the following two models which coordinate with the ones given in the previous section and have some corrections connected with unboundedness of representations. Let λ ∈ Rˆ be an equivalence class of a closed irreducible representation π(λ) of R defined on a dense domain D(λ) ⊆ H (λ) such that π(λ)(r), r ∈ R, and D(λ) satisfy conditions (1), (2), (3). We retain the notations K λ , K˜ λ , Pλ from the previous section.  λ (Mun 1 ): Assume that K is trivial and Pλ = ∅. Put H = g∈P˜λ H (gλ), where  H (gλ) = H (λ) for any g ∈ P˜λ , and D = g∈P˜λ D(gλ), where D(gλ) = D(λ). We define (rf )(gλ) = π(λ)(σ1i1 . . . σnin (r))f (gλ),

f ∈D

for g = (i1 , . . . , in ) ∈ Zn , r = r|D and partial isometries Ui , i = 1, . . . , n as follows:  0, if π(gλ)(σi (ti )) = 0, (Ui f )(gλ) = f (gi gλ), otherwise, The operators Xi are defined by Xi = Ui (εi ti )1/2 .  λ ˜ (Mun g∈P˜λ H (gλ), where 2 ): Assume that K is nontrivial, Pλ = ∅. Put H = H˜ (gλ) = H (gλ) ⊗ K(gλ) and H (gλ), K(gλ) are separable Hilbert spaces such that K(gλ) = K(λ), H (gλ) = H (λ) for any g ∈ K˜ λ . We define D = g∈P˜λ D(gλ) ⊗Igλ, where D(gλ) = D(λ) and Igλ is the identity operator in K(gλ). The closed operators r are defined by requiring (r(f ⊗ h))(gλ) = π(λ)(σ1i1 . . . σnin (r))f (gλ) ⊗ h(gλ),

f ⊗h∈D

for g = (i1 , . . . , in ) ∈ Zn , and r = r|D . Partial isometries Ui , i = 1, . . . , n are defined by imposing the following conditions: Ui H˜ (gλ) ⊂ H˜ (gi−1 gλ),

Ui∗ H˜ (gλ) ⊂ H˜ (gi gλ)

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and Uiδkk . . . Uiδ11 |H (λ)⊗K(λ) = 0 if there exists 1
s
k such that either π(g s λ)(ti ) = 0 and δs = 1, or −δ 1 (by Uiδi , where π(gi g s λ)(ti ) = 0 and δs = −1, where g s = gis−1s−1 . . . gi−δ 1 ∗ δi = −1, we mean Ui ), otherwise (Uiδkk . . . Uiδ11 (f ⊗ h))(gλ)  f (λ) ⊗ h(λ), = k k 1 1 . . . gi−δ )f (λ) ⊗ S(gi−δ . . . gi−δ )h(λ), W (gi−δ k k 1 1

g∈ / K˜ λ , g ∈ K˜ λ .

k 1 . . . gi−δ ) is a unitary operator such that W (g)D(λ) ⊆ D(λ) and Here W (gi−δ k 1

W (g)π(λ)(r)W −1 (g)ϕ = π(λ)(σiδkk . . . σiδ11 (r))ϕ,

r ∈ R, ϕ ∈ D(λ). (18)

S(·) is a unitary irreducible projective representation of K˜ λ on the space K(λ) with a multiple c(k1 , k2 ) = W (k1 )W (k2 )W −1 (k1 k2 ), i.e. S(k1 k2 ) = c(k1 , k2 )S(k1 )S(k2 ) for any k1 , k2 ∈ K˜ λ . THEOREM 6. Assume that there exists a Borel set N ⊂ . such that N is invariant with respect to σi , i = 1, . . . , n, the dynamical system (N, (σi )ni=1 ) is simple and for any representation of AεR1 ,...,εn the corresponding measure µ is based essentially on N (i.e. supp µ ⊂ N). Then any irreducible representation is unitarily equivalent to one given in the models Mun i , i = 1, 2. Proof. Let π be an irreducible representation of AεR1 ,...,εn . Then by Corollary 2, the family {x, Ui , Ui∗ | x ∈ Rπ , i = 1, . . . , n} is irreducible. As in the proof of Theorem 3 it follows that any irreducible family {x, Ui , Ui∗ | x ∈ Rπ , i = 1, . . . , n} is unitarily equivalent to one given in the models Mun i with Rπ and Rπ (λ) instead of π(R) and π(λ)(R) respectively. The statement of the theorem ✷ now follows from the fact that Rπ (λ) = Rπ(λ) for any irreducible π(λ).

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Note added in proofs: Recently, some generalizations of the results presented in this paper were obtained in: Mazorchuk, V. and Turowska, L.: Involutions on generalized Weyl algebras preserving the principal grading, Rep. Math. Phys. 48 (2001), 343–351. Mazorchuk, V., Ponomarenko, M. and Turowska, L.: Some associative algebras related to U (g) and twisted generalized Weyl algebras, Preprint 00-063, Bielefeld University, 2000, to appear in Math. Scand.