A Generalization of Mackey's Theory of Induced Representations

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1 Introduction

In this work we present a generalization of Mackey’s theory of induced unitary representations. The generalization lies in extension of Mackey’s theory on a wide class of Krein isometric representations of a separable topological group in Krein spaces, which are induced by Krein-unitary representations of the closed subgroup. Krein space is understood as the ordinary Hilbert space endowed 2 with a unitary involutive operator J: J∗ = J, J = 1. Operator acting in the Krein space is called Krein-isometric if it is defined on a dense domain D, and is closable and invertible on this domain and preserves the Krein-inner product (J , ), where ( , ) is the ordinary Hilbert space inner product (positive definite). Representation· · · · is called Krein-isometric in case there exists a common dense domain D for all representors of the representation, and all representors are closable Krein-isomeric operators on the common domain. Krein isometric representation of a separable topological group is called Krein-unitary if all representors are (separately) bounded, with almost uniform bound (with resect to the Hilbert space norm and topology of the group). The point is that the Krein-isometric representations we encounter in practice are induced by Krein- unitary representations of the subgroups. Perhaps the most important examples include all representaions acting in the single particle Hilbert spaces as well as in the Fock spaces of the fundamental free quantum fields, such as the free electromagnetic potential field (in the local Lorentz gauge) or the gauge mass less free fields underlying the Standard Model (in the BRST approach). We give the formulation and proof of the subgroup theorem and of the Kronecker product theorem for the induced Krein-isometric representations in the Krein space. We have been trying, whenever possible, to preserve the original Mackey’s proof of the said theorems, but because our representations are defined through unbounded operators (say representors), the proof cannot be exactly the same as that presented by Mackey. Presented extension of his theory allows us to reduce the problem of decomposition of the tensor product of induced Krein-isometric representations to an essentially geometric problem concerned with the coset and double coset spaces and to the Fubini theorem, at least for the wide range of topological groups which allow decomposition into product spaces of the subgroup and the coset and double coset spaces. In case of the representations we encouner in Quantum Field Theory (QFT),

2 where the groups are Lie groups, the common domain D has the reach linear- topological structure of the so-called standard nuclear space. In this case we have much more effective tools at our disposal, and the achieved decompositions can be given the meaining of generalized spectral decompositions in the sense of Neumark-Lanze, with respect to the generalized spectral measures. The same can be said of differential operators associated with pseudo-riemannian structures with non trivial symmetry groups where the presented generalization can also be applied to the spectral analysis of these operators. Although theoretical physicists believe that such an extension of Mackey theory should exist, it has never been worked out explicitly. In particular besides the claims (which we give in the furher part of this work) which conjecture existence of such an extension, it was also stressed the lack of rigorous construction of the most important free fields which would be based on the representation theory, compare e. g. [21]. Our work allows, among other things, to complete this lack [55]. Of course Quantum Field Theory is not the only domain of potential applicability of the presented extension of Mackey’s theory, but any analytic situation involving preservation of a not necessary positive definite inner product with non trivial symmetry group, will be closely related to our work.

2 Preliminaries

It should be stressed that the analysis we give here is inapplicable for general linear spaces with indefinite inner product. We are concerned with non-degenerate, decomposable and complete inner product spaces in the terminology of [2], which have been called Krein spaces in [9], [52], [2] and [54] for the reasons we explain below. They emerged naturally in solving physical problems concerned with quantum mechanics ([7], [45]) and quantum field theory ([20], [1]) in quantization of electromagnetic field and turned up generally to be very important (and even seem indispensable) in construction of quantum fields with non-trivial gauge freedom. Similarly we have to emphasize that we are not dealing with general unitary (i. e. preserving the indefinite inner product in Krein space) representations of the double cover G = T4sSL(2, C) of the Poincarégroup, but only with the exceptional representations ofLopuszański-typ e, which naturally emerge in construction of the free photon field, which have a rather exceptional structure of induced representations, and allow non-trivial analytic constructions of tensoring and decomposing, which is truly exceptional among Krein- unitary (preserving the indefinite product) representations in Krein spaces. The non-degenerate, decomposable and complete indefinite inner product space , hereafter called Krein space, may equivalently be described as an or- dinaryH Hilbert space with an ordinary strictly positive inner product ( , ), together with a distinguishedH self-adjoint (in the ordinary Hilbert space sense)· · fundamental symmetry (Gupta-Bleuler operator) J = P+ P , where P+ and P are ordinary self-adjoint (with respect to the Hilbert− space− inner product − ( , ))) projections such that their sum is the identity operator: P+ + P = I. · · −

3 The indefinite inner product is given by ( , )J = (J , ) = ( , J ). Recall that in our previous paper [54] the indefinite product· · was· designated· · · by ( , ) and the ordinary Hilbert space inner product associated with the fundamen·tal· symmetry J was designated by ( , )J. The indefinite and the associated definite inner product play symmetric roles· · in the sense that one may start with a fixed indefinite inner product in the Krein space and construct the Hilbert space associated with an admissible fundamental symmetry, or vice versa: one can start with a fixed Hilbert space and for every fundamental symmetry construct the indefinite inner product in it, both approaches are completely equivalent provided the fundamental symmetry being admissible (in the sense of [52]) and fixed. We hope the slight change of notation will not cause any serious misunderstandings and is introduced because our analytical arguments will be based on the ordinary Hilbert space properties, so will frequently refer to the standard literature on the subject, so we designated the ordinary strictly definite inner product by ( , ) which is customary. · · Let an operator A in be given. The operator A† in is called Krein- adjoint of the operator A inH in case it is adjoint in the senseH of the indefinite H inner product: (Ax, y)J = (JAx, y) = (Jx, A†y) = (x, A†y)J for all x, y , ∈ H or equivalently A† = JA∗J, where A∗ is the ordinary adjoint operator with 1 respect to the definite inner product ( , ). The operator U and its inverse U − · · isometric with respect to the indefinite product ( , )J,e. g. (Ux,Uy)J = (x, y)J 1 · · for all x, y (same for U − ), equivalently UU † = U †U = I, will also be called unitary∈ H (sometimes J-unitary or Krein-unitary) trusting to the context or explanatory remarks to make clear what is meant in each instance: unitarity for the indefinite inner product or the ordinary unitarity for the strictly definite Hilbert space inner product. In particular we may consider J-symmetric representations x A of in- 7→ x volutive algebras, i. e. such that x∗ Ax†, where ( ) ( )∗ is the involu- tion in the algebra in question. A fundamental7→ role for· 7→ the· spectral analysis in Krein spaces is likewise played by commutative (Krein) self-adjoint, or J- symmetric weakly closed subalgebras. However their structure is far from being completely described, with the exception of the special case when the rank of P+ or P is finite dimensional (here the analysis is complete and was done by Neumark).− Even in this particular case a unitary representation of a separable locally compact group in the Krein space, although reducible, may not in general be decomposable, compare [37, 38, 39, 40]. In case the dimension of the rank + = P+ or = P of P+ or P is finite we get the spaces analysed by Pontrjagin,H H KreinH− and Neumark−H , compare− e. g. [46], [25] and the literature in [2]. The circumstance that the Krein space may be defined as an ordinary Hilbert space with a distinguished non-degenerate fundamental symmetry (or Gupta- H 2 Bleuler operator) J = J∗, J = I in it , say a pair ( , J), allows us to extend the fundamental analytical constructions on a wide classH of induced Krein-isometric representations of G = T4sSL(2, C) in Krein spaces. In particular we may define a Krein-isometric representation of T4sSL(2, C) induced by a Krein-unitary representation of a subgrup H corresponding to a particular class of SL2, C)-

4 orbits on the dual group T4 of T4 (in our case we consider the class corresponding to the representation with the spectrum of the four-momenta concentrated on the “light cone”) word forc word as in the ordinary Hilbert space by replacing the representation of the subgroup H by a Krein-unitary representation L in L a Krein space ( L, JL). This leads to a Krein-isometric representation U in a Krein space (H L, JL) (see Sect. 3). Application of Lemma 7, Section 5, leads to the ordinaryH direct integral = dµ (q) of Hilbert spaces H G/H Hq G/H q = L over the coset measure space G/H with the measure induced by the H H R 1 L Haar measure on G. One obtains in this manner the Krein space ( ,U − J U) given by the ordinary Hilbert space equal to the above mentionedH direct in- H tegral of the ordinary Hilbert spaces q all of them equal to L together with 1 HL H the fundamental symmetry J = U − J U equal to the ordinary direct integral

G/H Jq dµG/H (q) of fundamental symmetries Jq = JL as operators in q = L 1 L H H andR with the representation U − U U of G in the Krein space ( , J) (and U given by a completely analogous formula as that in Lemma 7 ofH Section 5) of Wigner’s form [59] (imprimitivity system). This is the case for the indecomposable (although reducible) representation of G = T4sSL(2, C) constructed byLopuszański with H = T4 Gχ, with the ˜ C · ”small” subgroup Gχ ∼= E2 of SL(2, ) corresponding to the “light-cone” orbit in the spectrum of four-momenta operators. One may give to it the form 1 L L of representation U − U U equivalent to an induced representation U , because the representors of the normal factor (that is of the translation subgroup T4) of the semidirect product T4sSL(2, C) as well as their generators, i. e. four-momentum operators P0,...,P3, commute with the fundamental symmetry J = G/H Jq dµG/H (q), so that all of them are not only J-unitary but unitary with respect to the ordinary Hilbert space inner product (and their R generators P0,...,P3 are not only Krein-self-adjoint but also self-adjoint in the ordinary sense with respect to the ordinary definite inner product of the Hilbert space ), so the algebra generated by P0,...,P3 leads to the ordinary direct integralH decomposition with the decomposition corresponding to the ordinary spectral measure, contrary to what happens for general Krein-selfadjoint commuting operators in Krein space ( , J) (for details see Sect. 5). This in case H 1 L of G = T4sSL(2, C), gives to the representation U − U U of G the form of Wigner [59] (viz. a system of imprimitivity in mathematicians’ parlance) with the only difference that L is not unitary but Krein-unitary in ( L, JL). Another gain we have thanks to the above mentioned circumstanceH is that we can construct tensor product ( , J ) ( , J ) of Krein spaces ( , J ) and H1 1 ⊗ H2 2 H1 1 ( 2, J2) as( 1 2, J1 J2) where in the last expression we have the ordinary tensorH productsH ⊗ of H Hilbert⊗ spaces and operators in Hilbert spaces (compare Sect. 6). L Similarly having any two such (J1- and J2-)isometric representations U and M U induced by (JL and JM -unitary) representations L and M of subgroups G1 and G2 in Krein spaces ( L, JL) and ( M , JM ) respectively we may construct the tensor product U L HU M of Krein-isometricH representations in the tensor product Krein space ( ⊗, J ) ( , J ), which is likewise (J J -)isometric. It H1 1 ⊗ H2 2 1 ⊗ 2

5 L M L M turns out that the Kronecker product U U and U × are (Krein-)unitary equivalent (see Sect. 6) as representations× of G G. Because the tensor product U L U M as a representation of G is the restriction× of the Kronecker product U L ⊗ U M to the diagonal subgroup of G G we may analyse the representa- × L M × L M tion U U by analysing the restriction of the induced representation U × to the diagonal⊗ subgroup exactly as in the Mackey theory of induced representations in Hilbert spaces. Although in general for J J -unitary repre- 1 ⊗ 2 sentations in Krein space ( 1, J1) ( 2, J2) ordinary decomposability breaks H ⊗ H L M down, we can nonetheless still decompose the representation U × restricted to the diagonal into induced representations which, by the above mentioned Krein-unitary equivalence, gives us a decomposition of the tensor product representation U L U M of G. Indeed, it turns out that the whole argument of Mackey [31] preserves⊗ its validity and effectiveness in the construction of decomposition of tensor product of induced representations for the case in which the representations L and M of the subgrups G1 and G2 are replaced with (specific) unitary (or J - and J -unitary) representations in Krein spaces ( , J ) L M HL L and ( M , JM ) respectively. We give details on the subject below in Section 9. BecauseH theLopuszański representation is (Krein-unitary equivalent to) an induced representation in a Krein space (Sect. 5), we can decompose the tensor product ofLopuszański representations. The specific property of the group T4sSL(2, C) is that this decomposition may be performed explicitly into indecomposable sub-representations. The Krein-isometric induced representations of T4sSL(2, C) which we de- scribe here cover all representations important for QFT. All the representations which act on single particle states of local fields (including zero mass gauge fields) have three important properties: 1) They are strongly continuous on a common dense invariant subdomain. 2) Translations commute with the fundamental symmetry J, so that translations are unitary with respect to the Hilbert space inner product as well as are Krein-unitary, and thus compose ordinary (strongly continuous) unitary representation of the translation subgroup. 3) The representations are “locally” bounded with respect to the joint spectrum of translation generators. Let us formulate the requirement of “local boundedness” more precisey. Let us denote the translation representor U(a,1) just by T (a) and the representor U(0,α) of the SL(2, C) subgroup just by U(α). Let P 0,...,P 3 be the respective generators of the translations (they do exist by the strong continuity assumption posed on the Krein-isometric representation – physicist’s everyday computations involve the generators and thus our assumption is justified). Let be the commutative C∗-algebra generated by the functions f(P 0,...,P 3) of translationC generators P 0,...,P 3, where f is continuous on R4 and vanishes at infinity. Let

= dµ(p) (1) H Hp Spec(PZ0,...,P 3) be the direct integral decomposition of corresponding to the algebra (in the sense of [36] or [50]) with a spectralH measure µ on the joint spectrumC

6 Spec (P 0,...,P 3) of the translation generators. The requirement of “local boundedness” of the Krein-isometric representations of the semi-direct products T4sSL(2, C) which are preserved by the representations acting in the single particle and inFock spaces of free ﬁelds (of realistic QFT) is the following. The restriction U(α), α SL(2, C) to the second factor SL(2, C) is locally bounded with respect to∈ the direct integral decomposition (1) of the Hilbert space , determined by the restriction T (a), a G , of the representation of H ∈ 1 T4sSL(2, C) to the abelian normal factor T4. More precisely: let be the ordinary Hilbert space norm, then for every compact subset ∆k·k of the dual H G1 and every α G2 there exists a positive constant c∆,α (possibly depending on ∆ and α) such∈ that c U(α)f

3 Deﬁnition of the induced representation U L in Krein space ( L, JL) H Here by a Krein-unitary and strongly continuous representation L : G x Lx of a separable locally compact group G we shall mean a homomorphism∋ 7→ of G

7 into the group of all (Krein-)unitary transformations of some separable Krein space ( , J ) (i. e. with separable Hilbert space ) onto itself which is: HL L HL

(a) Strongly continuous: for each υ L the function x Lxυ is continuous with respect to the ordinary strictly∈ H deﬁnite Hilbert7→ space norm υ = (υ,υ) in . k k HL (b) Almostp uniformly bounded: there exist a compact neighbourhood V of unity e G such that the set L , x V G is bounded or, what is ∈ k xk ∈ ⊂ the same thing, that the set Lx with x ranging over a compact set K is bounded for every compact subsetk k K of G.

Because the strong operator topology in ( L) is stronger than the weak operator topology then for each υ, ϕ theB functionH x (L υ, ϕ) is continuous ∈ HL 7→ x on G. One point has to be noted: because the range and domain of each Lx equals , which as a Krein space ( , J ) is closed and non-degenerate, then HL HU U by Theorem 3.10 of [2] each Lx is continuous i. e. bounded with respect to the Hilbert space norm in L, and each Lx indeed belongs to the algebra ( ) of bounded operatorsk·k H in the Hilbert space (which is non-trivial B HL HL as an JL-isometric densely defined operator in the Krein space ( L, JL) may be discontinuous, as we will see in this Section, compare also [2]).H We also could immediately refer to a theorem which says that Krein-unitary operator is continuous, i. e. Hilbert-space-norm bounded (compare Theorem 4.1 in [2]). Besides in this paper will be considered a very specific class of Krein-isometric representations U of G in Krein spaces, to which the induced representations of G in Krein spaces, hereby defined, belong. Namely here by a Krein-isometric and strongly continuous representation of a separable locally compact group G we shall mean a homomorphism U : G x Ux of G into a group of Krein- isometric and closable operators of some separable∋ 7→ Krein space ( , J) with dense common domain D equal to their common range in and suchH that H U is strongly continuous on the common domain D: for each f D ∈ ⊂ H the function x Uxf is continuous with respect to the ordinary strictly definite Hilbert7→ space norm f = (f,f) in . k k H Let H be a closed subgroup of a separablep locally compact group G. In the applications we have in view the right H-cosets, i. e. elements of G/H, which are exceptionally regular, and have a “measure product property”. This is e. g. the case we encounter in decomposing tensor products of the representations of the double cover G of the Poincarégroup in Krein spaces encountered in QFT. Namely every element (with a possible exception of a subset of G of Haar measure zero) g G can be uniquely represented as a product g = h q, where ∈ · h H and q Q ∼= G/H with a subset Q of G which is not only measurable but,∈ outside a∈ null set, is a sub-manifold of G, such that G is the product H G/H measure space, with the regular Baire measure space structure on G/H× associated to the canonical locally compact topology on G/H induced by the natural projection π : G G/H and with the ordinary right Haar 7→

8 measure space structure (H, RH ,µH ) on H, which is known to be regular with the ring RH of Baire sets. (We will need the complete measure spaces on G, G/H but the Baire measures are sufficient to generate them by the Carathéodory method, because we have assumed the topology on G to fulfil the second axiom of countability.) In short (G, RG,µ) = (H G/H, RH Q,µH µG/H ). In our applications we are dealing with pairs H × G of Lie subgroups× × of the double ⊂ cover T4sSL(2, C) of the Poincarégroup P including the group T4sSL(2, C) itself, with a sub-manifold structure of H and Q ∼= G/H. This opportunities allow us to reduce the analysis of the induced representation U L in the Krein space defined in this Section to an application of the Fubini theorem and to the von Neumann analysis of the direct integral of ordinary Hilbert spaces. (The same assumption together with its analogue for the double cosets in G simplifies also the problem of decomposition of tensor products of induced representations of G and reduces it mostly to an application of the Fubini theorem and harmonic analysis on the ”small” subgroups: namely at the initial stage we reduce the problem to the geometry of right cosets and double cosets with the observation that Mackey’s theorem on Kronecker product and subgroup theorem of induced representations likewise work for the induced representations in Krein spaces defined here, and then apply the Fubini theorem and harmonic analysis on the ”small” subgroups.). Driving by the physical examples we assume for a while that the “measure product property” is fulfilled by the right H-cosets in G. (We abandon soon this assumption so that our results, namely the subgroup theorem and the Kronecker product theorem, hold true for induced representations in Krein spaces, without this assumption.) Let L be any (JL-)unitary strongly continuous and almost uniformly bounded representation of H in a Krein space ( L, JL). Let µH and µG/H be (quasi) invariant measures on H and on the homogeneousH space G/H of right H-cosets in G induced by the (right) Haar measure µ on G by the “unique factorization”. L Let us denote by the set of all functions f : G x fx from G to L such that H ∋ 7→ H (i) (f ,υ) is measurable function of x G for all υ . x ∈ ∈ HL (ii) f = L (f ) for all h H and x G. hx h x ∈ ∈ (iii) Into the linear space of functions f fulfilling (i) and (ii) let us introduce L L the operator J by the formula (J f)x = LhJLLh−1 (fx), where x = h q is the unique decomposition of x G. Besides (i) and (ii) we require · ∈ ( J ((JLf) ),f ) dµ < , L x x G/H ∞ Z where the meaning of the integral is to be found in the fact that the integrand is constant on the right H-cosets and hence defines a function on the coset space G/H.

(Note that L is written here as superscript! The L with the lower case of the index L is reserved for the space of the representationH L of the subgroup H G.) ⊂

9 Because every x G has a unique factorization x = h q with h H and ∈ · L ∈ q Q ∼= G/H, then by “unique factorization” the functions f as well as the∈ functions x (f ,υ) with υ , on G, may be treated∈ as H functions on 7→ x ∈ HL the Cartesian product H Q = H G/H = G. The axiom (i) means that the × ∼ × ∼ functions (h, q) (fh q,υ) for υ L are measurable on the product measure 7→ R · ∈ H R space (H G/H, H G/H ,µH µG/H ) ∼= (H Q, H Q,µH µG/H ). In particular× let W : q × W × be a function× on Q such× that×q (W ,υ) 7→ q ∈ HL 7→ q is measurable with respect to the standard measure space (Q, RQ, dq) for all υ L, and such that (Wq, Wq) dµG/H (q) < . Then by the analysis of∈ [36] H (compare also [41], 26.5) which is by now∞ standard, the set of such functions W (when functionsR § equal almost everywhere are identiﬁed) compose the direct integral L dµG/H (q) Hilbert space with the inner product (W, F ) = (W , F ) dµ H(q). For every such W dµ (q) the func- q q GR/H ∈ HL G/H tion (h, q) fh q = LhWq fulﬁls (i) and (ii). (ii) is trivial. For each υ L the R7→ · R ∈ H function (h, q) (fh q,υ) = (LhWq,υ) is measurable on the product measure 7→ · space (H Q, RH Q,µH µG/H ) = (G, RG,µ) because for any orthonormal × × × ∼ basis en n N of the Hilbert space L we have: { } ∈ H

(fh q,υ) = (fh q, JLJLυ) = (LhWq, JLJLυ) = (JLLhWq, JLυ) · · = (JLWq,Lh−1 JLυ)= (JLWq,en)(en,Lh−1 JLυ) n N X∈ where each summand gives a measurable function (h, q) (J W ,e )(e ,L J υ) 7→ L q n n h L on the product measure space (H Q, RH Q,µH µG/H ) by Scholium 3.9 of × × × [51]. On the other hand for every function (h, q) (fh q,υ) measurable on the 7→ · product measure space the restricted functions q (fh q,υ) and h (fh q,υ), i. e. with one of the arguments h and q ﬁxed, are7→ measurable,· which7→ follows· from the Fubini theorem (compare e. g. [51], Theorem 3.4) and thus q (f ,υ) 7→ q is measurable (i.e. with the argument h ﬁxed and equal e in (h, q) (fh q,υ)). Because a simple computation shows that 7→ ·

L L ( JL((J f)x),fx ) dµG/H = ( JL((J f)h q),fh q ) dµG/H (q) · · Z Z = (fq,fq ) dµG/H (q), Z one can see that when functions equal almost everywhere are identiﬁed L becomes a Hilbert space with the inner product H

L (f,g)= ( JL((J f)x),gx ) dµG/H . (2) Z L (In fact because the values of f are in the fixed Hilbert space L we do not have to tangle into the the whole∈ H machinery of direct integral HilbHert spaces of von Neumann. It suffices to make obvious modifications in the corresponding proof that L2(G/H) is a Hilbert space, compare [41], , 26.5.) §

10 A simple verification shows that JL is a bounded self-adjoint operator in the Hilbert space L with respect to the definite inner product (2) and that (JL)2 = I. ThereforeH ( L, JL) is a Krein space with the indefinite product H

L f,g JL = (J f,g)= ( JL(fx),gx ) dµG/H (3) Z which is meaningful because the integrand is constant on the right H-cosets, i. e. it is a function of q Q ∼= G/H. Let the function [x∈] λ([x],g) on G/H be the Radon-Nikodym derivative 7→ λ( ,g) = d(Rgµ) ( ), where [x] stands for the right H-coset Hx of x G (µ · dµ · ∈ stands for the (quasi) invariant measure µG/H on G/H induced by the assumed “factorization” property from the Haar measure µ on G and Rgµ stands for the right translation of the measure µ: Rgµ(E)= µ(Eg)). For every g G let us consider a densely deﬁned operator U L . Its domain 0 ∈ g0 D(U L ) is equal to the set of all those f L for which the function g0 ∈ H

x f ′ = λ([x],g ) f 7→ x 0 xg0 has ﬁnite Hilbert space norm (i. e. ordinaryp norm with respect to the ordinary deﬁnite inner product (2)) in L: H

L f ′,f ′ = ( JL((J f ′)x),fx′ ) dµG/H Z = ( J L J L −1 λ([x],g ) f , λ([x],g ) f ) dµ (x) < , L h(x) L h(x) 0 xg0 0 xg0 G/H ∞ Z p p 1 where h(x) H is the unique element corresponding to x such that h(x)− x Q; and whenever∈ f D(U L ) we put ∈ ∈ g0 L (Ug0 f)x = λ([x],g0) fxg0 .

U L, after restriction to a suitablep sub-domain, becomes a group homomorphism of G into a group of densely deﬁned JL-isometries of the Krein space ( L, JL). Let us formulate this statement more precisely in a form of a Theo- rem:H

L L THEOREM 1. The operators U , g0 G, are closed and J -isometric with g0 ∈ dense domains D(U L ), dense ranges R(U L ) and dense intersection D(U L )= g0 g0 g0 G g0 ∈ ] R(U L ). U L is equal to the closure of the composition U L U L = U L g G g0 g0k0 g0 Tk g k 0∈ 0 0 0 of the restrictions U L and U L of U L and U L to the domain D(U L ), i. T g0 k0 g0 k0 g0 G g0 g∈ g L e. the map g0 U is a Krein-isometric representation of G. There exists a 7→ gg0 g T dense sub-domain D D(U L ) such that U L D = D, U L is the closure g0 G g0 g0 g0 g⊂ ∈ L L L of the restriction Ug0 ofTUg0 to the sub-domain D, and g0 Ug0 is strongly continuous Krein-isometric representation of G on its domain7→D. g g 11 L L Let us introduce the class C00 of functions h q fh q = LhWq with ⊂ H · 7→ · q W continuous and compact support on Q = G/H. Of course each 7→ q ∈ HL ∼ such function W is an element of the direct integral Hilbert space L dµG/H . One easily veriﬁes that all the conditions of Lemma 4 of (the next)H Sect. 4 are L L L R true for the class C00. Therefore C00 is dense in . Let h q fh q = LhWq L H · 7→ · be an element of C00 and let K be the compact support of the function W . Using the “unique factorization” let us introduce the functions (q,h , q ) 0 0 7→ h′ H and (q,h0, q0) q′ Q = G/H in the following way. Let q,h0,q0 ∈ 7→ q,h0,q0 ∈ ∼ g0 = q0 h0. We deﬁne h′ H and q′ Q G to be the elements, · q,h0,q0 ∈ q,h0 ,q0 ∈ ⊂ uniquely corresponding to (q,h0, q0), such that

q h0 q0 = h′ q′ . (4) · · q,h0 ,q0 · q,h0,q0

′ Finally let cK,g0 = supq K Lh , which is ﬁnite outside a null set, on ∈ q,h0,q0 account of the almost uniform boundedness of the representation L, and because q h′ is continuous outside a µG /H -null set (easily seen by the “measure q,h0 ,q0 product7→ property”, but it holds true even if the “measure product property is not assumed” – compare the comments below in this Section.).

L 2 L L Uh q f = Uh q f,Uh q f k 0· 0 k 0· 0 0· 0 L L L = ( JL((J Uh q f)h q), (Uh q f)h q ) dµG/H (q) 0· 0 · 0· 0 · Z

= ( Lh′ fq′ ,Lh′ fq′ ) dµG/H (qq,h′ ,q ) (5) q,h0,q0 q,h0,q0 q,h0,q0 q,h0 ,q0 0 0 Z 2 cK,g ( fq′ ,fq′ ) dµG/H (qq,h′ ,q ) ≤ 0 q,h0,q0 q,h0 ,q0 0 0 Z 2 2 = c f , g0 = h0 q0 G. K,g0 k k · ∈ L D L G Thus it follows that C00 (Ug0 ) for every g0 . Similarly it is easily L L⊂ ∈ verifiable that C R(U ) whenever the Radon-Nikodym derivative λ([x],g0) 00 ⊂ g0 is continuous in [x]. It follows from definition that for f L being a member D L ∈ HR L of g G (Ug0 ) is equivalent to being a member of g G (Ug0 ). 0∈ 0∈ L L We shall show that U † = U −1 , where T † stands for the adjoint of the T g0 g0 T operator T in the sense of Krein [2], page 121: for any linear operator T with L dense domain D(T ) the vector g belongs to D(T †) if and only if there exists a k L such that ∈ H ∈ H (JLTf,g) = (JLf, k), for all f D(T ), ∈ and in this case we put T †g = k, with the unique k as D(T ) is dense (i. e. same definition as for the ordinary adjoint with the definite Hilbert space inner product ( , ) given by (2) replaced with the indefinite one (JL , ), given by (3)). · · · · D L † L † Now let g be arbitrary in Ug0 , and let Ug0 g = k. The inclusion L L L U † U −1 is equivalent to the equation U −1 g = k. By the definition of g0 ⊂ g0 g0 12 the Krein adjoint of an operator, for any f D U L we have ∈ g0 JL L JL Ug0 f,g = ( f, k), i. e. J U L f ,g dµ (x)= (J f , k ) dµ (x); L g0 x x G/H L x x G/H Z Z L which by the definition of Ug0 and quasi invariance of the measure µG/H means that

1 dµG/H (xg0− ) −1 JL fx, gxg0 dµG/H (x) s dµG/H (x) Z L = (JL fx, kx) dµG (x) for all f D U ; /H ∈ g0 Z i. e. the function u

1 dµG/H (xg0− ) −1 x ux = gxg0 kx 7→ s dµG/H (x) − is JL-orthogonal to all elements of D U L : (JLf,u) = 0 for all f D U L . g0 ∈ g0 Because D U L is dense in L, and JL is unitary with respect to the ordinary g0 H Hilbert space inner product (2) in L it follows that JLD U L is dense in L. H g0 H Therefore u must be zero as a vector orthogonal to JLD U L in the sense of g0 the Hilbert space inner product (2). Thus

1 dµG/H (xg0− ) −1 gxg0 = kx s dµG/H (x) almost everywhere, and because by deﬁnition (k, k) < , we have shown that L ∞ U −1 g = k. g0 L L L Next we show that U † U −1 . Let g be arbitrary in D(U −1 ) and let g0 g0 g0 L ⊃ L L L L U −1 g = k. It must be shown that for any f D U , (J U f,g) = (J f, k). g0 g0 g0 This is the same as showing that ∈ J U L f ,g dµ (x)= (J f , k ) dµ (x), L g0 x x G/H L x x G/H Z Z L which again easily follows from deﬁnition of Ug0 and quasi invariance of the

13 measure dµG/H (x):

L dµG/H (xg0) JL U f ,gx dµG (x)= JL fxg ,gx dµG (x) g0 x /H dµ (x) 0 /H Z Z s G/H 1 1 dµG/H (xg0− g0) dµG/H (xg0− ) J −1 −1 = 1 L fxg0 g0 ,gxg0 dµG/H (x) s dµG/H (xg0− ) dµG/H (x) Z 1 dµG/H (xg0− −1 = JL fx, gxg0 dµG/H (x) s dµG/H (x) Z L = J f , U −1 g dµ (x)= (J f , k ) dµ (x). L x g0 x G/H L x x G/H Z Z L L Thus we have shown that U † = U −1 . g0 g0 L L L L L D − D − − † Because C00 (U 1) then (U 1 ) is dense, thus U 1 , equal to Ug0 , ⊂ g0 g0 g0 is closed by Theorem 2.2 of [2] (Krein adjoint T † is always closed, as it is equal L L J T ∗J with the ordinary adjoint T ∗ operator, and because the fundamental symmetry JL is unitary in the associated Hilbert space L, compare Lemma 2.1 in [2]). H In order to prove the second statement it will be sufficient to show that L L L U † = U −1 because the homomorphism property of the map g0 U g0 g0 7→ g0 D L L restricted to g G (Ug ) is a simple consequence of the definition of Ug0 . But ∈ g L L the proof of the equality U † = U −1 runs exactly the same way as the T g0 g0 L L L † − D proof of the equality Ug0 =U 1 , with the trivial replacement of Ug0 g g0 by D, as it is valid for any dense sub-domain D contained in D U L instead g0 L L L of D U . Then by Theorem 2.5 of [2] it follows that U †† = U −1 † is g0 g0 g0 L L L L equal to the closure U of the operator U . Because U −1 † = U , we get g0 g0 gg0 g0 U L = U L . g0 g0 g g L L L L Byg the above remark we also have Ug0 = Ug0 for any restriction Ug0 of Ug0 L to a dense sub-domain D D(Ug0 ) In order to prove the third⊂ statement, letg us introduce a dense sugb-domain L L C0 C00 of continuous functions with compact support on G/H. Its full ⊂ L L definition and properties are given in the next Section. In particular Ug0 C0 = L C0 whenever the Radon-Nikodym derivative λ([x],g0) is continuous in [x]. For 0 L each element f of C0 we have the inequality shown to be valid in the course of proof of Lemma 1, Sect. 4:

0 0 2 L,V L,V 2 1 f f sup f f 2 sup µH (Kx− H) x1 x2 (h,e) (e,x1) (h,e) (e,x2) k − k ≤ h H · − · x G ∩ ∈ ∈ where f L,V is a function depending on f 0, continuous on the direct product group H G and with compact support K V with V being a compact × H ×

14 L,V neighbourhood of the two points x1 and x2. Because any such function f must be uniformly continuous, the strong continuity of U L on the sub-domain L L L L D C0 follows. Because Ug0 C0 = C0 , the third statement is proved with = L C0 (In case the Radon-Nikodym derivative was not continuous and “measure product property” not satisfied it would be sufficient to use all finite sums U L f 1 + ...U L f n,f k CL as the common sub-domain D instead of CL). g1 gn ∈ 0 0 REMARK 2. By definition of the Krein-adjoint operator and the properties: L L † L 1) Ug D = D, g G, 2) Ug = Ug−1 , g G, it easily follows that for each g G ∈ ∈ ∈ L † L L L † Ug Ug = I and Ug Ug = I (6) on the domain D. We may easily modify the common domain D so as to achieve the additional property: 3) JLD = D together with 1) and 2) and thus with (6). Indeed, to achieve this one may define D to be the linear span of the set

L m1 L L L mn+1 L (J ) U ...U (J ) f : with gk ranging over G, f C , n N and g1 gn ∈ 0 ∈ kn mk over the sequences with omk equal 0 or 1. In case the Radon-Nikodym 7→ L derivative λ is continuous and the “measure product property” fulﬁlled, D = C0 meets all the requirements.

L COROLLARY 1. For every Ug0 there exists a unique unitary (with respect to L the deﬁnite inner product (2)) operator Ug0 in and unique selfadjoint (with H L respect to (2)) positive operator Hg0 , with dense domain D(Ug0 ) and dense range L such that Ug0 = Ug0 Hg0 . Immediate consequence of the von Neumann polar decomposition theorem L and closedness of Ug0 .

Of course the ordinary unitary operators Ug0 of the Corollary do not compose any representation in general as the operators Ug0 and Hg0 of the polar decomposition do not commute if Ug0 is non normal.

L L THEOREM 2. L and JL commute if and only if U and J commute. If L L U and J commute, then L is not only JL-unitary but also unitary in the L ordinary sense for the deﬁnite inner product in the Hilbert space L. If U and JL commute then U L is not only JL-isometric but unitary withH respect to L L the ordinary Hilbert space inner product (2) in , i . e. the operators Ug0 are bounded and unitary with respect to (2). TheH representation L is uniformly bounded if and only if the induced representation U L is Krein-unitary (with each L Ug0 bounded) and uniformly bounded.

Using the functions (q,h0, q0) h′ H and (q,h0, q0) q′ 7→ q,h0,q0 ∈ 7→ q,h0,q0 ∈ Q ∼= G/H defined by (4), one easily verifies that L and JL commute (and thus L is not only JL-unitary but also unitary in the ordinary sense for the definite L L inner product in the Hilbert space L) if and only if U and J commute (i. e. when U L is not only JL-isometricH but unitary with respect to the ordinary Hilbert space norm (2) in L). To this end we utilize the fact that for each H

15 L L ﬁxed x, fx with f ranging over C00 has as their closed linear span. We leave details to the reader. H

COROLLARY 2. If N H G is a normal subgroup of G such that the ⊂ ⊂ restriction of L to N is uniformly bounded (or commutes with JL) then the restriction of U L to the subgroup N is a Krein-unitary representation of the L L subgroup with each Un ,n N bounded uniformly in n (or U restricted to N commutes with JL and is an∈ ordinary unitary representation of N in the Hilbert space L). H In the proof of the strong continuity of U L on the dense domain D we have L L used a specific dense subspace C0 of . In the next Subsection we give its precise definition and provide the remainingH relevant analytic underpinnings which we introduce after Mackey. In the proof of strong continuity we did as in the classical proof of strong continuity of the right regular representation of G in L2(G) or in L2(G/H) (of course with the obvious Radon-Nikodym factor in the latter case), with the necessary modifications required for the Krein space. In our proof of strong continuity on D the strong continuity of the representation L plays a much more profound role in comparison to the original Mackey’s theory. The additional assumption posed on right H-cosets, i. e. “measure product property” is unnecessary. In order to give to this paper a more independent character we point out that the above construction of the induced representation in Krein space is possible without this assumption which may be of use for spectral analysis for (unnecessary elliptic) operators on manifolds uniform for more general semi-direct product Lie groups preserving indefinite pseudo- riemann structures. Namely for any closed subgroup H G (with the “measure product property” unnecessary fulfilled) the right action⊂ of H on G is proper and both G and G/H are metrizable so that a theorem of Federer and Morse [14] can be applied (with the regular Baire (or Borel) Haar measure space structure (G, RG,µ) on G) in proving that there exists a Borel subset B G such that: (a) B intersects each right H-coset in exactly one point and (b)⊂ for each com- 1 pact subset K of G, π− (π(K)) B has a compact closure (compare Lemma 1.1 of [31]). In short B is a “regular∩ Borel section of G with respect to H”. In particular it follows that any g G has unique factorization g = h b, h H,b B. Using the Lemma and extending∈ a technique of A. Weil used· in studying∈ ∈relatively invariant measures Mackey gave in [31] a general construction of quasi invariant measures in G/H (all being equivalent). REMARK. The general construction of quasi invariant (standard) Baire (or Borel) measures on the locally compact homogeneous space G/H was proposed in a somewhat shortened form in 1 of [31], where the technique of A. Weil was adopted and developed into a ρ-§ and λ-functions construction. Today it is known as a standard construction of the quotient of a measure space by a group, detailed exposition can be found e.g. in [4]. Only for sake of completeness let us remind the main Lemmas and Theorem of 1 of [31] (details omitted in §

16 the exposition of [31] are to be found e. g. in [4] with the trivial interchang- ing of left and right). Let Lgµ and Rgµ be the left and right translations of a measure µ on G/H: Lgµ(E)= µ(gE) and Rgµ(E)= µ(Eg). Let µ be the right Haar measure on G. Denoting the the constant Radon-Nikodym derivative of the right Haar measure Lgµ with respect to µ by ∆G(g), and similarly deﬁned constant Radon-Nikodym derivative for the closed subgroup H by ∆H (g) we have the the following Lemmas and Theorems.

LEMMA. Let µ be a non-zero measure on G/H and µ0 = µG be the right Haar measure on G. The following conditions are equivalent: a) µ is quasi invariant with respect to G;

1 b) a set E G/H is of µ-measure zero if and only if π− (E) is of µ0-measure zero; ⊂

♯ c) the “pseudo-counter-image” measure µ is equivalent to µ0. Assume one (and thus all) of the conditions to be fulﬁlled and thus let ♯ µ = ρ µ0, where ρ is a Baire (or Borel) µ-measurable function non zero everywhere· on G. Then for every s G the Radon-Nikodym derivative λ( ,s) of the measure R µ with respect∈ to the measure µ is equal to · s d(R µ) λ(π(x),s)= s (π(x)) = ρ(xs)/ρ(x) dµ almost µ-everywhere on G.

THEOREM. a) Any two non zero quasi invariant measures on G/H are equivalent.

b) If µ and µ′ are two non zero quasi invariant measures on G/H and d(Rsµ)/dµ = d(Rsµ′)/dµ′ almost µ-everywhere (and thus almost µ′-everywhere), then µ′ = c µ, where c is a positive number. · ♯ LEMMA. Measure ρ µ0 has the form µ if and only if for each h H the equality · ∈ ∆ (h) ρ(hx)= H ρ(x) ∆G(h)

is fulﬁlled almost µ0-everywhere on G.

THEOREM. a) There exist functions ρ fulﬁlling the conditions of the preceding Lemma, for example

∆ (h(x)) ρ(x)= H , ∆G(h(x)) where h(x) H is the only element of H corresponding to x G such 1 ∈ ∈ that h(x)− x B. ∈

17 b) ρ can be chosen to be continuous. c) One may chose the regular section B to be continuous outside a discrete countable set in G/H whenever G is a topological manifold with H as closed topological sub-manifold; thus x h(x) becomes continuous outside a set of measure zero in G. 7→ d) Given such a function ρ one can construct a quasi invariant measure µ on G/H such that µ♯ = ρ µ . · 0 e) ρ(xs)/ρ(x) with s, x G does not depend on x within the class π(x) and determinates a function∈ (π(x),s) λ(π(x),s) on G/H G equal to the 7→ × Radon-Nikodym derivative d(Rsµ)/dµ(π(x)). f) Given any Baire (or Borel) function λ( , ) on G/H G fulfilling the general properties of Radon-Nikodym derivative:· · (i) for× all x,s,z G, λ(π(z), xs) = λ(π(zx),s)λ(π(z), x), (ii) for all h H, λ(π(e),h∈) = ∈ ∆H (h)/∆G(h), (iii) λ(π(e),s) is bounded on compact sets as a function of s, one can construct a quasi invariant measure µ on G/H such that d(Rsµ)/dµ(π(x)) = λ(π(x),s), almost µ-everywhere with respect to s, x on G. Thus every non zero quasi invariant measure µ on G/H gives rise to a ρ- function and λ-function and vice versa every “abstract Radon-Nikodym derivative” i.e. λ-function (or equivalently every ρ-function) gives rise to a quasi invariant measure µ on G/H determined up to a non zero constant factor. Ev- ery quasi invariant measure µ on G/H is thus a pseudo-image of the right Haar measure µ on G under the canonical projection π in the terminology of [4]. In particular if the groups G and H are unimodular (i. e. ∆G =1G and ∆H =1H ) then among quasi invariant measures on G/H there exists a strictly invariant measure. The measure space structure of G/H uniform for the group G may be trans- R ferred to B together with the uniform structure, such that (G/H, G/H ,µG/H ) ∼= (B, RB,µB). The set B plays the role of the sub-manifold Q in the “measure product property”. This however would be insufficient, and we have to prove a kind of regularity of right H-cosets instead of “measure product property”. Namely let us define h(x) H, which corresponds uniquely to x G, such that 1 ∈ ∈ 1 h(x)− x B. We have to prove that the functions x h(x) and x h(x)− x are Borel∈ (thus in particular measurable), which however7→ was carried7→ through in the proof of Lemma 1.4 of [31]. Now the only point which has to be changed is the definition of the fundamental symmetry operator JL in L. We put H L (J f)x = Lh(x)JLLh(x)−1 fx.

L We deﬁne as the set of functions G L fulﬁlling the conditions (i), (ii) and such thatH 7→ H ( J ((JLf) ),f ) dµ < . L x x ∞ BZ

18 The proof that L is a Hilbert space with the inner product H (f,g)= ( J ((JLf) ),g ) dµ = (f ,g ) dµ (b), where b B, L x x b b B ∈ Z BZ is the same in this case with the only diﬀerence that the regularity of H-cosets is used instead of the Fubini theorem in reducing the problem to the von Neu- mann’s direct integral Hilbert space construction. Namely we deﬁne a unitary map V : f W f = f from the space L to the direct integral Hilbert space 7→ |B H L dµB of functions b Wb L by a simple restriction to B which is “onto”H in consequence of the7→ regularity∈ H of H-cosets. Its isometric character is R W W −1 trivial. V has the inverse W f with f x = Lh(x)Wh(x) x. In particular W 7→ f is measurable on G as for an orthonormal basis en n N of the Hilbert space ∈ and any υ we have: { } HL ∈ HL W W (fx ,υ) = (fx , JLJLυ) = (Lh(x)Wh(x)−1x, JLJLυ) = (JLLh(x)Wh(x)−1x, JLυ)

= (JLWh(x)−1x,Lh(x)−1 JLυ)= (JLWh(x)−1x,en)(en,Lh(x)−1 JLυ) n N X∈ which, as a point-wise convergent series of measurable (again by Scholium 3.9 of [51]) functions in x is measurable in x. We have to prove in addition that the induced representations U L in Krein spaces ( L, JL) corresponding to diﬀerent choices of regular Borel sections B are (Krein-)unitaryH equivalent. Namely let B1 and B2 be the two Borel sections in question. The Krein-unitary operator U : (U f) = L f , where h (x) H transforms the intersection point 12 12 x h12(x) x 12 ∈ of the right H-coset Hx with the section B1 into the intersection point of the same coset Hx with the Borel section B2, gives the Krein-unitary equivalence. The proof is similar to the proof of Lemma 7 of Sect. 5. Therefore from now on everything which concerns induced representations in Krein spaces, with the group G not explicitly assumed to be equal T4sSL(2, C), does not assume “measure product property”. Also Theorems 1 and 2 and Corollaries 1 and 2 remain true without the “measure product property” for any locally compact and separable G and its closed subgroup H. Indeed using the regular Borel section B of G the functions (4): (q,h0, q0) h′ and 7→ q,h0,q0 (q,h0, q0) q′ may likewise be deﬁned in this more general situation. 7→ q,h0 ,q0 Moreover, by Lemma 1.1 and the proof of Lemma 1.4 of [31], h′ ranges q,h0,q0 within a compact subset of H, whenever q ranges within in a compact subset of G, so that the proofs remain unchanged. The construction of the induced representation in Krein space has also another invariance property: it does not depend on the choice of a quasi invariant measure µ on G/H in the unique equivalence class, provided the Radon- dµ′ Nikodym derivative dµ corresponding to measures µ′ and µ in the class is essentially “upper” and “lower” bounded: there exist two positive numbers δ and ∆ such that dµ dµ ess sup ′ < ∆ and esssup < δ. dµ dµ′

19 This boundedness condition of Radon-Nikodym derivative is unnecessary in case of the original Mackey’s theory of induced representations, which are unitary in the ordinary sense. Introducing the left-handed-superscript µ in µ L and µU L for indicating the measure used in the construction of L and U L,H we may formulate a Theorem: H

THEOREM 3. Let µ′ and µ be quasi invariant measures in G/H with Radon- dµ′ Nikodym derivative ψ = dµ essentially “upper” and “lower” bounded. Then ′ there exists a Krein-unitary transformation V from µ L onto µ L such that µ L 1 µ′ L H µ HL µ′ L V Uy V − = Uy for all y G; that is the representations U and U are Krein-unitary equivalent. ∈ Let f be any element of µ L and let π be the canonical map G G/H. Boundedness condition ofH the Radon-Nikodym derivative ψ ensures7→ (√ψ π f, √ψ πf) to be finite, i. e. ensures √ψ πf to be a member of ′ µ L◦as √ψ ◦π is measurable; and moreover the Krein-square-inner◦ product H ◦ µ′ L µ L (√ψ π f, √ψ πf)JL in is equal to that (f,f)JL in . Moreover ◦ ◦ H ′ H boundedness of ψ guarantees that every g in µ L is evidently of the form √ψ πf for some f µ L. Let V be the operatorH of multiplication by √ψ π. ′ Then◦ V defines a Krein-unitary∈ H map of µ L onto µ L. The verification that◦ µ L 1 µ′ L H H V U V − = U is immediate. Finally we mention the following easy but useful THEOREM 4. Let L and L′ be Krein-unitary representations in ( L, JL), which are Krein-unitary and unitary equivalent, then the induced representationsH ′ U L and U L are Krein-unitary equivalent.

4 Certain dense subspaces of L H We present here some lemmas of analytic character which we shall need later and which we have used in the proof of Thm. 1 of Sect. 3. Let µH be the right invariant Haar measure on H. Let CL denote the set of all functions f : G x f , which are continuous with respect to the Hilbert space ∋ 7→ x ∈ HL norm = ( , ) in the Hilbert space L, and with compact support. Let us denotek·k the support· · of f by K . H p f L 0 LEMMA 1. For each f C there is a unique function f from G to L ∈ 0 H such that (JLLh−1 fhx,υ) dµH (h) = (JLfx ,υ) for all x G and all υ L. This function is continuous and it is a member of L.∈ The function G∈/H H [x] J (RJLf 0) ,f 0 as well as the function G/HH [x] f 0 has a compact∋ 7→ L x x ∋ 7→ x support. Finally sup J (JLf 0) ,f 0 = sup (f 0,f 0) < , where B is a x G L x x b B b b regular Borel section of∈ G with respect to H of Sect.∈ 3. ∞ Let f CL. For each ﬁxed x G consider the anti-linear functional ∈ ∈

υ F (υ)= (J L −1 f ,υ) dµ (h) 7→ x L h hx H Z

20 on . From the Cauchy-Schwarz inequality for the Hilbert space inner product HL ( , ) in the Hilbert space L and unitarity of JL with respect to the inner product· · ( , ) in , one getsH · · HL

F (υ) (J L −1 f ,υ) dµ (h) J L −1 f υ dµ (h) | x |≤ | L h hx | H ≤ k L h hxkk k H Z Z

= J L −1 f dµ (h) υ = L −1 f dµ (h) υ ; k L h hxk H k k k h hxk H k k Z Z where the integrand in the last expression is a compactly supported continuous function of h as a consequence of the strong continuity of the representation L and because f is compactly supported norm continuous. Therefore the integral in the last expression is ﬁnite, so that the functional Fx is continuous. Thus by Riesz’s theorem (in the conjugate version) there exists a unique element gx of L (depending of course on x) such that for all υ L : Fx(υ) = (gx,υ). H 0 0 ∈ H We put fx = JLgx, so that Fx(υ) = (JLfx,υ), υ L. We have to show that 0 0 ∈ H f : x fx has the desired properties. 7→ 0 0 That f ′ = L ′ f for all h′ H and x G follows from right invariance of h x h x ∈ ∈ the Haar measure µH on H:

0 0 (JLLh′ fx ,υ) = (JLfx ,Lh′−1 υ)= (JLLh−1 fhx,Lh′−1 υ) dµH (h) Z = (JLLh′ Lh−1 fhx,υ) dµH (h)= (JLL(hh′−1)−1 fhx,υ) dµH (h) Z Z = (JLL(hh′h′−1)−1 fhh′x,υ) dµH (hh′)= (JLL(hh′h′−1)−1 fhh′x,υ) dµH (h) Z Z 0 = (JLL(h)−1 fhh′x,υ) dµH (h) = (JLfh′x,υ), Z for all υ L, h′ H, x G. Denote∈ H the compact∈ support∈ of f by K. From the strong continuity of the representation L it follows immediately that the function

L (h, x) f = L −1 f 7→ (h,x) h hx is a norm continuous function on the direct product group H G and compactly supported with respect to the ﬁrst variable, i. e. for every x× G the function L 1 ∈ h f(h,x) has compact support equal Kx− H. It is therefore uniformly norm continuous7→ on the direct product group H G∩ with respect to the ﬁrst variable. × For any compact subset V of G let φV be a real continuous function on G with compact support equal 1 everywhere on V (there exists such a function because G as a topological space is normal). For f CL and any compact V G we introduce a norm continuous function on the∈ direct product group H ⊂G as a L × product f φV : (h, x) f L,V = f L φ (x), 7→ (h,x) (h,x) V which in addition is compactly supported and has the property that L,V f(h,x) = Lh−1 fhx

21 for (h, x) H V H G. In particular f L,V as compactly supported is not only∈ norm× continuous⊂ × but uniformly continuous on the direct product group H G (i. e. uniformly in both variables jointly). Let en n N be an × { } ∈ orthonormal basis in the Hilbert space L and let G be any open set H O ⊂ 0 containing x1, x2 G with compact closure V . From the deﬁnition of f it follows that ∈ 0 0 2 0 0 2 0 0 2 f f = JL(f f ) = (JL(f f ),en) k x1 − x2 k k x1 − x2 k x1 − x2 n N X∈ 2 = (J L −1 (f f ),e ) dµ (h) L h hx1 − hx2 n H n N Z X∈ 2 (J L −1 (f f ),e ) dµ (h) ≤ L h hx1 − hx2 n H n N Z X∈ 2 = (J L −1 (f f ),e ) dµ (h) L h hx1 − hx2 n H Z n N X∈ 2 2 = J L −1 (f f ) dµ (h)= L −1 (f f ) dµ (h) L h hx1 − hx2 H h hx1 − hx2 H Z Z 2 1 1 − − sup Lh 1 fhx1 Lh 1 fhx2 µH (Kx1− H) (Kx 2− H) ≤ h H − ∩ ∪ ∩ ∈ 1 L,V L,V 2 1 G (x − G x ) sup f f 2 sup µ (Kx− H). 2 0 1 0 (h,e) (e,x1) (h,e) (e,x2) H ∩ ≤ h H · − · x G ∩ ∈ ∈ Because the function f L,V is norm continuous on H G and the continuity is 1 × uniform and supx G µH (Kx− H) < ([31], proof of Lemma 3.1) the norm continuity of f 0 is∈ proved. ∩ ∞ Similarly we get

0 2 2 1 fx sup Lh−1 fhx µH (Kx− H) k k ≤ h H ∩ ∈ L,V x 2 1 = sup f(h,e) ( e,x) µH ( Kx− H) < , h H · ∩ ∞ ∈ x because K is compact and f L,V is norm continuous on H G and compactly x × 0 supported, where V is a compact neighbourhood of x G. Therefore fx =0 for all x / HK. Thus as a function on G/H:[x] (J∈Lf 0) ,f 0 and ak fortiorik ∈ 7→ x x the function [x] J (JLf 0) ,f 0 vanishes outside the compact canonical L x x image of HK in G7→/H. Finally let us note that if h(x) is the element of H deﬁned in Sect. 3 corresponding to x G, then ∈ L 0 0 0 0 JL(J f )x,fx = (fb ,fb ) 1 with b = h(x)− x – the unique intersection point of the coset Hx with the Borel section B. Because f is continuous with compact support, then the last assertion of the Lemma follows from Lemma 1.1 of [31].

We shall denote the class of functions f 0 for f CL of Lemma 1 by CL. ∈ 0

22 0 0 L LEMMA 2. For each fixed x G the vectors fx for f C0 form a dense linear subspace of . ∈ ∈ HL 0 L Note that if f C0 and Rsf is defined by the equation (Rsf)x = fxs ∈ 0 0 L for all x and s in G then Rsf = (Rsf) so that for all f C and s G, 0 L 0 0 ∈L ∈ R f C . Therefore the set ′′ of vectors f for f C and x fixed is s ∈ 0 HL x ∈ 0 independent of x. Let ′ be the J -orthogonal complement of ′′ , i. e. the HL L HL set of all υ L such that (JLg,υ) = 0 for all g L′′ . Then if υ L′ 0 ∈ H 0 L ∈ H 0 ∈ H we have (fx ,υ) = 0 for all f C0 and all x G. Therefore (JLfhx,υ) = 0 0 ∈ L ∈ (J f ,L −1 υ) = 0 for all f in C , all x in G and all h H. Hence ′ is L x h 0 ∈ HL invariant under the representation, as L is JL-unitary. Let L′ be the restriction 0 L′ of L to L′ . Suppose that there exists a non zero member f of C0 . Thus a H0 L 0 fortiori f C0 and we have a contradiction since the values of f are all in ∈ L L ′ , so that we would have in (J , ): HL H 0 L 0 (f ,g)JL = (J f ,g)

0 = (JLfx ,gx dµG/H =0 Z for all g L, which would give us f 0 = 0, because the Krein space (JL, L) of the induced∈ H representation U L is non degenerate (or JL invertible). ThusH in order to show that L′ = 0 we need only show that when L′ = 0 there exists H0 L′ H 6 a non zero member f of C0 . But if none existed then

(JLLh′ −1 fhx,υ) dµH (h) Z ′ would be zero for all x, all υ in and all f in CL . In particular the integral HL would be zero for f = uυ′, for all continuous complex functions u on G of compact support and all υ′ ′ , i .e ∈ HL

u(hx)(JLLh′ −1 υ′,υ) dµH (h) Z would be zero for all x, all υ in , all υ′ in ′ and all complex continuous u of HL HL compact support on G, which, because L (and thus L′) is strongly continuous, would imply that

(JLLh′ −1 υ′,υ)=0 for all υ in , all υ′ in ′ and all h H. This is impossible because the Krein HL HL ∈ space (J , ) of the representation L is non degenerate and L′ −1 non-singular L HL h as a Krein-unitary operator. Thus we have proved that ′ = 0. This means HL that J ′′ is dense in the Hilbert space , and because J is unitary in LHL HL L HL with respect to the ordinary deﬁnite inner product ( , ), this means that L′′ is dense in the Hilbert space . · · H HL

LEMMA 3. Let C be any family of functions from G to such that: HL

23 (a) C L. ⊂ H (b) For each s G there exists a positive Borel function ρs such that for all f C, ρ R∈f C where (R f) = f . ∈ s s ∈ s x xs (c) If f C then gf C for all bounded continuous complex valued functions g on∈G which are∈ constant on the right H-cosets. (d) There exists a sequence f 1,f 2,... of members of C and a subset P of G 1 2 of positive Haar measure such that for each x P the members fx ,fx,... of have as their closed linear span. ∈ HL HL Then the members of C have L as their closed linear span. H Choose f 1,f 2,... as in the condition (d). Let u be any member of L which is JL-orthogonal to all members of C: H

L f,u JL = (J f,u)= ( JL(fx),ux ) dµG/H =0 Z for all f C. Then ∈

L j j (J (ρsg)(Rsf ),u)= ( JL((ρsg)(x)(Rsf )x),ux ) dµG/H =0 Z for every j N, all s and every bounded continuous g on G which is constant on ∈ N j the right H-cosets. It follows at once that for all s and all j (JLfxs,ux)=0 for almost all x G. Since x (J f j,u ) is a Borel function∈ on G the function ∈ 7→ L x x (x, s) (J f j ,u )= (J f j ,e )(e ,u ) 7→ L xs x L xs n n x n N X∈ is Borel on the product measure space G G on repeating the argument of Sect. 3 (Scholium 3.9 of [51]) and joining× it with the fact that composition of a measurable (Borel) function on G with the continuous function G G (x, s) xs G is measurable (Borel) on the product measure space G× G∋ (compare7→ e. g.∈ [51]). Thus we may apply the Fubini theorem (Thm. 3.4 in [51])× j and conclude that for almost all x, (JLfxs,ux) is zero for almost all s. Since j runs over a countable class we may select a single null set N G such that for j N ⊂ each x / N, (JLfxs,ux) is, for almost all s, zero for all j . It follows that for ∈ 1 j ∈ each x / N there exists s x− P such that (J f ,u )=0for j N and hence ∈ ∈ L xs x ∈ that ux = 0 because JL is unitary with respect to the ordinary deﬁnite Hilbert space inner product in the Hilbert space L. Thus u is almost everywhere zero and JLC must be dense in L. BecauseH JL is unitary in the ordinary sense with respect to the deﬁnite innerH product (eq. (2) of Sect. 3) in L, C must be dense in L. H H

LEMMA 4. Let C1 be any family of functions from G to such that: HL

24 (a) For each f C1 there exists a positive Borel function ρ on G such that ∈ 1 1 1 J f ,υ = J f ,υ = J f ,υ L ρ(x) x ρ(x) L x ρ(x) L x is continuous as a function of x for all υ . ∈ HL (b) C1 L. ⊂ H (c) For each s G there exists a positive Borel function ρs such that for all f C1, ρ R∈ f C1 where (R f) = f . ∈ s s ∈ s x xs (d) If f C1 then gf C1 for all bounded continuous complex valued functions∈g on G which∈ are constant on the right H-cosets and vanish outside 1 of π− (K) for some compact subset K of G/H.

1 (e) For some (and hence all) x G the members fx of L for f C have as their closed linear span.∈ H ∈ HL Then the members of C1 have L as their closed linear span. H 1 2 1 1 2 Choose f ,f ,... in C so that fe ,fe ,... have L as their closed linear span; e being the identity of G. Let u be any memberH of L which is JL- orthogonal to all members of C1. Then H

L j j (J (ρsg)(Rsf ),u)= ( JL((ρsg)(x)(Rsf )x),ux ) dµG/H =0 Z for every j N, all s and every bounded continuous g on G which is constant on ∈ N j the right H-cosets. It follows at once that for all s and all j (JLfxs,ux)=0 j ∈ for almost all x G. Since (x, s) (JLfxs,ux) is a Borel function on the product measure∈ space G G (compare7→ the proof of Lemma 3) we may apply the Fubini theorem as in× the preceding Lemma and conclude that for almost j all x, (JLfxs,ux) is zero for almost all s. Since j runs over a countable class we may select a single null set N in G such that for each x / N, (J f j ,u ) is ∈ L xs x for almost all s zero for all j. Suppose that ux1 = 0 for some x1 / N. Then (J f j,u ) = 0 for some j as J is unitary with respect6 to the ordinary∈ Hilbert L e x1 6 L space inner product ( , ) in L (as in the proof of the preceding Lemma). · · H j But for some positive Borel function ρ, (JLfx,ux1 ) ρ(x) is continuous in x. J j 1 Hence ( Lfx1s,ux1 ) ρ(x1s) = 0 for s in some neighbourhood of x1− . Thus j 6 1 (JLf ,ux ) = 0 for s in some neighbourhood of x− . But this contradicts x1s 1 1 6 J j G the fact that ( Lfx1s,ux1 ) is zero for almost all s . Therefore ux is zero almost everywhere. Thus only the zero element is orthogonal∈ (in the ordinary positive inner product space in L) to all members of JLC1 and it follows that JLC1 must be dense in L. BecauseH JL is unitary with respect to the ordinary deﬁnite inner product (H, ) in L, it follows that C1 is dense in L. · · H H

LEMMA 5. CL is dense in L. 0 H

25 The Lemma is an immediate consequence of Lemmas 2 and 4.

REMARK. For the reasons explained in Sect. 8 we are interesting in complete measure spaces on G/H and on all other quotient spaces encountered later in this paper. But the Baire or Borel measure is pretty sufficient in the investi- 2 L gation of the associated Hilbert spaces L (G/H,µG/H ) or as all measurable sets differ from the Borel sets just by null sets, and theH space of equivalence 2 classes of Borel square summable functions in L (G/H,µG/H ) is the same as the space of equivalence classes of square summable measurable functions. Re- call that the Baire measure space may be completed to a Lebesgue-type measure space, e. g. using the Carathéodory method. In other words the Baire or Borel (the same in this case) measure space may be completed such that any subset of measurable null set will be measurable.

1 2 L L LEMMA 6. There exists a sequence f ,f ,... of elements C0 such that k ⊂ H for each fixed x G the vectors fx , k =1, 2,... form a dense linear subspace of . ∈ HL We have seen in the previous Sect. that as a Hilbert space L is unitary equivalent to the direct integral Hilbert space dµ over theHσ-finite and HL G/H regular Baire (or Borel) measure space (G/H, RG/H ,µG/H ) with separable L. Because G/H = X is locally compact metrizableR and fulfils the second axiomH of countability its minimal (one point or Alexandroff) compactification X+ is likewise metrizable (compare e. g. [13], Corollary 7.5.43). Thus the Banach algebra C(X+) is separable, compare e. g. [25], Thm. 2 or [17]). Because C(X+) is equal + to the minimal unitization C0(X) of the Banach algebra C0(X) of continuous functions on X vanishing at infinity (compare [41]), thus by the construction of minimal unitization it follows that C0(X) is separable (of course with respect to + the supremum norm in C0(X)) as a closed ideal in C0(X) of codimension one. Because the measure space (G/H, RG/H ,µG/H ) is the regular Baire measure space, induced by the integration lattice C (X) C0(X) of continuous functions with compact support (compare [51]),K it follows⊂ from Corollary 4.4.2 of 2 [51] that the Hilbert space L (G/H,µG/H ) of square summable functions over X = G/H is separable. Let en n N be an orthonormal basis in L. Using standard – by now – Hilbert space{ } ∈ ([41]) and measure space (e. g.H Fubini theorem, compare eq. (28) of Sect.8) techniques and the results of [36] one can prove that

dµ = Ce dµ HL G/H n G/H n N Z M∈ Z = where = L2(G/H,µ ). Hn Hn ∼ G/H n N M∈ L Thus L dµG/H itself must be separable and therefore is separable. Thus we mayH choose a sequence f 1,f 2,... of elements CL HL such that for each R 0 ⊂ H

26 L n1 n2 f C0 there exists a subsequence f ,f ,... which converges in norm of ∈ L to f. Then a slight and obvious modiﬁcation of the standard proofk·k of theH Riesz-Fischer theorem (e. g. [51], Thm. 4.2) gives a sub-subsequence nm nm f 1 ,f 2 ,... which, after restriction to the regular Borel section B ∼= G/H converges almost uniformly to the restriction of f to B (where B ∼= G/H is locally compact with the natural topology induced by the canonical projection R R π, with the Baire measure space structure (G/H, G/H ,µG/H ) ∼= (B, B,µB)) obtained by Mackey’s technique of quotiening the measure space G by the group H recapitulated shortly in Sect. 3. As f k,f are continuous and compactly supported as functions on B ∼= G/H, the convergence is uniform on B. The Lemma now, for x B, is an immediate consequence of Lemma 2. Because for ∈ k k 1 each x G we have f = L f −1 ,f = L f −1 with h(x)− x B ∈ x h(x) h(x) x x h(x) h(x) x ∈ and because Lh is invertible (and bounded) for every h H, the Lemma is proved. ∈

5Lopusza´nski representation as an induced representation

Let G be a separable locally compact group and H its closed subgroup. In this section we shall need Lemma 7 (below), which we prove assuming the “measure product property”, because it is sufficient for the analysis of theLopuszański representation of the double covering of the Poincarégroup. However it can be proved without this assumption, as the reader will easily see by recalling the respective remarks of Sect. 3. R Thus we assume (for simplicity) that the right Haar measure space G , G , µG be equal to the product measure space H G/H, R , µ µ × H×G/H H × G/H R with H, H , µH equal to the right Haar measure space on H and with R the Mackey quotient measure space G/H, G/H , µG/H on G/H (described briefly in Sect 3). In most cases of physical applications both G and H are unimodular. Let g = h q be the corresponding unique factorization of g G with h H and q Q · G representing the class [g] G/H. Uniqueness of∈ the factorization∈ allows∈ us⊂ to introduce the following functions∈ ((already mentioned in Sect. 3) (q,h0, q0) h′ H and (q,h0, q0) q′ Q = G/H, 7→ q,h0 ,q0 ∈ 7→ q,h0,q0 ∈ ∼ where for any g0 = q0 h0 G we define h′ H and q′ Q G to · ∈ q,h0 ,q0 ∈ q,h0,q0 ∈ ⊂ be the elements, uniquely corresponding to (q,h0, q0), such that

q h0 q0 = h′ q′ . · · q,h0 ,q0 · q,h0,q0

In particular if g = hq, then q represents [g] G/H, and q′ represents [gg0], ∈ q,h0 ,q0 i.e. the right action of G on G/H. It is easily veriﬁable that (q,h0, q0) h′ 7→ q,h0 ,q0 behaves like a multiplier, i.e. denoting h′ and q′ just by hq,g′ and qq,g′ q,h0,q0 q,h0 ,q0 0 0

27 we have h′ h′ = h′ . q, g q′ , g q, g g 0 q,g 1 0 1 · 0 Let U L be the Krein isometric representation of G induced by an almost uniformly bounded Krein-unitary representation of H in the Krein space ( L, JL), deﬁned as in Sect. 3. Let us introduce the Hilbert space H

= dµ (7) H HL G/H GZ/H and the fundamental symmetry J

J = JL dµG/H (8) GZ/H in , i.e. operator decomposable with respect to the decomposition (7) whose H all components in its decomposition are equal JL. Because JL∗ = JL and JL∗ JL = JLJL∗ = I, then by [36] the same holds true of the operator J, i.e. it is unitary 2 and selfadjoint, i.e. J∗ = J and J∗J = JJ∗ = I, so that J = I and J is a fundamental symmetry. We may therefore introduce the Krein space ( , J). H LEMMA 7. Let G be a separable locally compact group and H its closed subgroup. Assume (for simplicity) that the ”measure product property” is fulﬁlled by G and H. Then the operators

U : L, and S : L , H 7→ H H 7→ H deﬁned as follows

UW = LhWq , and Sf = L −1 fh·q , h q q h · for all W and f L, are well deﬁned operators, both are isometric and Krein-isometric∈ H between∈ H( , J) and ( L, JL) and moreover US = I and SU = I and moreover H H 1 L U − J U = J, so that U and S are unitary and Krein-uinitary. We have

1 L Vg W = U − Ug UW = λ(q,g0 )Lh′ Wq′ ; 0 0 q,g q,g q q 0 0 p or equivalently

1 L Vg W = U − U UW = λ([g],g0 )L ′ W[g·g ] . 0 g0 h 0 [g] [g] [g],g0 p In short: U L is unitary and Krein unitary equivalent to the Krein-isometric representation V of G in ( , J). H

28 1 L That the functions UW , W , and U − f, f fulfil the required measurability conditions has been already∈ H shown in Sect.∈ H 3). Verification of the isometric and Krein-isometric character of both U and S is easy, and we leave it to the reader. Checking US = I and SU = I as well as the last equality is likewise simple. Now let us turn our attention to the construction of semi-direct product groups and their specific class of Krein-isometric representations to which the Lopuszański representation belong together with the related systems of imprimitivity in the Krein space ( , J), say of Lemma 7. Let G and G be separable H 1 2 locally compact groups and let G1 be abelian (G1 plays the role of four translations subgroup T4 and G2 plays the role of the SL(2, C) subgroup of the double covering G = T4sSL(2, C) of the Poincarégroup). Let there be given a homomorphism of G into the group of automorphisms of G and let y[x] G be 2 1 ∈ 1 the action of the automorphism corresponding to y on x G1. We assume that (x, y) y[x] is jointly continuous in both variables. We∈ define the semi-direct product7→ G = G sG as the topological product G G with the multiplica- 1 2 1 × 2 tion rule (x1,y1)(x2,y2) = (x1y1 [x2],y1y2). G = G1sG2 under this operation is a separable locally compact group. Recall that the subset of elements (x, e) with x G and e being the identity is a closed subgroup of the semi direct ∈ 1 product G naturally isomorphic to G1 and similarly the set of elements (e,y), y G is a closed subgroup of G = G sG naturally isomorphic to G . Let us ∈ 2 1 2 2 identify those subgroups with G1 and G2 respectively. Since (x, e)(e,y) = (x, y) it follows at once that any Krein-isometric representation (x, y) V(x,y) of G = G sG in the Krein space ( , J) is determined by its restrictions7→ N and 1 2 H U to the subgroups G1 and G2 respectively: V(x,y) = NxUy. Conversely if N and U are Krein-isometric representations of G1 and G2 which act in the same Krein space ( , J) and with the same core invariant domain D, and moreover if the representationH N commutes with the fundamental symmetry J and is therefore unitary, then one easily checks that (x, y) N U defines a Krein- 7→ x y isometric representation if and only if UyNxUy−1 = Ny[x]. Indeed the “if” part is easy. Assume then that V(x,y) = NxUy is a representation. Then for any (x, y), (x′,y′) G sG one has N U N ′ U −1 U ′ = N N ′ U ′ on the core ∈ 1 2 x y x y yy x y[x ] yy dense set D. Because Nx is unitary it follows that UyNx′ Uy−1 Uyy′ = Ny[x′]Uyy′ on D. Because U D = D for all y y G and U U −1 = I on D, then y ∈ ∈ 2 y y it follows that U N ′ U −1 = N ′ on D for all x, x′ G and all y G . y x y y[x ] ∈ 1 ∈ 2 Because the right hand side is unitary, then UyNx′ Uy−1 can be extended to a unitary operator, although U is in general unbounded. Now assume (which is the case for representations of translations acting in one particle states in QFT, for example this is the case for the restriction of theLopuszańs ki representation to the translation subgroup) that the representation N of the abelian subgroup G1 commutes with the fundamental symmetry J in , and thus it is not only Krein-isometric but unitary in in the usual sense.H Moreover the restrictions N of representations acting inH one particle states are in fact of uniform (even finite) multiplicity. Because N is a unitary representation of a separable locally compact abelian group G1 in the Hilbert space the Neumak’s theorem is applicable, which says that N is determined by a projection valued (spectral)

29 measure S ES (which as we will see may be associated with the direct integral decomposition7→ (7) with the appropriate subgroup H), deﬁned on the Borel

(or Baire) sets S of the character group G1 of G1:

Nx = χ(xc) dE(χ).

GZ c1 It is readily veriﬁed that N and U satisfy the above identity if and only if the spectral measure E and the representation U satisfy UyESUy−1 = E[S]y, for all y G2 and all Borel sets S G1; where the action [χ]y of y G2 on χ G1 ∈ ⊂ 1 ∈ ∈ is deﬁned by the equation [χ]y, x = χ,y− [x] (with χ, x denoting the value h i h i h i of the character χ G on the elementc x G ). Indeed: c ∈ 1 ∈ 1 c UyNxUy−1 = χ(x) d(UyE(χ)Uy−1 )= Ny[x] = χ(y[x]) dE(χ)

GZ GZ c1 c1 1 = [χ]y− (x) dE(χ)= χ(x) dE([χ]y). (9)

GZ GZ c1 c1 We call such E, N, and U a system of imprimitivity in the Krein space ( , J), after Mackey [32] who deﬁned the structure for representations N andHU in Hilbert space which are both unitary in the ordinary sense. H Consider now the action of G2 on G1. If the spectral measure E is concentrated in one of the orbits of G1 under G2 let χ0 be any member of this orbit O and let G be the subgroup of allc y G for which [χ ]y = χ . Then χ0 χ0 ∈ 2 0 0 y [χ0]y deﬁnes a one-to-onec Borel set preserving map between the points of 7→ O this orbit χ0 and the points of the homogeneous space G2/Gχ0 = G/H, where

H = G1 Gχ0 . In this way E, N, U, becomes a system of imprimitivity based on the homogeneous· space G/H. Now when E is concentrated on a single orbit the assumption of uniform multiplicity of N would be unnecessary, but instead we may require U to be “locally bounded”: Uyf

30 separable locally compact groups G1 and G2 with G1 abelian in a Krein space ( , J) and with the representation N commuting with J and thus being unitary inH , for which the following assumptions are satisﬁed: H O 1) The spectral measure is concentrated on a single orbit χ0 in G1 under G2. c 2) The representation U (equivalently the representation V ) is “locally bounded” with respect to E. Then the representation N (and equivalently the spectral measure E) is of uniform multiplicity. The fundamental symmetry J is decomposable with respect to the decomposition of associated (in the sense of [36]) to the spectral measure E of the system, and hasH a decomposition of the form (8). Assume moreover that:

3) The representation N has ﬁnite multiplicity.

Then V is unitary and Krein-unitary equivalent to a Krein-isometric representation U L induced by a Krein unitary representation L of the subgroup H = G G associated to the orbit. 1 · χ0 This theorem may be given a more general form by discarding 3), but the given version is sufficient for the representations acting in one particle states of free fields with non trivial gauge freedom, and thus acting in Krein spaces (with the fundamental symmetry operator J called Gupta-Bleuler operator in physicists parlance), where the representations L act in Krein spaces ( L, JL) of finite dimension. H Consider for example the double covering G = T4sSL(2, C) of the Poincaré group with the semi direct product structure defined by the following homomorphism: α[t ]= αxα∗, where the translation t : (a ,a ,a ,a ) (a ,a ,a ,a )+ x x 0 1 2 3 7→ 0 1 2 3 (x0, x1, x2, x3) is written as a hermitian matrix

x + x x ix x = 0 3 1 2 x + ix x − x 1 2 0 − 3 in the formula αxα∗ giving α[t ] and α∗ is the hermitian adjoint of α SL(2, C). x ∈ Characters χ T of the group T have the following form p ∈ 4 4

i( p0a0+p1x1+p2x2+p3x3) cχp(tx)= e − ,

4 for p = (p0,p1,p2,p3) ranging over R . For each character χp T4 let us consider the orbit O passing through χ , under the action χ [χ∈]α, α SL(2, C), χp p p 7→ p ∈ where [χp]α is the character given by the formula c

[χp]α 1 1 1 T t [χ ]α (t )= χ (α− [t ]) = χ (α− xα∗− )= χ ∗ (x)= χ ∗ (t ), 4 ∋ x −−−→ p x p x p αpα αpα x 31 1 1 where in the formulas αpα∗ and α− xα∗− , x and p are regarded as hermitian 2 2 matrices: × x + x x ix p + p p ip x = 0 3 1 2 and p = 0 3 1 2 . x + ix x − x p + ip p − p 1 2 0 − 3 1 2 0 − 3 Let G be the stationary subgroup of the point χ T . Let H = H = χp p ∈ 4 χp T G , and let L′ be a Krein-unitary representation of the stationary group 4 · χp Gχp . Then L given by c

Ltx g = χp(tx)Lg′ , tx T4,g Gχp · ∈ ∈ is a well deﬁned Krein-unitary representation of H = T G because G is χp 4 · χp χp the stationary subgroup for the point χp. The functions (q,h0, q0) h′ 7→ q,h0,q0 ∈ H and (q,h0, q0) q′ Q = G/Hχp corresponding to the respective 7→ q,h0,q0 ∈ ∼ O H = Hχp or the respective orbits χp are known for all orbits in T4 under SL(2, C) and may be explicitly computed. For example for p = (1, 0, 0, 1) lying on the light cone in the jointc spectrum sp(P0,...P3) of the canonical generators of one parameter subgroups of translations, the stationary subgroup Gχ = Gχ is equal to the group of p (1,0,0,1) matrices iφ/2 iφ/2 e e z C iφ/2 , 0 φ< 4π, z 0 e− ≤ ∈ isomorphic to (the double covering of) the symmetry group E2 of the Euclidean plane and with the orbit Oχ equal to the forward cone with the apex (1,0,0,1) removed. 4 Consider then the Hilbert space L to be equal C with the standard inner product and with the fundamental symmetryH equal

1 0 0 0 −0 100 J = . L  0 010   0 001      Finally let L′ be the following Krein-unitary representation

1+ 1 z 2 1 z 1 z 1 z 2 2 | | √2 √2 − 2 | | iφ/2 iφ/2 ′ 1 iφ iφ 1 iφ e e z L(z,φ) e− z e− 0 e− z  √2 √2  (10) iφ/2 1 eiφz 0 eiφ 1 eiφz 0 e− −−−−→ √2 √2  1 2 1 1 − 1 2   z z z 1 z   2 | | √2 √2 − 2 | |    of Gχ = E2 in the Krein space ( L, JL) and deﬁne the Krein-unitary (1,0,0,1) ∼ H Ltx·(z,φ) representation L: H = T4 Gχ tx (z, φ) χ (tx)L′ f · (1,0,0,1) ∋ · −−−−−→ (1,0,0,1) (z,φ) corresponding to the Krein-unitary representation L′ of Gχ . Then one (1,0,0,1) obtains in this way the system of imprimitivity with the representation V of

32 the Lemma 7 equal to theLopuszański representation acting in the one particle states of the free photon field in the momentum representation, having exactly Wigner’s form [59] with the only difference that L is not unitary but Krein- unitary.

Several remarks are in order. 4 1) In case of G = T4sSL(2, C), T4 = R with the natural smooth action of C O R4 SL(2, ) giving it the Lorentz structure. The possible orbits χp T4 = are: the single point (0, 0, 0, 0, ) – the apexc of “the light-cone”, the upper/lower⊂ half of the light cone (without the apex), the upper/lower sheet of thecparaboloid, and the one-sheet hyperboloid. Thus all of them are smooth manifolds (with the exclusion of the apex, of course). Joining this with the Mackey analysis of quasi invariant measures on homogeneous G/H spaces one can see that the spectral measures of the translation generators (for representations with the joint spectrum sp(P0,...P3) concentrated on single orbits) are equivalent to measures 4 induced by the Lebesgue measure on R = T4 (of course with the exclusion of the representations corresponding the the apex – the single point orbit, with the zero (0, 0, 0, 0) as the only value of the jointc spectrum sp(P0,...P3). 2) Note that for the system of imprimitivity E, N, U in the Krein space the condition:

V(x,y)ES V(x,y)−1 = NxUyES Uy−1 Nx−1

= N E N −1 = E for all (x, y) G sG and all Borel sets S G x [S]y x [S]y ∈ 1 2 ⊂ 1 holds, and is essentially equivalent to the condition: c

U E U −1 = E , for all y G , and all Borel sets S G . y S y [S]y ∈ 2 ⊂ 1 We may write it as V(x,y)ES V(x,y)−1 = E[S](x,y), with the trivial actionc [χ](x, e)= χ, x G1 and [χ](e,y) = [χ]y. It is more convenient to relate the system of imprimitivity∈ immediately to V and inspired by Mackey put the following more general deﬁnition. Let V be a Krein-isometric representation of a separable locally compact group G in a Krein space ( , J). By a system of imprimitivity for V , we mean the system E, B, ϕ consistingH of a) an analytic Borel set B; b) an anti-homomorphism ϕ of G into the group of all Borel automorphisms of B such that (y,b) (y, [b]y) is a Borel automorphism of G B; here we have written [b]y for7→ the action of the automorphism ϕ(y) on×b B. ∈ c) The spectral measure E consists of selfadjont and Krein selfadjoint projec- 1 tions commuting with J in ( , J), and is such that V E V − = E −1 . H y S y [S]y d) The representation V is “locally bounded” with respect to E.

33 Any induced Krein-isometric representation µU L possesses a canonical system of imprimitivity in ( L, JL) related to it. Namely let S be a Borel set on H G/H, and let S′ be its inverse under the quotient map G G/H. Let 1 ′ be → S ES L the characteristic function of S′. Then f 1S′ f, f is a self adjoint −−→ ∈L H and Krein self adjoint projection, which commutes with J . Thus S ES is a spectral measure based on the analytic Borel space G/H. By the inequality7→ (5) in the proof of Theorem 1 the representation µU L is “locally bounded”, i.e. fulﬁls condition 3) of Theorem 5 or condition d). The representation V of Lemma 7 in the Krein space ( , J) together with H the spectral measure E′ on B = G/H associated with the decomposition (7) is a system of imprimitivity in Krein space which by Lemma 7 is Krein-unitary and unitary equivalent to the canonical system of imprimitivity U L, E, ϕ deﬁned above. That V, E′ of Lemma 7 with ϕg (q) = qq,g′ composes a system of im- 0 0 primitivity can be checked directly using the multiplier property of the function

(q,g0 ) hq,g′ . 7→ 0 3) The plan for further computations is the following. First we start with the systems of imprimitivity fulfilling the conditions 1)-3) of Theorem 5 sufficient for accounting for the representations acting in one particle states of free fields. Then we prove the “subgroup” and “Kronecker product theorems” for the induced representations in order to achieve decompositions of tensor products of these representations into direct integrals of representations connected with imprimitivity systems concentrated on single orbits (using Mackey double- coset-type technics). The component representations of the decomposition will not in general have the standard form of induced representations (contrary to what happens for tensor products of induced representations of Mackey which are unitary in ordinary sense). But then we back to Theorem 5 applied again to each of the component representations in order to restore the standard form of induced representation in Krein space to each of them separately. In this way we may repeat the procedure of decomposing tensor product of the component representations (now in the standard form) and continue it potentially in infinitum. It turns out that the condition 3) of finite multiplicity will have to be abandoned in further stages of this process, but we have all the grounds for the condition 2) of “local boundedness” to be preserved in all cases at all levels of the decomposition. Indeed recall that the spectral values (p0,...p3) of the translation generators (four-momentum operators) in the tensor product of representations corresponding to imprimitivity systems concentrated on single O O orbits ′, ′′ T4, are the sums (p0′ ,...p3′ ) + (p0′′,...p3′′), with the spectral ⊂ O O values (p0′ ,...p3′ ) and (p0′′,...p3′′) ranging over ′ and ′′ respectively. Now the geometry of the orbitsc in case of G = T4sSL(2, C) is such that the sets of all values (p0′ ,...p3′ ) and (p0′′,...p3′′) for which (p0,...p3) ranges over a compact set, are compact (discarding irrelevant null sets of (p0,...p3) not belonging to the joint spectrum of momentum operators of the tensor product representation – the light cones – in the only case of tensoring representation corresponding to the positive energy light cone orbit with the representation corresponding to the negative energy light cone).

34 4) In fact the representation of one particle states in the Fock space (with the Gupta-Bleuler or fundamental symmetry operator) is induced by the following representation L′′ in the above deﬁned Krein space ( L, JL) of the double covering of the symmetry group of the Euclidean plane: H

1 2 1 i 1 2 1+ 2 z 2 (z + z) 2 (z z) 2 z 1 iφ | | iφ − 1 −iφ| | iφ 2 (e− z + e z) cos φ sin φ 2 (e− z + e z L(′′z,φ) =  i iφ iφ −1 iφ φ  , (11) (e z e− z) sin φ cos φ (e z e− z) 2 − − − 2 −  1 z 2 1 (z + z) i (z z) 1 1 z 2   2 | | 2 2 − − 2 | |    compare e.g. [57, 58], or [29, 30]. But the operator

10 0 0 0 1/√2 i/√2 0   . 0 1/√2 i/√2 0  00− 0 1      which is Krein-unitary and unitary in ( , J ) sets up Krein-unitary and uni- HL L tary equivalence between the representation L′ of (10) and the representation L′′ of (11) as well as between the associated representations L. By Theorem 4 it makes no difference which one we use, but for some technical reasons we prefer the representation L associated with (10). 5) The representation which we have called by the name ofLopusza ński have appeared in physics rather very early, compare [60], and then in relation to the Gupta-Bleuler quantization of the free photon field: [57, 58], [22], [26]. But it wasLopuszański [29, 30] who initiated a systematic study of the relation of the representation with the Gupta-Bleuler formalism. That’s why we call the representation after him.

6 Kronecker product of induced representations in Krein spaces

In this Section we deﬁne the outer Kronecker product and inner Kronecker product of Krein isometric (and Krein unitary) representations and give an important theorem concerning Krein isometric representation induced by a Kronecker product of Krein-unitary representations. The whole construction is based on the ordinary tensor product of the associated Hilbert spaces and operators in the Hilbert spaces. We recapitulate shortly a speciﬁc realization of the tensor product of Hilbert spaces as trace class conjugate-linear operators, in short we realize it by the Hilbert-Schmidt class of conjugate-linear operators with the standard operator L2-norm, for details we refer the reader to the original paper by Murray and von Neumann [35]. Alternatively one may consider linear Hilbert-Schmidt class operators, but replace one of the Hilbert spaces in question by its conjugate space, compare [31], 5. §

35 Let 1 and 2 be two separable Hilbert spaces over C (recall that by the proof ofH LemmaH 6 the Hilbert space L of the Krein-isometric representation U L of a separable locally compact groupH G induced by a Krein-unitary representation L of a closed subgroup G G is separable). A mapping T of to 1 ⊂ H2 1 is conjugate-linear iff T (αf + βg) = αT (f)+ βT (g) for all f,g 2 and allH complex numbers α and β, with the “over-line” sign standing for∈ complex H conjugation. For any such conjugate-linear operator T we define the conjugate ⊛ ⊛ version of its adjoint T , namely this is the operator fulfilling (Tg,f) = (T f,g) for all f 1 and all g 2. In particular if T is bounded, conjugate-linear, ∈ H ∈ H ⊛ finite-rank operator so is its conjugate adjoint T . If U1 and U2 are bounded operators in 1 and 2 respectively then U1TU2 is a finite rank operator from H H ⊛ ⊛ 2 into 1. One easily verifies that (AT B) = B∗T A∗, where A and B are H H 1 2 linear operators in and with A∗ and B∗ equal to their ordinary adjoint H H operators. If U1 and U2 are densely defined operators in 1 and 2 respectively on linear domains D and D and T is finiteH rankH operator with 1 ⊂ H1 2 ⊂ H2 the rank contained in D1 and supported in D2, then U1TU2 is a well defined finite rank operator. Let ′ = ′ be the linear space of finite rank H H1 ⊗ H2 conjugate-linear operators T of 2 into 1. For any two such operators T and ⊛ H H ⊛ S the operator TS is linear from 1 into 1 and of finite rank (similarly T S is linear and finite rank from Hinto H). We may therefore introduce the H2 H2 following inner product in ′: H ⊛ ⊛ T,S = Tr [ TS ]= (TS e ,e ) h i n n n X ⊛ ⊛ = (T en,S en)= (Tεm,Sεm) n m X X ⊛ ⊛ (T Sεm,εm) = Tr [ T S ], m X where en n N and εm m N are orthonormal bases in 1 and 2 respectively. { } ∈ { } ∈ H H The completion of ′ with respect to this inner product composes the tensor H product = 1 2. Let AHand HB be⊗ bounded H operators in and . Their tensor product A B H1 H2 ⊗ acting in is defined as the operator T AT B∗, for T . In H1 ⊗ H2 7→ ∈ H1 ⊗ H2 particular if for any f 1 and g 2 we define the finite rank conjugate- linear operator T : w ∈ Hf (g, w) supported∈ H on the linear subspace generated f,g 7→ · by g with the range generated by f, then Tf,g 1 2 is written as f g ∈ H ⊛⊗ H ⊗ and we have (f g ,f g ) = Tr T T = Tr T T = 1 1 2 2 f1,g1 f2,g2 f1,g1 g2,f2 ⊗ ⊗ ⊛ (f ,f ) (g ,g ) because T = T . 1 2 1 2 f2,g2 g2,f2 If ( · , J ) and ( , J ) are two Krein spaces, then we define their tensor H1 1 H2 2 product as the Krein space ( 1 2, J1 J2); verification of the self-adjointness H ⊗H 2 ⊗ of J1 J2 and the property J1 J2 = I is immediate. We⊗ say an operator T from ⊗ into is supported by finite dimensional 2 1 (or more generally: closed) linearH subspaceHM or by the projection P , in ⊂ H2 M case T = T PM , where PM is the self adjoint projection with range M. Similarly

36 we say an operator T from into has range in a finite dimensional (or H2 H1 more generally: closed) linear subspace N H1, in case T = PN T , where PN is the self adjoint projection with range N⊂. One easily verifies the following tracial property. Let B be any finite rank and linear operator from into H1 H1 supported on a finite dimensional linear subspace of the domain D1 and with the range also finite dimensional and lying in D1. Then for any linear operator defined on the dense domain D and preserving it, i. e. with D contained 1 ⊂ H1 1 in the common domain of A and its adjoint A∗, we have the tracial property

Tr [BA] = Tr [AB]. T Indeed any such linear B is a finite linear combination of the operators f,f′ T defined as follows: f,f′ (w) = (w,f) f ′. By linearity it will be sufficient to establish the tracial property for the linear· operator B of the form B = T + f1 ,f2 T with fi D1, i = 1, 2, 3, 4. Using the Gram-Schmidt orthogonalization f3 ,f4 ∈ we construct an orthonormal basis en n N of 1 with en D1. We have in this case { } ∈ H ∈

T T T T Tr [BA] = Tr f ,f + f ,f A = Tr f ,f A + Tr f ,f A 1 2 3 4 1 2 3 4 h i h h i = T Aen,en + T Aen,en f1 ,f2 f3 ,f4 n n X X = (Ae ,f ) (f ,e )+ (Ae ,f ) (f ,e ) n 1 · 2 n n 3 · 4 n n n X X = (e , A∗f ) (f ,e )+ (e , A∗f ) (f ,e ) = (f , A∗f ) + (f , A∗f ) n 1 · 2 n n 3 · 4 n 2 1 4 3 n n X X = (Af ,f ) + (Af ,f ) < , (12) 2 1 4 3 ∞ because by the assumed properties of the operator A the vectors f ,f D 1 3 ∈ 1 are contained in the domain of A∗ and likewise the vectors f2,f4 D1 lie in the domain of A. Similarly we have: ∈

T T T T Tr [AB] = Tr A f ,f + f ,f = Tr A f ,f + Tr A f ,f 1 2 3 4 1 2 3 4 h T i h T h i = A f ,f en,en + A f ,f en,en 1 2 3 4 n n X X = (e ,f ) (Af ,e )+ (e ,f ) (Af ,e ) = (Af ,f ) + (Af ,f ) < . n 1 · 2 n n 3 · 4 n 2 1 4 3 ∞ n n X X (13)

Comparing (12) and (13) we obtain the tracial property. L M Now let U1 = Ux and U2 = Uy be densely deﬁned and closable Krein isometric operators of the respective Krein isometric induced representations of the groups G and G in = L and = M respectively with linear 1 2 H1 H H2 H domains Di i, i = 1, 2, equal to the corresponding domains D of Theorem ⊂ H L 1 and Remark 2 and with the respective fundamental symmetries J1 = J ,

37 M i J2 = J . Therefore by Theorem 1 and Remark 2 U (Di) = Di and Ji(Di) = Di, i = 1, 2, so that Di is contained in the domain of Ui∗ and Ui∗(D1) = Di. Finally let T,S be any finite rank operators in the linear subspace D12 = linear span Tf,g ,f D1,g D2 of finite rank operators supported in D2 and { ∈ ∈ } ⊛ with ranges in D . In particular for each S D , S is supported in D 1 ∈ 12 1 and has rank in D2. By the known property of Hilbert Schmidt operators D1 D2 = D12 is dense in 1 2. We claim that U1 U2 is well defined on D⊗ D = D . Indeed, byH the⊗ H Gram-Schmidt orthonormalization⊗ we may 1 ⊗ 2 12 construct an orthonormal base en n N of 1 with each en being an element of the linear dense domain D . For{ any} ∈f ,f H D and g ,g D we have 1 1 2 ∈ 1 1 2 ∈ 2 2 U U f g + f g 1 ⊗ 2 1 ⊗ 1 2 ⊗ 2 = U1 Tf ,g + Tf ,g U2∗ ,U1 Tf ,g + Tf ,g U2∗ 1 1 2 2 1 1 2 2 D ⊛ E ⊛ = Tr U1 Tf ,g U2∗ U1 Tf ,g U2∗ +Tr U1 Tf ,g U2∗ U1 Tf ,g U2∗ 1 1 1 1 1 1 2 2 h ⊛ i h ⊛ i +Tr U1 Tf ,g U2∗ U1 Tf ,g U2∗ +Tr U1 Tf ,g U2∗ U1 Tf ,g U2∗ 2 2 1 1 2 2 2 2 h i h i = U f ,e g ,U ∗U g U ∗e ,f 1 1 n · 1 2 2 1 · 1 n 1 n X + U f ,e g ,U ∗U g U ∗e ,f 1 1 n · 1 2 2 2 · 1 n 2 n X + U f ,e g ,U ∗U g U ∗e ,f 1 2 n · 2 2 2 1 · 1 n 1 n X + U f ,e g ,U ∗U g U ∗e ,f . 1 2 n · 2 2 2 2 · 1 n 2 n X Because D1 is in the domain of U1∗ and U1∗(D1)= U1(D1)= D1 and similarly for U2, the last expression is equal to

U f ,e U g ,U g e ,U f 1 1 n · 2 1 2 1 · n 1 1 n X + U f ,e U g ,U g e ,U f 1 1 n · 2 1 2 2 · n 1 2 n X + U f ,e U g ,U g e ,U f 1 2 n · 2 2 2 1 · n 1 1 n X + U f ,e U g ,U g e ,U f 1 2 n · 2 2 2 2 · n 1 2 n X = U f ,U f U g ,U g + U f ,U f U g ,U g 1 1 1 1 · 2 1 2 1 1 1 1 2 · 2 1 2 2 + U f ,U f U g ,U g + U f ,U f U g ,U g < , 1 2 1 1 · 2 2 2 1 1 2 1 2 · 2 2 2 2 ∞

38 so that

2 U U f g + f g 1 ⊗ 2 1 ⊗ 1 2 ⊗ 2 = U1 T + T U2∗ ,U1 T + T U2∗ < f1 ,g1 f2 ,g2 f1 ,g1 f2 ,g2 ∞ D E and U U f g + f g is well defined. By induction for each T D , 1 ⊗ 2 1 ⊗ 1 2 ⊗ 2 ∈ 12 U U T = U TU ∗ is well defined conjugate-linear operator of Hilbert- 1 2 1 2 Schimdt⊗ class, so that U U is well defined on the linear domain D dense in 1 ⊗ 2 12 1 2. By the Proposition of Chap. VIII.10, page 298 of [47] it follows that UH ⊗U H is closable. Next, let T,S D , then by Theorem 1 and Remark 2 1 ⊗ 2 ∈ 12

Ji(Di)= Di and Ui(Di)= Di and and Ui∗(Di)= Di

Ui † Ui = JiUi∗JiUi = I and Ui JiUi∗Ji = I on Di.

Thus for each T,S D 12 the following expressions are well deﬁned and (e. g. for T = T and S∈= T ) f1,g1 f2,g2

(J J ) (U U ) (f g ) , (U U ) (f g ) 1 ⊗ 2 1 ⊗ 2 1 ⊗ 1 1 ⊗ 2 2 ⊗ 2 ⊛ = J1U1TU2∗J2 ,U1SU2∗ = Tr J1U1TU2∗J2 U1SU2∗ D ⊛ E h ⊛ i = Tr J1U1TU2∗J2U2S U1∗ = Tr J1U1T J2J2U2∗J2U2S U1∗ h i ⊛ h ⊛ i = Tr J U T J J U ∗J U S U ∗ = Tr J U T J S U ∗ 1 1 2{ 2 2 2 2} 1 { 1 1} 2 1 h ⊛ i ⊛h i = Tr T J S U ∗ J U = Tr T J S J J U ∗J U 2 1 { 1 1} 2 1{ 1 1 1 1} h ⊛ i h ⊛ i = Tr T J2S J1 = Tr J1T J2S h i h i = (J J ) (f g ) , f g , 1 ⊗ 2 1 ⊗ 1 2 ⊗ 2 because the tracial property is applicable to the pair of operators

⊛ B = T J2S U1∗ and A = J1U1 as well as to the pair of operators

⊛ B = T J2S and A = J1, as both the operators B are linear finite rank operators supported on finite dimensional subspaces contained in D1 and with finite dimensional ranges contained in D1 and for the operators A indicated to above the linear domain D1 is contained in the common domain of A and A∗; and moreover J1(U1)∗J1U1 and J2U2∗J2U2 are well defined unit operators on the domains D1 and D2 re- L M spectively. Therefore U1 U2 = U U is Krein-isometric on its domain D12 which holds by continuity⊗ for its closure.⊗

39 We may therefore define the outer Kronecker product Krein-isometric representation U L U M : G G (x, y) U L U M of the product group G G , × 1 × 2 ∋ 7→ x ⊗ y 1 × 2 which is Krein isometric in the Krein space L M , JL JM . All the more, if U and U are Krein-unitary representationsH ⊗H of G and⊗G , respectively in 1 2 1 2 ( 1, J1) and ( 2, J2), so is U1 U2 in the Krein space 1 2, J1 J2 . SimilarlyH one easilyH verifies that ×U U is almost uniformlyH bounded⊗ H whenever⊗ 1 × 2 U1 and U2 are. In particular if G1 and G2 are two closed subgroups of the separable locally compact groups G1 and G2 respectively and L and M their Krein unitary and uniformly bounded representations, then we may define the outer Kronecker product representation L M of the product group G G by × 1 × 2 the ordinary formula G1 G2 (ξ, η) Lξ Mη, which is Krein unitary and almost uniformly bounded× in the∋ Krein7→ space⊗ , J J whenever HL ⊗ HM L ⊗ M L and M are in the respective Krein spaces ( L, JL) and ( M , JM ). We may H µ µ HL M therefore define the Krein-isometric representation 1× 2 U × of the group L M G1 G2 in the Krein space × induced by the representation L M of the closed× subgroup G G , whereH µ are the respective quasi invariant× measures 1 × 2 i in Gi/Gi. Let us make an observation used in the proof of the Theorem of this Section. Let B1 be a Borel section of G1 with respect to G1 and respectively B2 a Borel section of G2 with respect to G2 defined as in Section 3 with the associated Borel 1 functions h1 : G1 x h1(x) G1 such that h1(x)− x B1 and h2 : G2 ∋ 7→ ∈ 1 ∈ ∋ y h (y) G such that h (y)− y B . Then B B is a Borel section of 7→ 2 ∈ 2 2 ∈ 2 1 × 2 G1 G2 with respect to the closed subgroup G1 G2 with the associated Borel × × 1 function h : (x, y) h(x, y) G1 G2 such that h(x, y)− (x, y) B1 B2, 7→ ∈ × L M∈ × equal to h(x, y)= h1 (x) , h2 (y) = h1 (x) h2 (y). Let w × . Thus the L M × ∈ H corresponding operator J × acts as follows L M J × w = (L M) J (L M) w h (x)×h (y) L M h (x)−1×h (y)−1 (x,y) (x,y) × 1 2 ◦ × ◦ × 1 2

= Lh (x) Mh (y) JL JM L −1 M −1 w(x,y) 1 ⊗ 2 ◦ ⊗ ◦ h1 (x) ⊗ h2 (y)

= Lh (x) JLL −1 Mh (y) JM M −1 w(x,y) . 1 h1 (x) ⊗ 2 h2 (y) L M Thus the vector JL M J × w in the integrand in the formula for the inner × (x,y) L M product in × H L M (w,u)= JL M J × w , u d(µ1 µ2)([(x, y)]) × (x,y) (x,y) × G ZG 1× 2 may be written as follows

L M L M JL M J × w = JL JM J × w × (x,y) ⊗ ◦ (x,y) = J L J L J M J M w = JL JM w , L h (x) L h (x)−1 M h (y) M h (y)−1 (x,y) x y (x,y) 1 1 ⊗ 2 2 ⊗ where we have introduced the following self-adjoint operators JL = J L J L and JM = J M J M x L h (x) L h (x)−1 y M h (y) M h (y)−1 1 1 2 2

40 acting in L and M , respectively, with the ordinary tensor product operator L MH H x J y J acting in the tensor product L M Hilbert space. Checking⊗ their self-adjointness is immediate.H ⊗ H Indeed, because L is Krein unitary in ( , J ) we have (and similarly for the rep. M): HL L

† ∗ L −1 = Lh (x) = JL Lh (x) JL. h1 (x) 1 1 Therefore L ∗ x J = JLL L JL, h1 (x) h1 (x) 2 L because JL = I. Because JL is self-adjoint, self-adjointness of x J is now M immediate (self-adjointness of y J follows similarly). We are ready now to formulate the main goal of this Section: THEOREM 6. Let L and M be Krein-unitary strongly continuous and almost uniformly bounded representations of the closed subgroups G1 and G2 of the separable locally compact groups G1 and G2, respectively, in the Krein spaces µ µ L M ( , J ) and ( , J ). Then the Krein isometric representation 1× 2 U × HL L HM M of the group G1 G2 with the representation space equal to the Krein space L M L M × ( × , J × ) is unitary and Krein-unitary equivalent to the Krein-isometric H µ1 L µ2 M representation U U of the group G1 G2 with the representation space equal to the Krein space× ( L M , JL JM×). More precisely: there exists a L M HL M⊗ H ⊗ map V : × which is unitary between the indicated Hilbert H ⊗ H 7→ H spaces and Krein-unitary between the Krein spaces ( L M , JL JM ) and L M L M H ⊗ H ⊗ ( × , J × ) and such that H

1 µ µ L M µ L µ M V − 1× 2 U × V = 1 U 2 U . (14) ×

Let T be any member of L M , regarded as a conjugate-linear operator H ⊗H ⊛ from µ2 M into µ1 L, with the corresponding linear operator TT on µ1 L having finiteH trace.H Let moreover T be a finite rank operator. Then there existH µ1 L µ2 M f1 ,f2 ,...,fn and g1 ,g2 ,...,gn such that T (g) = Tf ,g (g)+ ∈ H ∈ H 1 1 ...+T (g)= f g , w +...+f g ,g . For each (x, y) G1 G2 we may fn ,gn 1 · 1 n · n ∈ × define a conjugate-linear finite rank operator V (T ) from M into L as (x,y) H H follows. Let υ M , then we put V (T ) (υ)= f1 g1 ,υ +...+fn gn ,υ . ∈ H (x,y) · · Note, please, that V (T ) =Lξ V(T ) (Mη)∗for all (x, y) G1 G2 (ξx,ηy) (x,y) ∈ × and all (ξ, η) G G , so that the function V (T ): G G (x, y) ∈ 1 × 2 1 × 2 ∋ 7→ V (T ) L M fulfils V (T ) = Lξ Mη V (T ) for all (x,y) ∈ H ⊗ H (ξx,ηy) ⊗ (x,y) (x, y) G1 G2 and all (ξ, η) G1 G2. ∈ × ∈ × L M We shall show that the function V (T ) is a member of × and moreover, that V is unitary. To this end we observe first, that V His isometric (for the ordinary definite inner products), i. e. V (T ) = T . Indeed, let ek k N be an orthonormal basis in . Using thek observationk k wek have made{ just} before∈ HL

41 L M the formulation of the Theorem, self-adjointness of the operators x J and y J and Scholium 3.9 and 5.3 of [51], we obtain:

T 2 = f g + ... + f g , f g + ... + f g k k 1 ⊗ 1 n ⊗ n 1 ⊗ 1 n ⊗ n n = Tr Tf ,g + ... + Tf ,g Tg ,f + ... + Tg ,f = (fi,fj ) (gi,gj ) 1 1 n n 1 1 n n · i,j=1 h i X n L M = JL J fi , fj dµ1([x]) JM J gi , gj dµ2([y]) x x · y y i,j=1 Z Z X G1 G2 n L M = JL J fi , fj JM J gi , gj d(µ1 µ2)([(x, y)]) x x · y y × G ZG i,j=1 1× 2 X n L M = J fi , fj J gi , gj d(µ1 µ2)([(x, y)]) x x x · y y y × G ZG i,j=1 1× 2 X n L M = x J fi , ek ek, fj y J gi , gj d(µ1 µ2)([(x, y)]) x · x · y y × i,j=1 k N Z X X∈ n M L = ek, fj gi , y J gj x J fi , ek d(µ1 µ2)([(x, y)]) x · y y · x × i,j=1 k N Z X X∈ n L M = ek, fj J fi gi , J gj , ek d(µ1 µ2)([(x, y)]) x · x x · y y y × i,j=1 k N Z X X∈ n L M = ek, fj x J T(f ) ,(g ) y J gj , ek d(µ1 µ2)([(x, y)]) x · ◦ i x i y ◦ y × Z i,j=1 k N X X∈

n L M = x J T(f ) ,(g ) y J T(g ) ,(f ) ek , ek d(µ1 µ2)([(x, y)]) ◦ i x i y ◦ ◦ j x j y × i,j=1 k N Z X X∈ n L M = Tr x J T(f ) ,(g ) y J T(g ) ,(f ) d(µ1 µ2)([(x, y)]) ◦ i x i y ◦ ◦ j x j y × i,j=1 Z X h i n ⊛ L M = Tr x J T(f ) ,(g ) y J T(f ) ,(g ) d(µ1 µ2)([(x, y)]) ◦ i x i y ◦ ◦ j x j y × Z i,j=1 X h i L M = x J y J V (T ) , V (T ) d(µ1 µ2)([(x, y)]) ⊗ (x,y) (x,y) ! × G ZG 1× 2

L M 2 = JL M J × V (T ) , V (T ) d(µ1 µ2)([(x, y)]) = V (T ) . × (x,y) (x,y) ! × k k G ZG 1× 2

42 (The unspecified domain of integration in the above formulas is of course equal L M G1 G2.) Therefore V is isometric and V (T ) × for the indicated T , as the required× measurability conditions again easily∈ Hfollow from Scholium 3.9 of [51]. Now by the properties of Hilbert-Schmidt operators, the finite rank conjugate- linear operators T : µ2 M µ1 L are dense in µ1 L µ2 M (compare e. g. [35], Chap. II or [51],H Chap.7→ 14.2H or [49]). Thus theH domain⊗ H of the operator V is dense. In order to show that the range of V is likewise dense, consider the closure 1 L M C under the norm in × of the linear set of all functions V (T ), where T = H L L Tf ,g +...+Tf ,g with fi ranging over C and gj over the corresponding 1 1 n n 0 M M ⊂ H 1 set C0 . Because V is isometric it can be uniquely extended so that C lies in the⊂H range of this unique extension. Let us denote the extension likewise by V . (For a densely defined Krein-isometric map this would in general be impossible because V could be discontinuous, this is the reason why we need to know if V is continuous, i. e. bounded for the ordinary positive definite inner products.) Now by the property of Hilbert-Schmidt operators (mentioned above) the linear span of operators T : with υ and v ranging over dense υ,v HM 7→ HL subsets of L and M , respectively, is dense in L M . This property of Hilbert-SchmidtH operatorsH together with a repeatedH application⊗ H of Lemma 2 and 5 of Sect. 4 and Scholium 3.9 and 5.3 of [51] will show that all the conditions, 1 L M (a)-(e), of Lemma 4 are satisfied for C × . Let π be the canonical quotient map⊂G H G (G G )/(G G ). In 1 × 2 7→ 1 × 2 1 × 2 particular if ψ is a complex valued continuous function on G1 G2 which is 1× constant on the right G G cosets and vanish outside of π− (K) for some 1 × 2 compact subset K of (G1 G2)/(G1 G2), then it is measurable and ψ 2 × × ∈ L ((G1 G2)/(G1 G2),µ1 µ2) and by Scholim 3.9 and 5.3 of [51] it is an 2 × × × L -limit of continuous such functions of “product form” φ ϕ : G1 G2 (x, y) φ(x) ϕ(y). Thus the condition (d) of Lemma 4 follows.· The× above∋ 7→ · L mentioned property of Hilbert-Schmidt operators and Lemma 2 applied to C0 L M M ⊂ and to C0 , proves condition (e) of Lemma 4. Condition (b) follows H ⊂ H L M from the the fact that V (T ) × for finite rank operators T , proved in the first part of the proof. An∈ H application of the Lusin Theorem (Corollary 5.2.2 of [51], together with an obvious adaptation of the the standard proof of the Riesz-Fischer theorem already used in the proof of Lemma 5) proves condition (a) of Lemma 4. By the remark opening the proof of Lemma 2 the L M linear sets C0 and C0 of functions are closed with respect to right G1 and G2- translations, respectively. Thus it easily follows that the linear set of functions L M V T +...+T with fi C , gj C is closed under the right G1 G2- f1 ,g1 fn ,gn 0 0 ∈ ∈ 1 × translations. Then, a simple continuity argument shows that C is closed under right G G -translations. Thus condition (c) of Lemma 4 is satisfied with 1 × 2 trivial functions ρs all equal identically to the constant unit function. Thus Lemma 4 may be applied to C1 lying in the range of V , so that the L M 1 L M range is dense in × . Therefore C = × and V is unitary. We shall showH that V is Krein-unitary.H By the unitarity of V , it will be

43 suﬃcient by continuity to show that V is Krein-isometric on ﬁnite rank operators T L M . By self-adjointness of JL and JM we have the following equalities for∈T Hof⊗H the form indicated to above:

2 T = JL JM f g +...+f g , f g +...+f g k kJL⊗JM ⊗ 1 ⊗ 1 n ⊗ n 1 ⊗ 1 n ⊗ n ⊛ = Tr JL T + ... + T JM T +... + T f1 ,g1 fn ,gn f1 ,g1 fn ,gn h n i = (JLf ,f ) (JM g ,g ) i j · i j i,j=1 X n = JL fi , fj dµ1([x]) JM gi , gj dµ2([y]) x x · y y i,j=1 Z Z X G1 G2 n = JL fi , fj JM gi , gj d(µ1 µ2)([(x, y)]) x x · y y × G ZG i,j=1 1× 2 X

= Tr JL T(f ) ,(g ) + ... + T(f ) ,(g ) JM T(f ) ,(g ) + ... 1 x 1 y n x n y 1 x 1 y G ZG 1× 2 h ⊛ ... + T d(µ1 µ2)([(x, y)]) (fn )x ,(gn )y × i = JL JM V (T ) , V (T ) d(µ1 µ2)([(x, y)]) ⊗ (x,y) (x,y) ! × G ZG 1× 2

= JL M V (T ) , V (T ) d(µ1 µ2)([(x, y)]) × (x,y) (x,y) ! × G ZG 1× 2 2 = V (T ) . k kJL×M Recall that the domain D (common for all (x, y) G G ) of the 12 ∈ 1 × 2 operators U U = µ1 U L µ2 U M = µ1 U L µ2 U M representing 1 ⊗ 2 x ⊗ y × (x,y) (x, y) G G , is invariant for the operators U U = µ1 U L µ2 U M = ∈ 1 × 2 1 ⊗ 2 x ⊗ y µ1 U L µ2 U M . For each (x, y) let us denote the closure of µ1 U L µ2 U M = × (x,y) × µ1 L µ2 M µ1 L µ2 M Ux Uy likewise by U U . Note that V (T ), T D12 compose ⊗ × µ µ L M ∈ an invariant domain of the representation 1× 2 U × . Denote the closures of µ1 µ2 L M the operators × U(x,y× ) with the common invariant domain V (D12) likewise µ1 µ2 L M by × U(x,y× ) . The equality (14) is regarded as equality for the closures of the operators µ µ L M µ L µ M 1× 2 U × and 1 U 2 U . × µ µ L M By Theorem 1 and its proof the closures of 1× 2 U × do not depend on the choice of the dense common invariant domain. Therefore in order to show the equality (14) it is suﬃcient that the respective closed operators in (14) coincide on the domain of all ﬁnite rank operators T D . This however is immediate. ∈ 12

44 Indeed, let T = Tf ,g + ... + Tf ,g with fi D1 and gj D2. Then 1 1 n n ∈ ∈

µ1 L µ2 M µ1 L µ2 M µ1 L µ2 M ∗ U U (T )= Ux Uy (T ) = Ux T Uy × (x0,y0) 0 ⊗ 0 0 0

= λ1( , x0) λ2( ,y0) TR f ,R g + ... + TR f ,R g . (15) · · x0 1 y0 1 x0 n y0 n On the other handp we have:p

µ µ L M 1× 2 U × V (T ) = λ ([x], x ) λ ([y],y ) V (T ) (x0,y0) 1 0 2 0 (x,y) (x·x0,y·y0) p p = λ1([x], x0) λ2([y],y0) T(R f ) ,(R g ) + ...... + T(R f ) ,(R g ) , x0 1 x y0 1 y x0 n x y0 n y p p so that

1 µ µ L M V − 1× 2 U × V T = λ1( , x0) λ2( ,y0) T + ... + T . · · Rx0 f1 ,Ry0 g1 Rx0 fn ,Ry0 gn p p Comparing it with (15) one can see that (14) holds on D12. Thus the proof of (14) is complete now. The Theorem is hereby proved completely. Presented proof of Theorem 6 is an extended and modified version of the Mackey’s proof of Theorem 5.2 in [31]. µ µ L M Note, please, that the equality (14) for the closures of the operators 1× 2 U × and µ1 U L µ2 U M is non trivial. Indeed, recall that in general almost all kinds of pathology× not excluded by general theorems can be shown to exist for unbounded operators. In particular two distinct and closed operators may still coincide on a dense domain. This is why we need to be careful in proving (14). This in particular shows that the fundamental theorems of the original Mackey theory by no means are automatic for the induced Krein-isometric representations, where the representors are in general densely defined and unbounded. Here we saw it for the Theorem 6. But differences in the proofs arise likewise in the latter part of the theory. In particular if we want to prove the subgroup theorem and the so called Kronecker product theorem for the induced Krein-isometric representations with precisely the same assumptions posed on the group as in Mackey’s theory, then some additional analysis will have to be made in treating decompositions of non finite quasi invariant measures. Compare Sect. 8.

7 Subgroup theorem in Krein spaces. Prelimi- naries

This Section is a word for word repetition of the argument of 6 of [31]. That the general Mackey’s argument may be applied to induced represen§ tations in Krein spaces is the whole point. Although it is rather clear that his general argumentation is applicable in the Krein space, we restate it here because it lies at the very heart of the presented method of decomposition of tensor product

45 of induced representations, and will make the paper self contained. It should be noted however that it requires some additional analysis in decomposing non finite quasi invariant measures, which makes a difference in proving the existence of the corresponding direct integral decompositions. The circumstance that theLopuszański representation of G is equivalent to an induced representation in a Krein space greatly simplifies the problem of decomposing tensor product ofLopuszański representations and reduces it largely to the geometry of right cosets and double cosets in the group G and to a “Fubini-like” theorem, just like for the ordinary induced representations of Mackey. Similar decomposition method of quotiening by a subgroup in construction of complete sets of unitary representations of semi simple Lie groups was applied by Gelfand and Neumark, and by several authors in constructing harmonic analysis on classical Lie groups. The main gain is that the subtle analytic properties of theLopuszański representation (unboundedness) does not intervene dramatically after this reduction to geometry of cosets and double cosets. Our main theorem asserts the existence of a certain useful direct integral decomposition of the tensor product U L U M of two induced representations of a group G in a Krein space, whose construction⊗ is completely analogous to that of Mackey for ordinary unitary representations, compare [31]. By definition U L U M is obtained from the outer Kronecker product representation U L U M ⊗ L M × of G G by restricting U U to the diagonal subgroup G ∼= G of all (x, y)× G G with x = y. By× the Theorem of Sect. 6, U L U M is Krein- ∈ × L M L M × unitary equivalent to U × . Thus U U can be analysed by analysing the L M ⊗ restriction of U × to the diagonal subgroup G ∼= G. Our theorem on tensor product decomposition follows (just as in [31]) from these remarks and a theorem on restriction to a subgroup of an induced representation in a Krein space, say a subgroup theorem. Subgroup theorem gives a decomposition of the restriction of an induced representation (in a Krein space) to a closed subgroup, with the component representations in the decomposition themselves Krein-unitary equivalent to induced representations. Namely, let L be strongly continuous almost uniformly bounded Krein-unitary representation of the closed subgroup L L G1 of G and consider the restriction G2 U of U to a second closed subgroup G2. While G acts transitively on the homogeneous space G/G1 of right G1- cosets this will not be true in general of G2. Moreover, and this is the main advantage of induced representations, any division of G/G1 into two parts S1 and S2, each a Baire (or Borel) set which is not a null set (with respect to any, and hence every quasi invariant measure on G/G1), and each invariant under L G2 leads to a corresponding direct sum decomposition of G2 U . Indeed the L L L closed subspaces S1 and S2 of all f which vanish respectively outside 1 H1 H ∈ H of π− (S1) and π− (S2) are invariant and are orthogonal complements of each other with respect to the ordinary (as well as the Krein) inner product on L. Assume for a while, just for illustrative purposes, that there is a nullH set N in G/G1 whose complement is the union of countably many non null orbits C1, C2,... of G/G1 under G2. Then by the above remarks we obtain a direct L sum decomposition of G2 U into as many parts as there are non null orbits.

46 Our analysis reaches its goal after analysing the nature of these parts. Analysis of these parts is our goal of the rest of this Section. In our paper we shall consider a more general case in which all of the orbits can be null sets and the sum becomes an integral and we have to use the von Neumann theory of direct integral Hilbert spaces [36]. Of course according to the definition given above (with S1 or S2 equal to a G2 orbit C in G/G1), L C will be zero dimensional whenever the orbit C is a null set. However it isH possible to reword the definition so that it always gives a non zero Hilbert space (with the respective Krein structure) and so that when C is not a null set this definition is essentially the same as that already given, compare [31], 6. L § Indeed note that when C is a non null set then C may be equivalently defined as follows. Let x be any member of G such thatH π(x ) C and consider the c c ∈ set L ′ of all functions f from the double coset G x G to such that: (i) HC 1 c 2 HL x (fx,υ) is a Borel function for all υ L, (ii) fξx = Lξ(fx) for all ξ G1 and7→ all x G x G and (iii): ∈ H ∈ ∈ 1 c 2

f = ( J ((JLf) ),f ) dµ = (f ,f ) dµ (b) < , k kC L x x G/G1 b b B ∞ CZ (G x ZG ) B 1 c 2 ∩ where B is the regular Borel section of G with respect to G1 of Sect. 3 (we could use as well the sub-manifold Q of Sect. 3 but we prefer to proceed generally and independently of the “factorization” assumption). The operator JL in L HC is given by simple restriction, and its deﬁnition on L ′ is obvious: HC L,C (J f)x = Lh(x)JLLh(x)−1 fx; with the obvious deﬁnition of the Krein inner product in L ′ HC

f,g = (JL,C f,g)= ( J (f ),g ) dµ , f,g L ′. JL,C L x x G/G1 ∈ HC Z C Similarly we define the operator U L,C in L for ξ G as the restriction of U L ξ HC ∈ 2 ξ to L , i. e. to the functions supported by the orbit C, and its definition giving HC an equivalent representation on L ′ is likewise obvious: HC L,C (Uξ f)x = λ([x], ξ) fxξ, p with the λ-function of the quasi invariant measure µ restricted to C G2. Moreover, and this is the whole point, the measure in C need not× be defined by restricting µ = µG/G1 to C. There exists a non zero measure µC on C quasi invariant with respect to G2 determined up to a constant factor, whose Radon-Nikodym function d(RηµC )/dµC , η G2 (i. e. the associated λC - function) is equal to the restriction to the subspace∈ C G of the λ-function, × 2 i. e. Radon-Nikodym derivative d(Rηµ)/dµ , associated with µ = µG/G1 . Indeed, although C does not have the form of a quotient of a group by its

47 closed subgroup, it follows from Theorem 3, page 253 of [27] that the map x π(xcx) induces a Borel isomorphism ψ of the quotient space G2/Gxc onto 7→ 1 C, where Gxc = G2 (xc− G1xc) is the closed subgroup of all x G2 such that π(x x) = π(x ).∩ Thus C G G /G G as Borel spaces∈ under c c 2 ∼= 2 xc 2 the indicated isomorphism and× moreover if [x] ×G /G and [z] = π(x x) ∈ 2 xc c correspond under this isomorphism and η G2 then [x]η and [z]η do also, where [x]η = [xη] and [z]η = [zη] denote the∈ action of η G on [x] G /G ∈ 2 ∈ 2 xc and [z] C respectively. Thus the existence of the quasi invariant measure µC on C follows∈ from the general Mackey classification of quasi invariant measures on the quotient of a locally compact group by a closed subgroup, compare the respective Theorem of Sect. 3. Using the quasi invariant measure µC on C gives L ′ a non trivial space C for every orbit C, which in case of a non null orbit C is trivially equivalentH to L . HC REMARK. Recall that Borel ψ is a Borel isomorphism with respect to the Borel structure on C induced from the surrounding space G/G1: we define E C to be Borel iff E = E′ C for a Borel set E′ in G/G . However ⊂ ∩ 1 our assumptions concerning the group G and the subgroups G1 and G2 are exactly the same as those of Mackey, and they do not even guarantee the local compactness of the orbits C, compare Sect. 8.

We are now in a position to formulate the main goal of this Section:

LEMMA 8. Let C be any orbit in G/G1 under G2 and let xc be such that xc xc L ′ µ L π(xc) C. Let C be deﬁned as above. Let U be the representation of G2 induced∈ by the stronglyH continuous almost uniformly bounded Krein-unitary rep- xc 1 −1 resentation L : η Lxcηxc of G2 (xc− G1xc) with the representation space xc 7→ ∩ of L equal to Lxc = L and the fundamental symmetry JLxc = JL; and with H H x 1 the quasi invariant measure µ c in the homogeneous space G /(G (x − G x )) 2 2∩ c 1 c equal to the transfer of the measure µC in C over to the homogeneous space x x by the map ψ. Let µ c L c be the Krein space of the induced representation x x x x µ c U L c . We assume theH fundamental symmetry J in µ c L c to be deﬁned xc H −1 by the equation (Jxc g)t = Lh(xct)JLLh(xct) gt and the Krein inner product given by the ordinary formula

(J f˜, f˜) dµxc ([t]), t G . L t t ∈ 2 Z G G (x −1G x ) 2 2∩ c 1 c

xc xc L ′ µ L Then there is a Krein-unitary map Vxc of C onto such that if g xc xc xc H xc H ∈ µ L corresponds to f L ′ then µ U L g corresponds to U L,Cf where H ∈ HC s s (U L,Cf) = f λ ([x],s) for all x C and all s G . s x xs C ∈ ∈ 2 For eachp function f on G1xcG2 satisfying the conditions (i) and (ii) of L ′ ˜ ˜ ˜ the deﬁnition of C let f be deﬁned by ft = fxct for all t G2. Then (ft,υ) is H ∈ 1 a Borel function of t on G for all υ . If η G = G (x − G x ) then 2 ∈ HL ∈ xc 2 ∩ c 1 c

48 1 ˜ ˜ −1 ˜ −1 ˜ if ξ = xcηxc− we have fηt = fxc ξxct = fξxct = Lξft = Lxcηxc (ft); that is

˜ −1 ˜ fηt = Lxcηxc (ft) (16)

1 for all t G and all η G (x − G x ). Conversely let g be any func- ∈ 2 ∈ 2 ∩ c 1 c tion from G2 to L which is Borel in the sense that x (gx,υ) is a Borel function on G forH all υ and which satisfies (16).7→ We define the cor- 2 ∈ HL responding function f by the equation fξxct = Lξ(gt) for all ξ G1 and 1 1 1 ∈ t G . If ξ x t = ξ x t then ξ − ξ = x t t − x − so that g −1 = ∈ 2 1 c 1 2 c 2 2 1 c 2 1 c t2t1 t −1 Lξ2 ξ1 (gt). Therefore Lξ2 (gt2 )= Lξ1 (gt1 ) and f is well defined. Next we show that (f ,υ) is Borel function of x on G x G for all υ . Let f ′ be the x 1 c 2 ∈ HL function on G G defined by f ′(ξ, η) = L (g ) for all (ξ, η) G G . 1 × 2 ξ η ∈ 1 × 2 Choose now an orthonormal basis ϕi i N in L. Then we have (f ′(ξ, η),υ)= { } ∈ H (f ′(ξ, η), JLJLυ) = (JLf ′(ξ, η), JLυ) = (JLLξ(gη), JLυ) = (JLgη,Lξ−1 JLυ) = i∞=1 = (JLgη, ϕi)(ϕi,Lξ−1 JLυ). By Scholium 3.9 of [51] (f ′(ξ, η),υ) is a Borel function of (ξ, η) on G1 G2 regarded as the product measure space, for all υP . Let us introduce× after Mackey a new group operation in G G ∈ HL 1 × 2 putting (ξ1, η1)(ξ2, η2) = (ξ1ξ2, η2η1) and call the resulting group G3. Then 1 1 1 ξ1xcη1 = ξ2xcη2 if and only if (ξ2, η2)− (ξ1, η1) = (ξ2− ξ1, η1η2− ) has the 1 1 1 1 form (ξ, xc− ξ− xc). The set of all (ξ, xc− ξ− xc), ξ G1 is a subgroup G of G . Thus the map (ξ, η) ξx η sets up a one-to-one∈ correspondence 4 3 7→ c between the points of the homogeneous space G3/G4 of left G4-cosets and the points of the double coset G1xcG2. The map is continuous and on account of the assumed separability it follows again from Theorem 3, page 253 of [27] that the map sets up a Borel isomorphism. Moreover the function (ξ, η) (f ′(ξ, η),υ) is constant on left G4-cosets in G3, as an easy computation 7→ 1 1 shows that (f ′((ξ, η)ω ),υ) = (f ′(ξ, η),υ) for all ω = (ξ , x − ξ − x ) G . 0 0 0 c 0 c ∈ 4 Therefore (ξ, η) (f ′(ξ, η),υ) defines a function on G /G which by Lemma 7→ 3 4 1.2 of [31] must be Borel because (ξ, η) (f ′(ξ, η),υ) itself is Borel on G3. That (f ,υ) is a Borel function of x G x7→G now follows from the fact that the x ∈ 1 c 2 mapping of G3/G4 onto G1xcG2 is a Borel isomorphism and preserves Borel sets. Finally observe that f˜ = g. Therefore f f˜ is a one-to-one map of 7→ functions satisfying (i) and (ii) of the definition of L ′ onto Borel functions HC satisfying (16). Consider the mapping t π(xct) of G2 onto C. It defines 7→ 1 one-to-one and Borel set preserving map ψ from G /(G (x − G x )) onto 2 2 ∩ c 1 c C and such that if [t] = π′(t) and [z] = π(z) correspond under the map ψ and η G2 then [x]η and [z]η do also (π′ stands for the canonical projection ∈ 1 G2/(G2 (xc− G1xc)) G2). Finally z (JLfz,fz) and t (JLf˜t, f˜t) de- ∩ 7→ 7→ ˜ 7→˜ fine functions π(z) (JLfπ(z),fπ(z)) and π′(t) (JLfπ′(t), fπ′(t)) on C and 1 7→ 7→ G /(G (x − G x )) respectively which correspond under the same map ψ: 2 2 ∩ c 1 c ′ ′ ˜ ′ ˜ ′ (JLfψ(π (t)),fψ(π (t))) = (JLfπ(xct),fπ(xct)) = (JLfxct,fxct) = (JLfπ (t), fπ (t)). If we use this same map ψ to transfer the measure µC on C over to the homo- 1 x geneous space G /(G (x − G x )) we will get a quasi invariant measure µ c 2 2 ∩ c 1 c

49 there such that

(JLfz,fz) dµC([z]) = (JLf[z],f[z]) dµC ([z]) CZ CZ

˜ ˜ xc (JLfψ([t]),fψ([t])) dµC (ψ([t])) = (JLf[t], f[t]) dµ ([t]) CZ G /(G (Zx −1G x )) 2 2∩ c 1 c

xc = (JLf˜t, f˜t) dµ ([t]).

G /(G (Zx −1G x )) 2 2∩ c 1 c Thus by the polarization identity (compare e. g. [51], 8.3, page 222 or [2], ˜ § page 4) the map f f sets up the Krein-unitary transformation Vxc demanded 7→ L,C 1 µxc Lxc by the Lemma as the verification of Vxc Us Vx−c = Us , s G2, and L,C 1 ∈ Vxc J Vx−c = Jxc is almost immediate as Vxc is bounded, which we show below in Lemma 9. Similarly verification that Jxc Jxc = I and that Jxc is self adjoint with respect to the definite inner product

˜ ˜ xc (f, g˜)x0 = JL(Jx0 ft), g˜t dµ ([t]) (17)

−1 G2/(G2 (Zxc G1xc)) ∩

x x in the Hilbert space µ c L c , is likewise immediate. H Note that in general the norm and topology induced by the inner product

(17) deﬁned by Jxc is not equivalent to the norm

xc f˜ 2 = (f,˜ f˜)= J (JL f˜), f˜ dµxc ([t]) k k L t t Z G G (x −1G x ) 2 2∩ c 1 c

x and topology deﬁned by the ordinary fundamental symmetry JL c of Sect. 3 1 (of course with G and H replaced with G and G (x − G x )): 2 2 ∩ c 1 c xc L ˜ xc xc ˜ J ft = L JLL −1 ft, hxc (t) hxc (t)

1 where hxc (t) G2 (xc− G1xc) is deﬁned as in Remark 2 by a regular Borel ∈ ∩ 1 section B of G with respect to the subgroup G (x − G x ). However xc 2 2 ∩ c 1 c if for each t G2, h(xct) Gxc , then the two topologies coincide. Similarly ∈ ∈ 1 whenever the homogeneous space G2 G2 (xc− G1xc) is compact then the two topologies coincide (but this case is not∩ interesting).

LEMMA 9. The operators Vxc of the preceding Lemma are also isometric with xc xc L ′ µ L respect to the norms C in C and xc = ( , )xc in , where k·k H k·k · ·xc H ( , ) is deﬁned as by (17), giving the norm in µxc L induced by J . In xc p xc particular· · we have V =1 for all x . H k xc k c

50 1 Denote the subgroup G (x − G x ) by G . The Lemma is an imme- 2 ∩ c 1 c xc diate consequence of deﬁnitions of C , Vxc and (17) giving the norm in x x µ c L c : k·k k·k H V f 2 = (f,˜ f˜) = J (J f˜) , f˜ dµxc ([t]) k xc kxc xc L xc t t Z G2/Gxc 1 L,C 1 1 x = J (V − J V V − f) , (V − f) dµ c ([t]) L xc xc xc t xc t (18) Z G2/Gxc

1 L,C 1 xc = JL(Vx−c J f)t, (Vx−c f)t dµ ([t]) Z G2/Gxc and because Vxc is Krein-unitary, i. e. isometric for the Krein inner products

J ( ) , ( ) dµ ([z]) and J ( ) , ( ) dµxc ([t]), L · z · z C L · t · t CZ Z G2/Gxc the last integral in (18) is equal to

J (JL,Cf) ,f dµ ([z]) = f 2 . L z z C k kC Z C

Note, please, that the Lemmas of Sect. 4, i. e. Lemmas 1 – 6, are equally Lxc Lxc applicable to the Krein space ( , Jxc ), with J replaced by Jxc , and with H 1 the section Bxc replaced with the image of G2 G2 (xc− G1xc) under the inverse of the map t x t. We formulate this remark∩ as a separate 7→ c x LEMMA 10. The Lemmas 1 – 6 are true for the Hilbert space L c of the Lxc x H c x Krein space ( , Jxc ), i. e. with L replaced by L , JL replaced by JL c = JL, H L Lxc x Lreplaced with L c = L, J = J replaced by Jxc and ﬁnally with the H H H 1 section Bxc replaced with the image of G2 G2 (xc− G1xc) under the inverse of the map t x t. ∩ 7→ c The proofs remain unchanged.

x x x In Subsection 9 we explain why we are using J in µ c L c instead of JL c . xc H 8 Decomposition (disintegration) of measures

Let G, G1 and G2 be such as in Sect. 7. Because the base of the system of neighbourhoods of unity in G is countable, the uniform space X = G/G1 is metrizable (compare e. g. [56], 2) for any closed subgroup G G. The § 1 ⊂

51 right action of G on G is proper and the quotient map π : G G/G is 1 7→ 1 open, so that the space X = G/G1 of right G1 orbits (G1 cosets) automatically has the required regularity: measurability of the equivalence relation defined by the G1 orbits. In particular the quotient space X is Hausdorff, separable and locally compact and the measure ρ µ (with the ρ-function of Sect. 3 · 0 and right Haar measure µ0 on G) is decomposable into a direct integral of measures ρ µ = β dµ([x]) with the component measures β of the · 0 [x] [x] G/G1 decomposition concentratedR in the G1 orbit (right coset) [x] and with Radon- Nikodym derivative associated with the action of the subgroup G1 (i. e. λ[x]- function) corresponding to β[x] equal to the restriction to the orbit [x] and to the subgroup G1 of the Radon-Nikodym (i. e. λ-function) corresponding to the measure ρ µ . This in particular gives us the quasi invariant regular Baire (or · 0 Borel) measure µ = µG/G1 on the uniform space X corresponding to ρ, i. e. the factor measure of ρ µ (Mackey’s method of constructing general regular quasi · 0 invariant measure on the quotient space X = G/G1). This is not the case if we replace G with X = G/G1 acted on by a second closed subgroup G2 G. The quotient space X/G2 is in general a badly behaved non Hausdorff space⊂ with non measurable equivalence relation defined in X with the G2 orbits as equivalence classes. We require a regularity condition in order to achieve an effective tool for constructing effectively a dual of the group G in question with the help of decomposition of tensor product of induced representations. Let X, for example X = G/G1, be any separable locally compact metrizable space with an equivalence relation R in X, for example with the equivalence classes given by right G2-orbits in X = G/G1 under the right action of a second closed subgroup G G. Let the equivalence classes form a set C and for each 2 ⊂ x X let πX(x) C denote the equivalence class of x. Let X be endowed with a ∈ ∈ regular measure µ (quasi invariant in case X = G/G1). We define following [48] the relation R to be measurable if there exists a countable family E0, E1, E2,... 1 of subsets of C such that πX− (Ei) is a Baire (or Borel) set for each i and such 1 that µ(πX− (E0)) = 0, and such that each point C of C not belonging to E0 is the intersection of the Ei which contain it. Strictly speaking in Rohlin’s definition of measurability of R, accepted by Mackey in [31], the set E0 is 1 empty and πX− (Ek), k 1, are just µ-measurable and not necessary Borel. But the difference is unessential≥ as we explain below in this Sect.. Under this assumption of measurability µ may be decomposed (disintegrated) as an integral µ = µC dν(C) over C of measures µC , with each µC concentrated on the C correspondingR equivalence class C, i .e G2 orbit in case X = G/G1, with a regular measure ν = µX/G2 on C = X/G2 i. e. the factor measure of µ, which we may call the “double factor measure” µ(G/G1)/G2 of µ0 = µG in case X = G/G1; and moreover in this case when X = G/G1 the Radon-Nikodym derivative (i. e. λC -function) corresponding to µC and associated with action of the subgroup G2 is equal to the restriction to the orbit C and to the subgroup G2 of the Radon Nikodym derivative (λ-function) corresponding to µ. In this

52 case we say after Mackey that the subgroups G1 and G2 are regularly related. In short: the orbits in G/G1 under the right action of G2 form the equivalence classes of a measurable equivalence relation. (Using literally Rohlin’s definition of measurability: almost all of the orbits in G/G1 under the right action of G2 form the equivalence classes of a Rohlin-measurable equivalence relation.) Let us explain the meaning of the regularity condition. Even if G1 and G2 were not regularly related we could of course find a countable set E1, E2,... of Borel unions of orbits which generate the σ-ring of all measurable unions of orbits. The unique equivalence relation R such that x G/G and y G/G are ∈ 1 ∈ 1 in the relation whenever x and y are in the same sets Ej will be measurable. This equivalence relation gives us a decomposition of the quasi invariant measure µ into quasi invariant component measures µP concentrated on subsets P C, but in this general non regular situation the subsets P are unions of many⊂ L orbits C C. This would give us decomposition of U restricted to G2, but in this decomposition∈ the component representations will not be associated with single orbits, i. e. with single double cosets G1x0G2 and will not be identifiable Lcc as “induced representations” U of G2 of Lemma 8 of Sect. 7. Little or nothing is known of such component representations related to non transitive systems of imprimitivity. In fact the regularity of the G2-orbits in G/G1 is essentially equivalent for the group G to be of type I. Because of the bi-unique correspondence between G2 orbits in G/G1 and double cosets G1xG2 in G, and because of the relation between Borel structures on X = G/G1 and on X/G2, we may reformulate the regularity condition as follows. We assume that there exists a sequence E0, E1, E2,... of measurable subsets of G each of which is a union of double cosets such that E0 has Haar measure zero and each double coset not in E0 is the intersection of the Ej which contain it (compare Lemma 20).

x REMARK. In fact the representations U L c of Lemma 8 of Sect. 7 do not have the standard form of induced representations deﬁned in Sect. 3 as Jxc = x JL c , but in relevant cases of representations encountered in QFT they may be6 shown to be Krein-unitary equivalent to standard induced representations (in the sense of Sect. 3). Anyway they are concentrated in single orbits. Note also that One may characterise the space of orbits by considering the respective group algebra or the associated universal enveloping C∗algebra. Connes developed a general theory of cross-product C∗-algebras and von Neumann algebras associated with foliations, strongly motivated by the Mackey theory of induced representations, compare [6] and references there in.

EXAMPLE 1. The equivalence relation on the two-torus X = R2/Z2 given by the leaves of the Kronecker foliation associated to an irrational number θ, i. e. given by the diﬀerential equation dy = θdx, is not measurable. The leaves, i. e. equivalence classes, can be viewed as orbits

53 of the additive group R on the two-torus X = R2/Z2.

′ In the original Mackey’s theory the induced representations µU L and µ U L are unitary equivalent whenever the quasi invariant measures µ and µ′ on G/G1 are equivalent, which is always the case, as all such measures are equivalent. In this case we may assume all measures µ in the induced representations µU L to be finite without any lost of generality. In particular, and this simplifies matter, we may restrict ourself to finite measures µ on G/G1, as Mackey did in [31], in constructing decomposition (disintegration) µ = µC dν(C) with each C of the measures µC concentrated on the corresponding orbitR C and the corresponding Radon-Nikodym derivative associated with µC under the action of G2 equal to the restriction to the orbit C and to the subgroup G2 of the Radon Nikodym derivative associated with µ (this is proved in 11 of [31]). This is not ′ the case for the induced representations µU L and µ U L §in Krein spaces defined here, as already indicated by Theorem 3 and its proof: they are (Krein-unitary) inequivalent whenever the quotient space G/G1 is not compact and the Radon- Nikodym derivative dµ′/dµ is not “lower” or “upper” bounded. Therefore we cannot restrict ourself to finite measures µ in construction of the decomposition µ = µC dν(C) with the above mentioned properties. Because Mackey’s C constructionR of decomposition of finite measure µ is sufficient for the theory of unitary group representations (as well as for the extension of the construction of induced representation to representations of C∗-algebras along the lines proposed by Rieffel) decomposition having the above mentioned properties of a quasi invariant measure µ which is not finite has not been constructed explicitly in the classical mathematical literature, at least the author was not able to find it (in the Bourbaki’s course on integration [4], Chap. 7.2.1-7.2.3 decomposition of this type is constructed but under stronger assumption than measurability of the equivalence relation given by right G2 action on X = G/G1 where it is assumed instead that the action is proper and moreover where it is assumed that the measure µ is relatively invariant and not merely quasi invariant – assumptions too strong for us). Because the required decomposition of not necessary finite quasi invariant measure µ on G/G1 is important for the decomposition of the restriction of the induced representation µU L in a Krein space to a closed subgroup (and a fortiori to a decomposition of tensor product of induced representations µU L and µU M in Krein spaces) we present here its construction explicitly only for the sake of completeness. The construction presented here uses a localization procedure in reducing the problem of decomposition to the Mackey-Godement decomposition ([31], 11) of a finite quasi invariant measure. § REMARK. Whenever the action of G2 on X = G/G1 is proper one can just replace the continuous homomorphism χ : G R in [4], Chap. 7.2.1-7.2.3, 2 7→ + by the Radon-Nikodym derivative associated with the measure µ on X = G/G1 in this case. Using the Federer and Morse theorem [14] one constructs a regular Borel section of X with respect to G2 which enables the construction of the

54 factor measure ν on the quotient C = X/G2 of the space X by the group G2 with the method of [4] changed in minor points only. Let X be the separable locally compact metrizable (in fact complete metric) space G/G1 equipped with a regular quasi invariant measure µ. Let R be the equivalence relation in X given by the right action of a second closed subgroup G with the associated quotient map πX : X X/R = X/G , and let K be a 2 7→ 2 compact subset of X. There is canonically defined equivalence relation RK on K induced by R on K with the associated quotient map π : K K/R equal K 7→ K to the restriction of πX to the subset K. Note please that for an equivalence relation R in the separable locally compact and metrizable space X = G/G1 the above mentioned (Rohlin’s [48]) condition of measurability of R is equivalent to the following condition: the family K of those compact sets K X for which ⊂ the quotient space K/RK is Hausdorff is µ-dense, i. e. one of the following and equivalent conditions is fulfilled: (I) For a subset A X to be locally µ-negligible it is necessary and sufficient that µ(A K) =⊂ 0 for all K K. ∩ ∈ (II) For any compact subset K0 of X and for any ǫ > 0 there exists a subset K K contained in K and such that µ(K K) ǫ. ∈ 0 0 − ≤ (III) For each compact subset A of X there exists a partition of A into a µ- negligible subset N and a sequence Kn n N of compact subsets belonging to K. { } ∈ (IV) For each compact subset K of X there exists an increasing H H ... 1 ⊆ 2 ⊆ sequence Hn n N of compact sets belonging to K contained in K and { } ∈ such that the set Z = K Hn is µ-negligible. − n N S∈ Indeed, because the the system of neighbourhoods of unity in G is countable, the uniform space X = G/G1 is completely metrizable and locally compact (compare e. g. [56], 2) for any closed subgroup G1 G. Therefore Propo- sition 3 of [3], Chap.§ VI, 3.4, is applicable. By this⊂ Proposition we need § only show that using the family K one can construct the sets E0, E1,... of the 1 Rohlin’s measurability condition of R, for which πX− (Ek), k 1, are not only µ-measurable but moreover Borel. This however follows from≥ the fact that X is countable at infinity: there exists a sequence of compact subsets K1 K2 ... of X such that X = K and moreover we may assume that they⊂ are regular⊂ ∪i i closed sets: cl int Km = Km. Indeed, let k k N be a countable base of the topology in X, such that the closure of each{O } ∈ is compact (there exists such a base because X is second Ok Ok countable and locally compact). For each k choose a sequence Kkl l N of O { } ∈ compact sets belonging to K and a µ-negligible subset Mk giving the partition k = Mk ˙ Kk1 ˙ Kk2 ˙ ... of k, existence of which is assured by the condition O(III). Define∪ the∪ µ-negligible∪ O set M = M and a maximal subset M of M ∪k k 0 invariant under the action of G2 on X.

55 By the condition (IV) we can construct for each K a sequence H m m1 ⊂ Hm2 Hm3 ... of compact subsets of Km belonging to K and a µ-negligible subset⊂Z such⊂ that K = Z ˙ K . Define the µ-negligible set Z = m m m∪ ∪n mn Zm and the maximal subset Z0 of Z invariant under the action of G2. n N ∈ S Let us define a countable family of sets E0 = πX(Z0 M0), Emn = πX(Kmn)= N ∪ πKmn (Kmn) = Kmn/RKmn , m,n in X/G2 = X/R, where Kmn/RKmn is Hausdorff by assumption. ∈ Now let x and x be two elements of X not in N = Z M such that 1 2 0 0 ∪ 0 πX(x ) = πX(x ). Then by construction there exists H K containing x′ 1 6 2 mn ∈ 1 and x′2 with πX(x′1)= πX(x1) and πX(x′2)= πX(x2).

Hmn/RHmn containing πX(x1) = πX(x′1) and πX(x2)= πX(x′2) is Hausdorﬀ by construction. Thus there exist two compact non intersecting neighbourhoods ′ ′ ′ ′ x 1 and x 2 of x′1 and x′2 respectively such that for Kx 1 = x 1 Hmn and O O 1 1 O ∩ Kx′ = x′ H we have π− (Kx′ ) π− (Kx′ ) = . By construction we 2 O 2 ∩ mn X 1 ∩ X 2 ∅ may choose Km1n1 Kx1 and Km2n2 Kx2 in K such that x′1 Km1n1 and ⊂ ⊂ 1 ∈1 x′ K . Of course we have E E = π− (K ) π− (K )= 2 ∈ m2n2 m1n1 ∩ m2n2 X m1n1 ∩ X m2n2 . Thus the intersection of all Emn K containing πX(x1) X/G2 is equal ∅ ∈1 1 ∈ πX(x1) . We have to show that πX− (Emn) = πX− (πX(Kmn)) are Baire (or { } 1 Borel) sets. To this end observe please that πX− (πX(Kmn)) is equal to the 1 saturation of Kmn, i. e. πX− (πX(Kmn)) = Kmn G2. Choose a compact 1· neighbourhood V of the unit in G2 such that V = V − . Then if G2 is connected n then G2 = V ; if G2 is not connected then it is still a countable sum of n N ∈ n connected componentsS of the form V ηm, with ηm G2 chosen from m-th n N ∈ ∈ connected component G2m of G2.S Thus in each case G2 is a countable sum 1 Vkl of compact sets Vkl. Therefore πX− (Emn)= Kmn G2 = Kmn Vkl k,l N · k,l N · beingS∈ a countable sum of compact sets is contained in the σ-ringS∈ generated by the compact sets and all the more it is a Borel set contained in the σ-ring generated by the closed sets. Thus both deﬁnitions of measurability of the equivalence relation R on X are equivalent.

LEMMA 11. There exists a Borel set B0 in X = G/G1 and a µ-negligible subset N X consisting of G orbits in X = G/G such that B intersects 0 ⊂ 2 1 0 each G2 orbit not contained in N0 in exactly one point. For the proof compare e. g. [3], Chap. VI, 3.4, Thm. 3. §

Adding to B0 any section of the µ-negligible set N0 we obtain a measurable section B00 for the whole space X. For equivalence relations R on smooth manifold X deﬁned by foliations on X (i. e. smooth and integrable sub-bundles of T X) existence of a measurable section is equivalent for the foliation to be of type I: i. e. the von Neumann algebra associated to the foliation is of type I iﬀ the foliation admits a Lebesgue measurable section, compare [6], Chap. I.4.γ, Proposition 5.

56 Because the Borel space X = G/G1 is standard it follows by the second Theorem on page 74 of [34] that the quotient Borel structure on X/G2 N0/G2 is likewise standard; i. e. there exits a Borel isomorphism ψ : (X N )/G− S 0 − 0 2 → 0 ⊂ onto a Borel subset S0 of a complete separable metric space S. The space (X N )/G however need not be locally compact and it is not − 0 2 if the action of G2 on X = G/G1 is not proper but only measurable, i.e. with measurable equivalence relation determined by the action of G2. Similarly G2- orbit C in X as a subset of a locally compact space X need not be closed if the action of G2 is not proper and thus need not be locally compact with the topology induced from the surrounding space X.

LEMMA 12. Let N0 be as in Lemma 11. A necessary and sufficient condition 1 that a subset E of X/G2 N0/G2 be a Borel set is that πX− (E) be a Borel set in X N . A necessary and− sufficient condition that a function f on X/G N /G − 0 2 − 0 2 be a Borel function is that f πX be a Borel function on X N . ◦ − 0 Let p0 be the Borel function ψ0 πX : X N0 S0. Let E′ be any 1 ◦ − → subset of S0 such that p0− (E′) is a Borel set. Let B0 be the Borel section of X N0 with respect to G2, existence of which has been proved in Lemma 11. − 1 Then p (p− (E′) B ) = E′, and thus E′ is a Borel set by Theorem 3, page 0 0 ∩ 0 253 of [27], compare likewise the Theorems on pages 72-73 of [34], because p0 is one-to-one Borel function on B0. Conversely: if E′ is Borel in S0 then because 1 p0 is a Borel function, so is the set p0− (E′). The first part of the Lemma follows now from this and from definition of the Borel structure induced on ψ0((X N0)/G2) and a fortiori on X/G2 N0/G2. The remaining part of the Lemma− is an immediate consequence of the− first part.

We have the following disintegration theorem for the (not necessarily ﬁnite) measure µ and any of its pseudo image measures ν on X/G2 (for deﬁnition of pseudo image measure ν compare e. g. [3], Chap. VI.3.2):

1 LEMMA 13. For each orbit C = π− (d ) X with d X/G there exists X 0 ⊂ 0 ∈ 2 a Borel measure µC in X concentrated on the orbit C, i. e. µC (X C) = 1 1 − µ (X π− (d ))=0. For any g L (X,µ) the set of all those G orbits C for C − X 0 ∈ 2 which g is not µC -integrable is ν-negligible and the function

C g(x)dµ (x) 7→ C Z is ν-summable and ν-measurable, and

dν(C) g(x) dµC (x)= g(x) dµ(x). (19) Z Z Z In short

µ = µC(x) dν(C). Z

57 REMARK 3. For each orbit C the measure µC may also be naturally viewed as a measure on the σ-ring RC of measurable subsets of C induced from the surrounding space X: E RC iﬀ E = E′ C for some E′ RX, i. e. with the subspace Borel structure.∈ ∩ ∈ For the proof we refer the reader e. g. to [3], Chap. VI, 3.5. §

We shall show that for each C the measure µC is quasi invariant and that d(RηµC ) for all η G2 the Radon-Nikodym derivative λC ( , η) = ( ) is equal ∈ · dµC · d(Rη µ) to the restriction of the Radon-Nikodym derivative λ( , η) = dµ ( ) to the orbit C. In doing so we prefer reducing the problem to· the Mackey-Godeme· nt decomposition of a ﬁnite measure ([31], 11) using a localization of the measure § space (X, RX,µ) and its disintegration. Toward this end we need some further Lemmas.

LEMMA 14. Let µ, µC and ν be as in the preceding Lemma. Let K be a compact subset of X. Then πX(K) is measurable on X/G2.

Let K be any compact subset of X and let Z,Kn be the subsets of condition (IV), i. e. K K is an increasing sequence of compact subsets of K, and n ∈ Z is µ-negligible subset of K such that K = Z ˙ K1 K2 ... . Let us deﬁne the subset (if any) Z Z consisting of intersections∪ ∪ of full∪ G -orbits with K, 0 ⊂ 2 i. e. the maximal subset of Z invariant under the action of G2 on X.

K Z

G2-orbits in X = G/G1

1 Then πX(K Z)= πX(K Z0). We shall show that µ(Z0 G2)= µ(πX− (πX(Z0)) = 0. Toward this− end observe− that because X is metrizable· and separable we may assume the elements , m N, of basis of topology to be the balls with com- Om ∈ pact closure m; and the σ-ring of Borel sets on X generated by the open m or closed Oballs. O Om

58 For each ǫ> 0 ǫ η, O · there exists open K η G2 Oǫ ⊃ ∈ with: µ( K) <ǫ Oǫ − by regularity of µ

Oǫ

Z G 0 · 2

By the regularity and quasi invariance of the measure µ it easily follows that the µ-measure of the intersection of Z0 G2 with any open set in X is equal zero, and thus again by the regularity of µ· and second countability of X it easily 1 follows that µ(Z G ) = µ(πX− (πX(Z ))) = 0. Thus πX(Z ) is a subset of 0 · 2 0 0 a measurable null set, and so must be a measurable set with ν(πX(Z0)) = 0, because ν is a pseudo-image measure of µ under πX. Moreover, we have:

πX(K Z)= πX(K Z )= πX(K) πX(Z ), − − 0 − 0 because Z0 consists of intersections of G2-orbits with K. On the other hand ψ πX(K Z) 0 ◦ − is a Borel set in S, and thus πX(K Z) is a Borel set in X/G2 as ψ0 is a Borel isomorphism. Indeed, because images− preserve the set theoretic sum operation we have ψ πX(K Z)= ψ πX(K ). 0 ◦ − 0 ◦ i n N [∈ Because K K then K /R is Hausdorﬀ and the quotient map π is closed j ∈ j Kj Kj and thus the quotient space Kj/RKj is homeomorphic to the compact space

πKj (Kj ), and moreover because Kj is compact and metrizable (as a subspace of the metrizable space X) the quotient space Kj/RKj is likewise metrizable ([13], Thm. 7.5.22). We can therefore apply the Federer and Morse Theorem 5.1 of [14] in order to prove the existence for each j of a Borel subset B K such j ⊂ j that πKj (Bj ) = πKj (Kj)(= πX(Kj )) and such that πKj is one-to-one on Bj . Therefore ψ0 πX is one-to-one Borel function on a Borel subset Bj of complete separable metric◦ space X to a complete separable metric space S. Therefore again by the Theorem on page 253 of [27] (compare likewise the Theorem on page 72 of [34]), it follows that ψ πX(B )= ψ πX(K ) is a Borel set. Because 0 ◦ j 0 ◦ j ψ0 is a Borel isomorphism it follows that πX(Kj ) is a Borel set in X/G2.

59 Thus πX(K) diﬀers from a Borel set πX(K Z) by a measurable ν-negligible − subset πX(Z ) πX(K); so we have shown that πX(K) is measurable. 0 ⊂ Note that the Lemma 14 is non trivial. By the well known theorem of Suslin – continuous image of a Borel set is not always Borel, but it is always measurable, compare e. g. [24], Lemm. 11.6, page 142 and Thm. 11.18, page 150, where the references to the original literature are provided. However this argument would be insuﬃcient for πX(K) to be measurable in X/G2 for any compact set K X. Indeed it would in addition require to be shown that the ⊂ quotient Borel structure on X/G2 is equal to the σ-ring of Borel sets generated by the closed (open) sets of the quotient topology on X/G2.

LEMMA 15. Let µ,µC ,ν be as in Lemma 13 and let K be a compact subset of X. Let η G and let R be the σ-ring of Borel subsets of K induced form ∈ 2 K the surrounding measure space X. Let (µ)K′ and (µC )K′ denote the restrictions of µ and µC to K deﬁned on the σ-ring RK respectively, and let Rηµ, RηµC denote their right translations; and similarly let (ν) be the restriction of π′ X(K) the measure ν to the subset πX(K). Then

(a)

(µ)K′ = (µC )K′ d(ν)π′ X(K)(C) Z with each (µ )′ concentrated on C K. C K ∩ (b)

Rηµ = RηµC dν(C). Z

(Note that the σ-ring of Borel sets with a regular measure on this ring is suﬃcient to recover all measurable subsets and their measures obtained by the standard completion of the Borel measure space.) Part (a) of the Lemma is an immediate consequence of Lemmas 13 and 14 with 1 g inserted for g in K · the formula (19), where 1K is the characteristic function of the compact set K. The only non-trivial part of the proof lies in showing that πX(K) is measurable, which was proved in Lemma 14. 1 1 For (b) observe that if Rη−1 g L (X,µ) g L (X, Rηµ), then by Lemma 13: ∈ ⇔ ∈

1 1 g(x) d(R µ)= g(x η− ) dµ = dν(C) g(x η− ) dµ (x) η · · C Z Z Z Z = dν(C) g(x) d(RηµC )(x), Z Z thus

Rηµ = RηµC dν(C). Z

Note that the operations of restriction ( )′ to K and right translation R ( ) · K η · do not commute. Indeed if we write R ( )′ for R (( )′ ), then R ( )′ = η ◦ · K η · K η ◦ · K ( )K′ η−1 Rη = (Rη( ))K′ η−1 i. e. ﬁrst restrict to K and then translate Rη is · · ◦ · · 1 the same as ﬁrst translate R and then restrict to K η− (and not to K). η ·

REMARK 4. Let Op(µ) denote a repeated application of several restrictions to compact sets and translations: ( )K′ 1 , Rη1 ( ),... performed on the measure µ. Then the repeated application of Lemma· 15 (a)· and (b) gives

Op(µ)= Op(µC ) dOp(ν)(C), Z where Op(ν) denotes the restriction ()′ withf the compact set K X which πX(K) ⊂ arises in the following way: ( )K′ is the restriction which arises from Op by commutingf all translations to· the right (so as to be performed ﬁrst) and all restrictions to the left (so as to be performed after all translations): Op = ( )′ R ( ) or Op( ) = (R ( ))′ . · K ◦ η · · η · K

LEMMA 16. Let K, (µ) , (µ ) , (ν) be as in the preceding Lemma. For K′ C K′ π′ X(K) 1 1 any bounded and (µ) -measurable function g and for any f L (π− (K), (ν) ) K′ X π′ X(K) the set of all those G orbits C having non empty intersection∈ C K for which 2 ∩ g is not µC-integrable is ν-negligible and the the function

C g(x)d(µ )′ (x) 7→ C K Z on this set of orbits C is (ν) -summable and (ν) -measurable, and π′ X(K) π′ X(K)

f(C) g(x) d(µC )K′ (x) d(ν)π′ X(K)(C)= f(πX(x))g(x) d(µ)K′ (x). (20) Z Z Z The Lemma is an immediate consequence of the preceding Lemma. The only non-trivial part of the proof is is to show that f is measurable on X/G2 if and only if f πX is measurable on X. But this is an immediate consequence of Lemma 12. ◦

In order to simplify notation let us denote the operation of restriction ( )′ · K to K just by ( )′ in the next Lemma and its proof. In all other restrictions ( )D′ the sets D will· be speciﬁed explicitly. ·

LEMMA 17. Let µ,µC be as in Lemma 13 and let K be a compact subset of X. Let η G2 and let C be any G2-orbit having non empty intersection 1 ∈ C K η− K. Then for the respective measures obtained by right translations ∩ · ∩ and restrictions performed on µ and µC respectively we have:

61 (a) The measures ((µC )′)K′ η−1 and (Rη(µC )′)′, deﬁned on measurable subsets 1 · of C K K η− , are equivalent. ∩ ∩ · (b)

d(RηµC ) λC ( , η)= ( ) · dµC · d (R (µ ) ) d (R µ ) = η C ′ ′ ( )= η ′ ′ ( ) d ((µC )′)K′ η−1 · d(µ′)K′ η−1 · · · d R µ = η ( )= λ( , η) dµ · ·

1 on C K K η− . ∩ ∩ · In addition to the operations of translation and restriction let us introduce after Mackey, [31], 11, one more operation defined on finite measures µ on § · 1 X, giving measures µ on X/G2. Namely we put µ(E) = µ(πX− (E)). µ′ is well defined for any quasi invariant measureeµ on X/G2 because µ′ is finite. More precisely µ′ ise defined on the σ-ring of measurablee subsets E of πXe(K) 1 1 by the formula: µ′(E) = µ′(πX− (E)) = µ(K πX− (E)). A simple verification e ∩ of definitions shows that µ′ is a pseudo image measure of the measure µ′ under πX, so that e e µ′ = µC′ dµ′(C), Z on measurable subsets of K and where thee integral is over the orbits C having non void intersection with K and with µC′ concentrated on C K. Similarly we have for the pairs of measures ∩ ^ ^ (µ′)Kη′ −1 , (µ′)Kη′ −1 and (Rηµ′)′ , (Rηµ′)′ : (21) ^ (µ′)Kη′ −1 = (µ′)Kη′ −1 d (µ′)Kη′ −1 (C) C Z and ^ (Rηµ′)′ = Rηµ′)′ d (Rηµ′)′ (C), Z C 1 both (µ′)Kη′ −1 and (Rηµ′)′ defined on measurable subsets of K η− K (instead · ∩ 1 of K): with the measure (R µ′)′ equal to the measure R µ restricted to K η− η η · ∩ K, and (µ′)Kη′ −1 = (µ)Kη′ −1 K equal to the measure µ restricted to the same 1 ∩ compact subset K η− K; and with the corresponding tilde measures both · ∩ 1 defined on measurable subsets of the measurable (Lemma 14) set πX(K η− K); namely · ∩

^ 1 1 (R µ ) (E) = (R µ′)′(π− (E)) = R µ′(K π− (E)) η ′ ′ η X η ∩ X 1 1 1 = (R µ)′ −1 (K π− (E)) = R µ(Kη− K π− (E)) η Kη ∩ X η ∩ ∩ X

62 and ^ ^ 1 1 (µ′)Kη′ −1 (E)= (µ)Kη′ −1 K (E)= µ(Kη− K πX− (E)). ∩ ∩ ∩ Note please that our Lemma 16 holds true for any pseudo-image measure ν of µ. By Lemma 14, any pseudo-image measure of the restriction µ′ is a restriction (ν) of a pseudo-image measure of µ. It follows that the Lemma π′ X(K) 16 is applicable to the pairs of measures (21). Indeed it is suﬃcient to insert 1 K η− K instead of K in the Lemma 16 and apply it to (µ′)Kη′ −1 = (µ)Kη′ −1 K · ∩ ∩ (or to (Rηµ′)′ = (Rηµ)Kη′ −1 K) instead of µ′, because for an appropriate ∩ ^ ν, (ν) − gives the pseudo-image measure (µ ) − (or respectively π′ X(K η 1 K) ′ Kη′ 1 ^ · ∩ (Rηµ′)′) of (µ′)Kη′ −1 (or respectively of (Rηµ′)′). We may thus apply Lemma 11.4 of [31], 11, to the pairs of measures (21). Because µ is quasi invariant, the § measures (µ′)Kη′ −1 = (µ)Kη′ −1 K and (Rηµ′)′ = (Rηµ)Kη′ −1 K are equivalent 1 ∩ ∩ as measures on K η− K, and thus by Lemma 11.4 of [31] it follows that · ∩ ^ ^ 1 (µ ) −1 and (R µ ) are equivalent as measures on πX(K η− K). Intro- ′ Kη′ η ′ ′ · ∩ ducing the corresponding measurable weight function f1 on X/G2 which is non 1 zero on πX(K η− K), we have · ∩ ^ ^ f d (µ ) − = d (R µ ) 1 · ′ Kη′ 1 η ′ ′ and ^ (Rηµ′)′ = f1(C) Rηµ′)′ d (µ′)Kη′ −1 (C), (22) C Z ^ (µ′)Kη′ −1 = (µ′)Kη′ −1 d (µ′)Kη′ −1 (C). (23) C Z Now applying again the Lemma 11.4 of [31] to the pairs of measures:

^ ^ (µ′)Kη′ −1 , (µ′)Kη′ −1 and (Rηµ′)′ , (µ′)Kη′ −1 with the respective decompositions (23) and (22) we prove that the measures

(µ′)Kη′ −1 and Rηµ′)′ are equivalent and C C

d Rηµ′)′ C d (Rηµ′)′ f1(C) ( )= ( ) · · d (µ )′ −1 · d (µ ) − ′ K η ′ Kη′ 1 · C d (R µ) = η ( )= λ( , η), d µ · ·

1 on C K η− K, where the last two equalities follow from deﬁnitions and where∩ · ∩ ^ d (R µ ) f = η ′ ′ . 1 ^ d (µ′)Kη′ −1

63 On the other hand it follows from Lemma 15 and Remark 4 that

−1 (Rηµ′)′ = (Rη(µC )′)′ d (ν)π′ X(Kη K)(C) Z ∩ and

−1 −1 −1 (µ′)Kη′ = ((µC )′)Kη′ d (ν)π′ X(Kη K)(C). Z ∩ ^ Thus both (R µ ) and (ν) −1 being pseudo-image measures of the mea- η ′ ′ π′ X(Kη K) ∩ 1 sure (Rηµ′)′ under πX (of course restricted to Kη− K)) are equivalent. Intro- ∩ 1 ducing the respective measurable, non zero on πX(Kη− K), weight function ∩ f2 we have ^ −1 f2 d (ν)π′ X(Kη K) = d (Rηµ′)′, · ∩ so that

d (Rη(µC )′)′ = f2(C) d Rηµ′)′ . · C Similarly because µ is quasi invariant, the measures (µ′)Kη′ −1 and (Rηµ′)′ are ^ equivalent, and thus again by Lemma 11.4 of [31] the measures (µ′)Kη′ −1 and

(ν) −1 are likewise equivalent. Introducing the respective non zero on π′ X(Kη K) 1 ∩ πX(Kη− K) and measurable weight function f we have ∩ 3 ^ − f3 d (ν)π′ X(Kη 1 K) = d (µ′)Kη′ −1 , · ∩ so that

d ((µC )′)Kη′ −1 = f3(C) d (µ′)Kη′ −1 . · C Joining the above equalities we obtain (the last two equalities follows from deﬁnition of λC and from deﬁnition of Radon-Nikodym derivative, i. e. its local character)

d Rηµ′)′ C 1 d (Rη(µC )′)′ λ( , η)= f1(C) ( )= f1(C) f3(C) · · · · f2(C) · · d ((µC )′)′ −1 d (µ′)Kη′ −1 Kη C d (Rη(µC )′)′ d (RηµC ) = = ( )= λC ( , η) d ((µC )′)Kη′ −1 d µC · · 1 on C K K η− , because by the known property of Radon-Nikodym derivatives (compare∩ ∩ e.· g. Scholium 4.5 of [51]) ^ ^ d (ν) −1 d (µ ) − 1 d (Rηµ′)′ π′ X(Kη K) ′ Kη′ 1 f1 f3 = ∩ =1, · f2 · ^ · ^ · d (ν)′ −1 d (µ ) − d (R µ ) πX(Kη K) ′ Kη′ 1 η ′ ′ ∩ 1 on all orbits C with non void intersection C K K η− . ∩ ∩ · We are are now in a position to formulate the main goal of this Section.

64 LEMMA 18. Let µ be any quasi invariant measure on X and let ν be any pseudo image measure of µ. Then the measures µC in the decomposition

µ = µC (x) dν(C) Z of Lemma 13 are also quasi invariant and for each η G2 the Radon-Nikodym d(RηµC ) ∈ derivative λC ( , η)= ( ) is equal to the restriction of the Radon-Nikodym · dµC · derivative λ( , η)= d(Rη µ) ( ) to the orbit C. · dµ · Indeed, let x be any point in X and η any element of G2. We show that on a neighbourhood of x the statement of the Theorem holds true. To this end let be a neighbourhood of x chosen from the basis of topology Om constructed above. Then m η is a neighbourhood of x η. Therefore the O · · 1 compact set K = m ( m η) has the property that K (K η− ) contains an open neighbourhoodO ∪ ofOx.· Now it is sufficient to apply∩ Lemma· 17 with this K in order to show that the equality of the Theorem holds true on some open neighbourhood of x. REMARK 5. It has been proved in Sect. 7 that for each orbit C there exists a measure µC , concentrated on C, with the associated Radon-Nikodym derivative equal to the restriction to the orbit C of the Radon-Nikodym derivative associated with µ. This however would be insufficient because we need to know that the measures µC conspire together so as to compose a decomposition of the measure µ. This is why we need Lemma 18. Although the Lemma was not explicitly formulated in [31], it easily follows for the case of finite µ from the Lemmas of [31], 11. §

Using Lemma 13 and the general properties of the integral and the algebra of measurable functions one can prove a slightly strengthened version of Lemma 13 which may be called a skew version of the Fubini theorem, because it extends the Fubini theorem to the case where we have a skew product measure µ with only one projection, i.e. the quotient map πX:

LEMMA 19 (Skew Version of the Fubini Theorem). Let µ, µC and ν be such as in Lemma 13. Let g be a positive complex valued and measurable function on X. Then C g(x)dµ (x) (24) 7→ C Z is measurable, and if any one of the following two integrals:

dν(C) g(x) dµC (x) and g(x) dµ(x), Z Z Z does exist, then there exists the other and both are equal in this case. In particular it follows that if g is integrable on (X, RX,µ) then

dν(C) g(x) dµC (x)= g(x) dµ(x). (25) Z Z Z

65 For the proof compare [3], Chap. VI, Remark of 3.4. Here we give only few comments: The Lemma holds for positive and continuous§ g with compact support as a consequence of Lemma 13. Next we note that the class of functions which satisfy (24) and (25) is closed under sequential convergence of increasing sequences. The Lemma follows by repeated application of the sequential continuity of the integral for increasing sequences; compare, please, the proof of Thm. 3.4 and Corollary 3.6.2 of [51].

Note that the integral

g(x)dµC (x) Z in (24) and (25) may be replaced with

C g (x)dµC (x), CZ

C where g is the restriction of g to the orbit C, because µC is concentrated on C. However just like in the ordinary Fubini theorem the whole diﬃculty in application of the skew version of the Fubini Theorem lies in proving the πX measurability of g on the “skew product”X X/G2 measure space (X, RX,µ). Indeed even if the orbits C were nice closed−−→ subsets and gC measurable on C (with respect to the measure structure induced from the surrounding space X) the function g still could be non measurable on (X, RX,µ); for simple examples we refer e. g. to [51] or to any other book on measure theory. More restrictive constrains are to be put on the separate gC as functions on the orbits C in order to guarantee the measurability of g on the measure space X. We face the same C 2 problem with the ordinary Fubini theorem. If in addition g L (C,µC ) for each C (or ν-almost all orbits C), the required additional requirement∈ is just the von Neumann direct integral structure put on C gC which is the necessary and suﬃcient condition for the existence of a function7→ f L2(X,µ) such that f C = gC for ν-almost all orbits C. Namely, consider the∈ space of functions C gC L2(C,µ ), which composes 7→ ∈ C 2 L (C,µC ) dν(C), (26)

X/GZ 2 then for every element C gC of direct integral (26) there exists a function f L2(X,µ) such that f C 7→= gC for ν-almost all orbits C. In short ∈

2 2 L (C,µC ) dν(C)= L (X,µ). (27)

X/GZ 2

We skip proving the equality (27) because in the next Section we prove a more general version of (27) for vector valued functions g L on X = G/G , ∈ H 1

66 compare Lemma 22 (a). This strengthened version (27) of the skew Fubini theorem lies behind harmonic analysis on classical Lie groups and provides also an eﬀective tool for tensor product decompositions of induced representations in Krein spaces. In practice the classical groups with the harmonic analysis relatively complete on them, have the structure of cosets and double cosets (corresponding to the orbits C) much more nice in comparison to what we have actually assumed, so that a vector valued version of the strengthened version of the ordinary Fubini theorem:

L2(Y,µ ) dµ = L2(X Y,µ µ ) (28) Y X × X × Y XZ would be suﬃcient for our applications. Namely the “measure product property” holds also in our practical applications for the double coset space:

G,RG,µG

= G1 G /G1 (G/G1)/G2 , R , µ µ µ × × G1×G/G1×(G/G1)/G2 G1 × G/G1 × (G/G1)/G2 with the analogous functions (4), measure µ = µ and the pseudo image G/G1 measure ν = µ eﬀectively computable. (G/G1)/G2 Note that (27) and (28) may be proved for more general measure spaces. In particular our proof of (27) may be adopted to general non-separable case, provided that the assertion of Lemma 19 holds true for the measures µ and ν. Here the measure spaces are not “too big”, so that the associated Hilbert spaces of square summable functions are separable.

At the end of this Section we transfer the measure structure on X/G2 over to the the set G1 : G2 of all double cosets G1xG2, using the natural bi-unique 1 correspondence C DC = π− (C) between the orbits C and double cosets D. Next we transfer it7→ again to a measurable section B of G cutting every double coset at exactly one point and give measurability criterion for a function on B with this measure structure inherited from X/G2. We shall use it in Sections 9 and 10.

DEFINITION 1. We put dν0(D) = dν(CD ) for the measure ν0 transferred over to measurable subsets of the set of all double cosets, where CD is the orbit 1 corresponding to the double coset, i. e. D = π− (C). Let B0 be a measurable section of X with respect to G2, existence of which has been proved in Lemma 11. Let B be a measurable (even Borel) section of G with respect to G1 (which 1 exists by Lemma 1.1 of [31]). Next we deﬁne the set B = π− (B0) B. We call B the section of G with respect to double cosets. ∩ B is measurable by Lemma 1.1 of [31] and by Lemmas 11, 12 of this Sec- tion. It has the property that every double coset intersects B at exactly one R point. We may transfer the measure space structure (X/G2, X/G2 ,ν) over to R get (B, B,νB ).

67 DEFINITION 2. For each double coset D there exists exactly one element xD B D. We define dνB (xD ) = dν0(D). The same holds for orbits C: ∈ ∩ 1 to each orbit C there exists exactly one element x B π− (C). We put c ∈ ∩ respectively dνB (xc) = dν(C). Note that xc = xD iff C and D correspond. LEMMA 20. A set E of orbits C is measurable iff the sum of the corresponding double cosets, regarded as subsets of G, is measurable in G. Thus in particular a function g on B is measurable iff there exists a function f measurable on G and constant along each double coset, such that the restriction of f to B is equal to g.

1 By Lemma 1.2 of [31] a set F X = G/G is measurable iff A = π− (F ) ⊂ 1 is measurable in G and by Lemma 12 a subset E X/G2 is measurable iff 1 ⊂ F = πX− (E) is measurable on X. Thus a set E of orbits C is measurable iff the sum of the corresponding double cosets, regarded as subsets of G, is measurable in G, (as already claimed at the beginning of this Section). This proves the Lemma. In particular if we define s(x) to be the double coset containing x, then we transfer the measure ν over to the subsets of double cosets correctly if we define 1 the set E of double orbits to be measurable if and only if s− (E) is measurable on G. Writing x for the variable with values in G, and writing [x] for π(x) varying over X = G/G1 we have

LEMMA 21. Let µ, µC and ν be such as in Lemma 13. Let g be a positive complex valued and measurable function on X. Let µ = µ = µ be the D xD CD measure concentrated on the orbit CD corresponding to the double coset D.Then:

D g([x]) dµD([x]) and B x g([x]) dµx ([x]) (29) 7→ ∋ D 7→ D Z Z are measurable, and

1) if any one of the following two integrals:

dν0(D) g([x]) dµD([x]) and g([x]) dµ([x]), Z Z Z does exist, then there exists the other and both are equal in this case.

In particular it follows that if g is integrable on (X, RX,µ) then

dν0(D) g([x]) dµD([x]) = g([x]) dµ([x]). (30) Z Z Z 2) Similarly if any one of the following two integrals:

dν (x ) g([x]) dµ and g([x]) dµ([x]), B D xD Z Z Z does exist, then there exists the other and both are equal in this case.

68 In particular it follows that if g is integrable on (X, RX,µ) then

dν (x ) g([x]) dµ ([x]) = g([x]) dµ([x]). (31) B D xD Z Z Z

Because by deﬁnition (with x X = G/G and x G) ∈ 1 ∈

g(x) dµC (x)= g([x]) dµD([x]), Z Z CD D the Lemma is an immediate consequence of deﬁnitions Def 1 and 2 and Lemma 19.

9 Subgroup theorem in Krein spaces

Let G1 and G2 be regularly related closed subgroups of G (for definition com- L pare Sect. 8). Consider the restriction U to the subgroup G2 G of the G2 ⊂ representation µU L of G in the Krein space µ L, induced from a representa- H tion L of the subgroup H = G1, defined as in Sect 3. For each G2-orbit C in L,C X = G/G1 let us introduce the Krein-isometric representation U , defined in Sect. 7, and acting in the Krein space ( L ′, JL,C). Let ν be any pseudo HC image measure of µ on X/G2, for its definition compare [3], Chap. VI.3.2. For simplicity we drop the µ superscript in µU L and µ L and just write U L and L. H H Let us remind the definition of the direct integral of Hilbert spaces after [50], but compare also [36]: R DEFINITION 3 (Direct integral of Hilbert spaces). Let (X/G2, X/G2 ,ν) be a measure space M, and suppose that for each point C of X/G2 there is a Hilbert L ′ L L ′ space C . A Hilbert space is called a direct integral of the C over M, symbolicallyH H H L = L ′ dν(C), (32) H HC Z if for each g L there is a function C gC on X/G to the disjoint union ∈ H 7→ 2 L ′ C L ′ C , such that g C for all C, and with the following properties 1) C X/G H ∈ H ∈ 2 and` 2): 1) If g and k are in L and if u = αg +βk, and if , is the inner product H · · C L ′ C C in C then C g , k is integrable on M , and the inner product H 7→ C (g, k) on L is equal to H C C (g, k)= g , k C dν(C), Z X/G2

69 and uC = αgC + βkC for almost all C X/G , and all α, β C. ∈ 2 ∈ C C L ′ C C 2) If C u is a function with u C for all C, if C g ,u C is 7→ L ∈ H C C 7→ measurable for all g , and if C u ,u C is integrable on M, ∈ H L 7→ then there exists an element u′ of such that H C C u′ = u almost everywhere on M.

The function C gC is called the decomposition of g and is symbolized by 7→ g = gC dν(C).

X/GZ 2

A linear operator U on L is said to be decomposable with respect to the direct integral Hilbert space decompositionH (32) if there is a function C U C 7→ on X/G with U C being a linear operator in L ′ for each C, and 2 HC 3) the property that for each g in its domain and all k in L, (Ug)C = U C gC C CH C almost everywhere on M and the function C U g , k C is integrable on M. 7→ If U is densely deﬁned the property 3) is equivalent to the following:

L C C C 3’) for all g, k in in the domain of U, C U g , k C is integrable on M and H 7→ C C C U g , k C dν(C) = (Ug,k). Z X/G2 The function C U C is then called the decomposition of U with respect to (32) and symbolized7→ by U = U C dν(C).

X/GZ 2 If C U C is almost everywhere a scalar operator, U is called diagonalizable with respect7→ to (32). The totality of all bounded operators diagonalizable with respect to (32) composes the commutative von Neumann algebra AG/G2 associated with the decomposition (32), compare [36]. A bounded operator U in L is decomposable with respect to (32) if and only if it commutes with all elementsH of ′ AG/G2 U AG/G2 . This condition may easily be extended on unbounded operators:⇔ e.∈ g. closable U is decomposable with respect to (32) if the spectral projectors of both the factors in its polar decomposition commute with all elements of AG/G2 ; or still more generally: U is decomposable with respect to (32) U is aﬃliated with the commutor A ′ of A , i.e. iﬀ it commutes ⇔ G/G2 G/G2 with every unitary operator in the commutor A ′′ = A of A ′. G/G2 G/G2 G/G2

70 Note that the map T which transforms g into its decomposition C gC may be viewed as a unitary operator decomposing U: 7→

1 C TUT − = U dν(C).

X/GZ 2

There are many possible realizations T : f T (f) of the Hilbert space L as the direct integral (32) all corresponding to the7→ same commutative decompo-H sition algebra A . However the difference between any two T : f T (f)= G/G2 7→ C C C f and T ′ : f T ′(f) = C f ′ of them is irrelevant: there 7→ 7→ 7→ exists for them a map C U C with each UC unitary in L ′ and such that: 7→ HC 1) U C f C = f C ′ for almost all C. 2) C f C,gC is measurable in realization T C U C f C ,U Cf C 7→ C ⇔ 7→ C is measurable in realization T ′. (Compare [36]). For the reasons explained in the footnote to Lemma 6 it is sufficient to R R consider the σ-rings X/G2 and X of Borel sets, with the Borel structure on X/G2 defined as in Sect. 8, in the investigation of the respective Hilbert and Krein spaces. We shall need a LEMMA 22. (a) L = L ′ dν(C). H ∼ HC X/GZ 2

(b) U L = U L,C dν(C). G2 ∼

X/GZ 2

The equivalences = are all under the same map (or realization) T : L ∼ H 7→ L ′ dν(C) giving the corresponding decomposition T (f): C f C HC 7→ X/G2 forR each f L, in which f C is the restriction of f to the double coset ∈ H 1 DC = G1xcG2 = π− (C) corresponding to C; i. e. we chose xc B G for which π(x ) C, compare Def. 1 and 2. ∈ ⊂ c ∈

71 In particular T is unitary and Krein-unitary map between the Krein spaces

( L, JL) and L ′ dν(C), JL,C dν(C) . H HC X/GZ 2 X/GZ 2

REMARK 6. The equivalences ∼= may be read in fact as ordinary equalities. Let ( , ) = 2 = J (JL,C ) , ( ) dµ (x) · · C k·kC L · x · x C Z C be defined on L ′ as in Sect. 7. Recall that for any element g of L ′ dν(C) HC HC X/G2 C R i. e. a function C g from the set of G2-orbits X/G2 to the disjoint 7→ 2 L ′ C L ′ C union C such that g C for all C, the function C g C = C X/G H ∈ H 7→ k k ∈ 2 C C (g ,g )Cìs ν-summable and ν-measurable and defines inner product for any g, k L ′ dν(C) by the formula ∈ HC X/G2 R L,C C C C C (g, k)= dν(C) JL(J g )x, kx dµC (x)= (g , k )C dν(C). Z Z Z X/G2 C X/G2 (33) We shall exhibit a natural unitary map T from L onto L ′ dν(C) or, H HC X/G2 what is equivalent, we shall show that the decomposition TR(f) = C f C corresponding to each f L, with f C equal to the restriction of f 7→to the ∈ H 1 double coset DC = G1xcG2 = π− (C) corresponding to C, has all the properties required in Definition 3. Let f and k be any functions in L. Then by Lemma 13 we have H dν(C) J (JLf) , k dµ (x)= J (JLf) , k dµ(x)= f 2 < , L x x C L x x k k ∞ Z Z Z X/G2 C X

L with the set of all G2 orbits C for which x JL(J f)x, kx is not µC -integrable being ν-negligible and the function 7→ C J (JLf) , k dµ (x) 7→ L x x C Z C being ν-summable and ν-measurable. Moreover, because for each orbit C the measure µC is concentrated on C (Lemma 13), the integral

L JL(J f)x, kx dµC (x) Z C

72 is equal

L C C L C C JL((J f) )x, (k )x dµC (x)= JL(J f )x, (k )x dµC (x) Z Z C C where f C (and similarly for kC ) is the restriction of f to the double coset 1 D = G xG = π− (C) corresponding to C. i.e. with any x for which π(x) C, C 1 2 ∈ say x = xc, with C xc B of Sect. 8. (We have chosen x = xc to belong to the measurable section7→ B∈of double cosets in G constructed in Sect. 8, but this C L ′ L,C is unnecessary here.) Because f C and likewise J are deﬁned as the ordinary restrictions, (JLf)C = JL∈f C H= JL,Cf is the restriction of JLf to the double coset DC = G1xcG2 corresponding to C. We thus obtain

L L,C C C JL(J f)x, kx dµC (x)= JL(J f )x, (k )x dµC (x). Z Z C C Therefore it follows that the map T : f C f C , where f C is the restriction of f to the double coset corresponding7→ to the7→ orbit C, fulfils the requirements of Part 1) of Definition 3; in particular T (f) = f and the range T ( L) is a Hilbert space with the inner product (33).k k k k H We shall verify Part 2) of the Definition 3: i. e. that the decomposition map T (f) = C f C defined as above has the properties indicated in 2) of Definition 3 on its7→ whole range T ( L). Toward this end let C uC fulfil the conditions required in 2) of Def. 3:H 7→

C J (JL,C uC) , (kC ) dµ (x)= uC, kC (34) 7→ L x x C C Z C is measurable for each k L and ∈ H C J (JL,CuC) , (uC ) dµ (x)= uC,uC (35) 7→ L x x C C Z C is measurable and integrable. Consider the space F of all functions C kC 7→ ∈ L ′ fulﬁlling the following conditions: HC

C J (JL,C gC) ,gC dµ (x)= kC, kC 7→ L x x C C Z C is measurable and integrable. Let X be the maximal linear subspace of F, where a subspace of F we have called linear, whenever it is closed under formation of ﬁnite linear combinations over C. X is not empty as it contains the subspace T ( L) itself, which is a Hilbert space. Moreover if C kC , C rC are any H 7→ 7→

73 two functions belonging to X the formula

h C kC , C rC = (kC , rC ) dν(C) 7→ 7→ C X/GZ 2

L,C C C = JL(J k )x, (r )x dµC (x) dν(C) Z Z X/G2 C deﬁnes a hermitian form on X. Thus by the Cauchy-Schwarz inequality we have:

2 (kC , rC ) dν(C) (kC , kC ) dν(C) (rC , rC ) dν(C) . C ≤ C · C X/GZ 2 X/GZ 2 X/GZ 2

(36) Now by the ﬁrst part of the proof, T ( L) is a Hilbert space with the inner product (33) and in particular a linearH subspace of F. We may thus insert for C kC in (36) any decomposition C f C of f L, with f C equal to the 7→ 7→ 1 ∈ H restriction of f to the double coset DC = π− (C) corresponding to C. Similarly we may insert the function C uC for the function C rC in (36). Indeed, because of the conditions (34)7→ and (35), fulﬁlled by the7→ function C uC , the function 7→

C f C + uC,f C + uC = f C ,f C + f C ,uC + uC ,f C + uC ,uC 7→ C C C C C is measurable and by the Cauchy-Schwarz inequality integrable, for all f L. Therefore C uC and T ( L) are both contained in one linear subspace∈ of HF, and thus by the7→ maximalityH of X they are contained in X, so that we can insert C uC for C rC in (36). Thus the indicated insertions in the inequality (36)7→ lead us to the7→ inequality

2 (f C ,uC) dν(C) (f C ,f C) dν(C) (uC ,uC) dν(C) C ≤ C · C X/GZ 2 X/GZ 2 X/GZ 2

for all C f C in T ( L). Therefore the linear functional 7→ H T (f) L T (f) = L C f C = h C f C , C uC , 7→ 7→ 7→ 7→ on T ( L) is bounded by the last inequality. Because the range T ( L) of T is a HilbertH space it follows by the Riesz theorem ((e. g. Corollary 8.3.2.H of [51]) applied to the linear functional L that there exists exactly one element T (f ′) in the range of T such that

(f,f ′)= T (f) ,T (f ′)

C C C C = (f ,f ′ ) dν(C)= h C f , C u C 7→ 7→ X/GZ 2

74 for all f L. Therefore ∈ H C C C C f ,f ′ C dν(C)= f ,u C dν(C) Z Z X/G2 X/G2 L L for all f and for a ﬁxed f ′ , or equivalently ∈ H ∈ H C C C f ,f ′ u dν(C)=0, − C Z X/G2 L C C C for all f . Inserting the deﬁnition of f ,f ′ u we get: ∈ H − C

L,C C C C J (J f ) , (f ′ u ) dµ (x) dν(C) L x − x C X/GZ 2 CZ

L C C = J (J f) , (f ′ u ) dµ (x) dν(C)=0, L x − x C X/GZ 2 CZ (37) for all f L. By Lemma 6 there exists a sequence f 1,f 2,... of elements L L∈ H k C0 such that for each ﬁxed x G the vectors fx , k = 1, 2,... form a dense⊂ linear H subspace of . By the proof∈ of the same Lemma 6 there exists a HL sequence g1,g2,... of continuous complex valued functions on X = G/G1 with 2 2 compact supports, dense in L (X,µ) with respect to the L norm 2 . For each k·kL gj deﬁne the corresponding function gj′ on G by the formula gj′ (x) = gj (π(x)), where π is the canonical quotient map G G/G = X. Note, please, that 7→ 1 L C C L C C J (J g′ f) , (f ′ u ) = (g′ ) J (J f) , (f ′ u ) for all j N L j · x − x j x · L · x − x ∈ L i and all f . Inserting now g′ f for f in (37) we get ∈ H j · L i C C g (C) J (J f ) , (f ′ u ) dµ (x) dν(C)=0, j · L x − x C X/GZ 2 CZ

2 for all i, j N. Because gj j N is dense in L (X,µ) and the function ∈ { } ∈ L i C C C J (J f ) , (f ′ u ) dµ (x) 7→ L x − x C CZ by construction belongs to L2(X,µ), it follows that outside a ν-negligible subset N of orbits C L i C C J (J f ) , (f ′ u ) dµ (x)=0, L x − x C CZ for all i N. Thus if C/ N, then ∈ ∈ L i C C J (J f ) , (f ′ u ) dµ (x)=0, (38) L x − x C CZ

75 for all i N. Applying Lemma 8 to this orbit C and the associated L ′ we x C get an isomorphism∈ of it with a Krein space L c of an induced representationH H (recall that xc B G with π(xc) C, compare Def. 2). Then (38) together ∈ ⊂ ∈ Lxc C C with Lemma 10 and Lemma 3 or 4 applied to gives f ′ u = 0. This shows that the decomposition T : f C f CH fulﬁls Part 2)− of Deﬁnition 3. We have thus proved Part (a) of the7→ Lemma.7→ Then we have to prove that the operators T U L T 1 and T JL T 1 are G2 − − decomposable with respect to (32) and C U L,C and C JL,C are their respective decompositions. Let η G . Writing7→ λ(η) for the7→λ-function [x] ∈ 2 7→ λ([x], η) corresponding to the measure µ and analogously writing λC (η) for the λ function [x] λ ([x], η) corresponding to µ we have: C 7→ C C L 1 C L T U T − C f = T U f G2 η 7→ G2 η = T λ(η)R f = C λ(η) R fC , η 7→ |C η p p where λ(η) C denotes the restriction of λ(η) to the orbit C. By Lemma 18 the restriction |λ(η) of λ(η) to the orbit C is equal to λ (η), so that |C C L 1 C C L,C C T U T − C f = C λC (η)Rηf = C U f , G2 η 7→ 7→ 7→ η p which means that U L = U L,C dν(C), G2 ∼

X/GZ 2 and proves (b). Similarly for the operator JL:

L 1 C L T J T − C f = T J f 7→ C = T JLf = C JLf . 7→ C By deﬁnition of the operator JL,C we have JLf = JLf C = JL,Cf C . There- fore L 1 C L,C C T J T − C f = C J f , 7→ 7→ which means that L L,C J ∼= J dν(C),

X/GZ 2 and proves (c). 2 L,C L ′ L,C Because for each C, J is unitary and self adjoint in C and J = I, then by [36], 14, the same holds true for the operator H §

JL,C dν(C) in L ′ dν(C), HC X/GZ 2 X/GZ 2

76 so that L ′ dν(C), JL,C dν(C) , HC X/GZ 2 X/GZ 2 is a Krein space. Finally we have to show that T is Krein unitary. To this end observe that for each f,g L ∈ H T (f),T (g) = JL,Cf C ,gC dν(C) R JL,C dν(C) C X/GZ 1

L,C 2 C C = JL (J ) f , g dµC (x) dν(C) x x C Z Z X/G1 C C C = JL f , g dµC (x) dν(C). x x Z Z X/G1 C C C Because f and g are the ordinary restrictions of f and g to G1xcG2 and the measure µC is concentrated on C (Lemma 13), the integrand in the last L integral may be replaced with JL f , g . Because f,g , the function x x ∈ H x JL f g is constant on the right G1-cosets and measurable and 7→ x x integrable on X = G /G1 as a function of right G1-cosets. Thus by Lemma 19 the last integral is equal to

JL f , g dµC (x) dν(C)= JL f , g dµ(x)= f,g L , x x x x J Z Z Z X/G1 C X so that

T (f),T (g) = f,g JL . R JL,C dν(C) L Actually we could merely use all g′ f, with g C (X) and f C0 instead i · ∈ K ∈ of its denumerable subset g′ f , i, j N in the proof of Lemma 22. Its j · ∈ denumerability shows that L is separable as the direct integral (a). This however is superﬂuous becauseH separability of L has been already shown within the proof of Lemma 6. H

LEMMA 23. Let B be the section of G with respect to double cosets of Def. 1 and let C x B, D x B be the bi-unique maps of Def. 2. Let ν be 7→ c ∈ 7→ D ∈ 0 the measure on the subsets of the set G1 : G2 of all double cosets D equal to the transfer of the measure ν on X/G2 over to the set of double cosets by the natural 1 bi-unique map C DC = π− (C). Let νB be the measure on the section B equal to the transfer of7→ν over to the section B by the map C x (or equivalently 7→ c equal to the transfer of ν0 by the map D x ). Let µD = µC , where C 7→ D D D

77 is the orbit corresponding to the double coset D, be the measure concentrated on CD , where µC is the measure of Lemma 13. Let us denote the space of L ′ functions C of Sect. 7, deﬁned on the double coset D corresponding to C just L H L,C L,C by D and similarly if U and J is the representation and the operator of H L,D L,C L,D L,C Sect. 7, then we put U = U D and J = J D ; analogously we deﬁne L,x L,C L,x L,C U D = U D and J D = J D . Then we have

(a)

L L ′ L L = C dν(C)= D dν0(D)= x dνB (xD ). H ∼ H H H D X/GZ 2 G1Z:G2 BZ

(b)

L L,C L,D L,x U = U dν(C)= U dν0(D)= U D dν (x ). G2 ∼ B D

X/GZ 2 G1Z:G2 BZ

(c)

L L,C L,D L,x J J dν(C)= J dν (D)= J D dν (x ). ∼= 0 B D

X/GZ 2 G1Z:G2 BZ

L L The equivalences ∼= are all under the same map T : D dν(C) H 7→ G1:G2 H C giving the corresponding decomposition T (f): D f D (orR respectively 7→ CD L CD T (f): xD f ) for each f , in which f is the restriction of 7→ ∈ H 1 f to the double coset D = DC = G1xD G2 = π− (CD ) corresponding to D

CD . In particular T is unitary and Krein-unitary map between the Krein spaces ( L, JL) H and L dν(C), JL,D dν (D) HD 0 G1Z:G2 G1Z:G2 or respectively

L L,xD x dνB (xD ), J dνB (xD ) . H D BZ BZ

The Lemma follows from Lemma 22 by a mere renaming of the points of the measure space X/G2 of G2-orbits C in X, with the preservation of the measure structure under the indicated renaming, which is guaranteed by Def. 1 and 2.

78 µxc Lxc LEMMA 24. Let , Jxc be the Krein space of the representation µxc Lxc H U of the subgroup G2 defined in Lemma 8 with the inner product ( , )xc µxc Lxc · · in defined by eq. (17) in the proof of Lemma 8. For each xD B we x x x x put µ HD L D = µ c L c , J = J , G = G and , = ( , )∈ with xD xc x xc x xc H H D · · D · · L the orbit C corresponding to D. For each fixed element f consider the following function ∈ H x D x x B x f µ D L D ∋ D 7→ ∈ H x D where for each xD , f is defined ase the function

x D D e G2 t f = f , t x ·t ∋ 7→ D with f D equal to the restriction of f toe D. The linear set of all such functions x D H x f with f ranging over the whole space L and with the inner product D 7→ H x x e D D (f, g)= f , g dνB (xD ), (39) xD Z B e e e e is equal to x x µ D L D dν (x ). H B D BZ

x Note, please, that by deﬁnition of the measures µ D and the operators J xD

x x x x D D D D x f , g = J J f , g dµ D ([t]) L xD xD t t Z G2/Gx D e e e e D D x = JLLh(x t)JLLh(x t)−1 f , g dµ D ([t]) D · D · x ·t x ·t Z D D G2/Gx D

L D D L = JL J f , g dµD([x]) = JL J f ,gx dµD([x]) x x x Z Z D D and because

L L G/G1 [x] JL J f ,g[x] = JL J f ,gx ∋ 7→ [x] x is measurable it follows from (24) of Lemma 21 that the function

x x D D xD f , g x 7→ D is measurable for all f,g L. Similarlye bye (25) of part 2) of Lemma 21 ∈ H

79 x x D D

(f, g)= f , g dνB (xD ) xD Z B x x e e e e D D xD = JL Jx f , g dµ ([t]) dνB (xD ) D t t x Z Z D B G2/Gx D e e L L JL J f ,gx dµD([x]) dνB (x )= JL J f ,gx dµ([x]) x D x Z Z Z B D G/G1 = (f,g).

Therefore is a Hilbert space with the inner product (39) as the isometric image of theH Hilbert space L. We need only show Part 2) of Def. 3 to be x H D x x µ D L D fulfilled. Toward this end let xD u be a function fulfilling the conditions of Part 2) of Def. 37→ (of course∈ withH the obvious replacements of x x L ′ µ D L D C with D and C with ). We have to show existence of a function x H H D L f ′ such that the function xD f ′ is equal almost everywhere to the x ∈ H D 7→ function xD u . We proceed exactly as in the proof of Part (a) of Lemma 22 by formation7→ of the analogous maximale linear subspace X in the space F of x D all functions x k for which D 7→ x x D D xD k , k x 7→ D is measurable and integrable and then using Riesz theorem and Lemma 3 or 4 in proving the existence of f ′ (in this case the proof is even simpler because the x D x D Lemma 8 is not necessary in proving f ′ u = 0 from the analogue of (38); indeed it is sufficient to apply Lemma 10 and− Lemma 3 or 4). From now on we identify the Hilberte space L with the direct integral: A H b L L = x dνB (xD ). H H D BZ

D with the realization T T (f) of the direct integral equal to T (f): xD f , where f D is the ordinary7→ restriction of f L to the double coset D. Similarly7→ by ∈ H x x µ D L D dν (x ), H B D BZ we understand the direct integral with the realization of Lemma 24.

LEMMA 25. For each orbit C let Vxc be the Krein-unitary map deﬁned in Lemma 8. For each x B (equivalently: each double coset D) let us put D ∈

80 V = V with C corresponding to D. Then x V is a decomposition of xD xc D xD a well deﬁned operator 7→

x x L L V µ D L D = x dνB (xD ) dνB (xD ): H H D −→ H BZ BZ x x D D x f x Vx f . D 7→ 7→ D 7→ D In short V = V dν (x ). xD B D BZ The operator V is unitary and Krein-unitary between the Krein spaces

x x µ D L D dν (x ) , Jx dν (x ) H B D D B D BZ BZ

L L,xD L L and x dνB (xD ) , J dνB (xD ) = ( , J ); H D H BZ BZ and moreover:

x x L 1 L,x 1 µ D L D V U V − = V U D dν (x ) V − = U dν (x ) G2 B D B D BZ BZ and

L 1 L,x 1 V J V − = V J D dν (x ) V − = J dν (x ). B D xD B D BZ BZ

Let f be any element of L and t G . By deﬁnition we have H ∈ 2 x x D D D Vx f = f = f , D x ·t t D t

x D with f deﬁned in Lemma 24. Thus by the realizatione of

x x e µ D L D dν (x ) H B D BZ given in Lemma 24, V is onto. Moreover, by the proof of Lemma 24

x x x x D D D D x V f , g = f , g D xD x x 7→ D D is measurable for all g L, and thus for all e ∈ H e e x D x x x g µ D L D dν (x ); D 7→ ∈ H B D Z B e 81 therefore V is a well deﬁned operator. Moreover, by the proof of Lemma 24

x x D D (Vf,Vg)= V f , V g dν (x ) xD xD B D xD Z B x x D D

= f , g dνB (xD ) = (f,g), xD Z B e so that V is unitary (it likewise follows from Lemmae 9). Again by Lemma 21 we have:

x x D D (Vf,Vg) = J f , g dν (x ) R Jx dν (x ) xD B D D B D xD Z B e D De = JL f , g dµD([x]) dνB (x ) x x D Z Z B D

= JLfx ,gx dµD([x]) dνB (xD ) = (f,g)JL

G/GZ 1 which shows that V is Krein unitary. Because by Lemma 8

x x L,x 1 µ D L D L,x 1 V U D V − = U and V J D V − = J , xD xD xD xD xD the rest of the Lemma is thereby proved. REMARK 7. By a mere renaming of points associated to the isomorphisms D x B = G1 : G2 = X/G2 of measure spaces, e.g introducing V = Vx , µ = µ D ∼ ∼ D D and the measure ν0 as in Def. 2 we may rephrase Lemma 25 as follows. D VD is a decomposition of a well deﬁned operator 7→

V = VD dν0(D):

G1Z:G2

D D L = L dν (D) V µ L dν (D): H HD 0 −→ H 0 G1Z:G2 G1Z:G2 D f D D V f D . 7→ 7→ 7→ D The operator V is unitary and Krein-unitary between the Krein spaces

D D µ L dν (D) , J dν (D) H 0 D 0 G1Z:G2 G1Z:G2 and L dν (D) , JL,D dν (D) = ( L, JL); HD 0 0 H G1Z:G2 G1Z:G2

82 and moreover:

D D L 1 L,D 1 µ L V U V − = V U dν (D) V − = U dν (D) G2 0 0

G1Z:G2 G1Z:G2 and

L 1 L,D 1 V J V − = V J dνB (xD ) V − = JD dν0(D).

G1Z:G2 G1Z:G2

DEFINITION 4. Let G1 and G2 be two closed subgroups of a separable locally compact group G. Let B be any Borel section of G with respect to G1 and for 1 each x G let h(x) be the unique element of G such that h(x)− x B. ∈ 1 · ∈ Let µ be any quasi invariant measure µ on G/G1 and let ν be any pseudo- image measure on (G/G1)/G2 of the measure µ under the quotient map πG/G1 : G/G (G/G )/G ; so that: 1 7→ 1 2

µ = µC dν(C).

(G/GZ1)/G2

Let us call any measure ν0 on measurable subsets of the set G1 : G2 of all double cosets admissible iﬀ it is equal to the transfer of ν over to G1 : G2 by the natural 1 map (G/G1)/G2 C π− (C) G1 : G2. Finally let x be any element of G with π(x) C. We∋ put7→µx for the∈ measure on G /G equal to the transfer of the ∈ 2 x measure µC over to G2/Gx by the map G2/Gx [y] [xy] C G/G1, where 1 ∋ 7→ ∈ ⊂ G = G (x− G x) and where [ ] denotes the respective equivalence classes. x 2 ∩ 1 · Summing up we have just proved the following

THEOREM 7 (Subgroup Theorem). Let U L be the isometric representation of the separable locally compact group G in the Krein space ( L, JL), induced by H the Krein-unitary representation L of the closed subgroup G1 of G and the quasi invariant measure µ on G/G1 and the Borel section B of G with respect to G1. Then U L is independent to within Krein-unitary equivalence of the choice of B. Let G2 be a second closed subgroup of G and suppose that G1 and G2 are regularly 1 related. For each x G consider the closed subgroup Gx = G2 (x− G1x) and Lx ∈ ∩Lx let U denote the representation of G2 in the Krein space ( , Jx ) induced x H by the Krein-unitary representation L : η L −1 of the subgroup G in the 7→ xηx x −1 Krein space ( L, JL), where Jx g = Lh(x t)JLL g and with the inner t h(x t) t Hx · x · product in L and Krein-inner product in ( L , J ) deﬁned respectively by the x formulas H H

x (f,g)x = JL Jx f , g dµ ([t]) t t Z G2/Gx

83 and x (f,g)J = (Jx f,g)x = JL f , g dµ ([t]); x t t Z G2/Gx x Lx and with the quasi invariant measure µ on G2/Gx given by Def. 4. Then U is determined to within Krein-unitary and unitary equivalence by the double D L Lx coset G1xG2 = s(x) to which x belongs and we may write U = U , where L D = s(x). Finally U restricted to G2 is a direct integral over G1 : G2 with respect to any admissible (Def. 4) measure in G1 : G2, of the representations D U L .

x x It may happen that all the component representations µ D U L D are bounded and thus Krein-unitary, although U L is unbounded. In this case the norms x x µ D L D U are unbounded functions of xD (resp. D). Unfortunately instead xD x x µ D L D of Jx we cannot use any standard fundamental symmetry in : D H x x x D D L D xD xD J f = L JLL −1 f , hx (t) hx (t) t D D t where hx (t) Gx is definede as in Sect. 3 by a regulare Borel section Bx of D ∈ D D G with respect to the subgroup G = G (x 1G x ). A difficulty will 2 xD 2 D − 1 D x ∩ x x arise with this JL D . Namely in general the norms µ D U L D are such that xD the operator V would be unbounded with the standard fundamental symmetries x x in µ D L D . It isH important that in practical computations, e.g. with G equal to the double covering of the Pouncarégroup, much stronger regularity is preserved, e. g. the “measure product property” (see the end of Sect. 8), with the measurable sections B and B as differential sub-manifolds (if we discard unimportant null subset), so that the function x h(x) and all the remaining functions – analogue of (4) – associated to the7→ measure product structure are effectively computable together with the measures µ and ν0. This is important because together with the theorem of the next Section give an effective tool for decomposing tensor product of induced representations of the double cover of the Poincarégroup in Krein spaces. Moreover the operator V of Lemma 25 and Remark 7 is likewise effectively computable in this case.

10 Kronecker product theorem in Krein spaces

Let µ1 U L and µ2 U M be Krein-isometric representations of the separable locally compact group G induced from Krein-unitary representations of the closed subgroups G G and G G respectively. The Krein-isometric representa- 1 ⊂ 2 ⊂ tion µ1 U L µ2 U M of G is obtained from the Krein-isometric representation ⊗ µ1 U L µ2 U M of G G by restriction to the diagonal subgroup G of all those (x, y) × G G for× which x = y, which is naturally isomorphic to G itself: ∈ ×

84 G = G, with the natural isomorphism (x, x) x. Thus by the natural iso- ∼ 7→ morphism the representation µ1 U L µ2 U M of G may be identified with the ⊗ restriction of the representation µ1 U L µ2 U M of the group G G to the diag- × × onal subgroup G. By Theorem 6, µ1 U L µ2 U M is Krein-unitary and unitary × µ µ L M equivalent to the Krein-isometric representation 1× 2 U × of G G induced × by the Krein-unitary representation L M of the closed subgroup G1 G2. Thus the Krein-isometric representation×U L U M of G is naturally equivalent× ⊗ µ µ L M to the restriction of the Krein-isometric representation 1× 2 U × of G G to the closed diagonal subgroup G. Thus we are trying to apply the Subgroup× Theorem 7 inserting G G for G, G for G , and the subgroup G G G G × 2 1 × 2 ⊂ × for G1 in the Subgroup Theorem. But the Subgroup Theorem is applicable in that way if the subgroups G1 G2 and G are regularly related. Mackey rec- ognized that they are indeed regularly× related in G G if and only if G and × 1 G2 are in G, pointing out a natural measure isomorphism between the measure spaces (G1 G2): G and G1 : G2 of double cosets respectively in G G and × ×1 G. The isomorphism is induced by the map G G (x, y) xy− G. However his argumentation strongly depends on the× finiteness∋ of7→ the quasi∈ invariant measures in the homogeneous spaces (G G)/(G1 G2) and G/G1 which slightly simplifies the construction of the σ-rings× of measurable× subsets in the 1 corresponding spaces of double cosets. Our proof that the map (x, y) xy− induces isomorphism of the respective spaces of double cosets must have7→ been slightly changed at this point by addition of Lemma 20. The rest of the proof of Theorem 8 of this Section follows from the Subgroup Theorem 7 in the same way as Theorem 7.2 from Theorem 7.1 in [31]. By the above remarks we shall show that the measure spaces (G G ): G 1 × 2 and G1 : G2 of double cosets constructed as in Sect. 8 are isomorphic, with 1 the isomorphism induced by the map G G (x, y) xy− G. Note first of all that the indicated map sets up a one-to-one× ∋ correspondenc7→ ∈ e between the double cosets in (G1 G2): G and double cosets in G1 : G2, in which the double × 1 coset (G1 G2)(x, y)G corresponds to the double coset G1xy− G2. Moreover in this mapping× a set is measurable if and only if its image is measurable and vice versa, a set is measurable if and only if its inverse image is measurable. Thus it is an isomorphism of measure spaces. Indeed (x1, x2) and (x2,y2) go into the same point of G under the indicated map if and only if they belong to the same left G coset in G G. Now by Lemma 20 of sect. 8 and by Lemma 1.2 of [31] (equally applicable× to left coset spaces) the indicated one-to-one map of double coset spaces is an isomorphism of measure spaces. Thus the Subgrop Theorem 7 µ1 µ2 L M is applicable to × U × with L replaced by L M, G replaced by G G, G1 replaced by G G and G replaced by G and the× function G x h(×x) G 1 × 2 2 ∋ 7→ ∈ 1 replaced by the function G G (x, y) h(x, y)= h1 (x),h2 (y) G1 G2, where the functions G x × h ∋(x) G7→and G y h (y) G ∈correspond× ∋ 7→ 1 ∈ 1 ∋ 7→ 2 ∈ 2 to the respective Borel sections of G with respect to G1 and G2 respectively used in the construction of the representations µ1 U L and µ2 U M (compare Sect. 3 and 6). In order to simplify formulation of the upcoming theorem let us give the following

85 12 DEFINITION 5. Let ν0 be the admissible measure on the set of double cosets G G ): G in G G given by Def. 4, where we have used the product quasi 1 × 2 × invariant measure µ = µ µ on the homogeneous space G G G G . 1 × 2 × 1 × 2 Let us deﬁne the measure ν12 on the space G1 : G2 of double cosets in G to be 12 1 equal to the transfer of ν0 by the map induced by G G (x, y) xy− G. (x,y) × ∋ 7→ ∈ If (µ1 µ2) is the quasi invariant measure on G/G(x,y) given by Def. 4 with × 1 x,y G = G (x, y)− (G G )(x, y) then we deﬁne µ to be the transfer of the (x,y) ∩ 1× 2 measure (µ µ )(x,y) over to the homogeneous space G x 1G x y 1G y 1 2 − 1 − 2 by the map (×x, x) x. ∩ 7→ Now we are ready to formulate the main goal of this paper:

THEOREM 8 (Kronecker Product Theorem). Let G1 and G2 be regularly related closed subgroups of the separable locally compact group G. Let L and M be Krein-unitary representations of G1 and G2 respectively in the Krein spaces ( L, JL) and ( M , JM ). For each (x, y) G G consider the Krein- H H x y ∈ × unitary representations L : s Lxsx−1 and M : s Mysy−1 of the subgroup 1 1 7→ 7→ (x− G1x) (y− G2y) in the Krein spaces ( L, JL) and ( M , JM ) respectively. Let us denote∩ the tensor product Lx M y Krein-unitaryH representationH acting in ⊗ x,y N x,y the Krein space ( L JM , JL JM ), by N . Let U be the Krein-isometric H ⊗ ⊗ x,y N x,y representation of G induced by N acting in the Krein space , Jx,y , x,y H where for each w N ∈ H

−1 −1 Jx,y w = Lh (xs) JL Lh (xs) Mh (ys) JM Mh (ys) w ; s 1 1 ⊗ 2 2 s x,y and with the inner product in Hilbert space N and the Krein-inner product x,y in the Krein space N , J given by theH formulas H x,y

x,y (w,g)x,y = JL JM Jx,y w , g dµ ([s]) ⊗ s s −1 Z −1 G x G1x y G2y ∩ and

(w,g) = (J w,g) Jx,y x,y x,y

x,y = JL JM w , g dµ ([s]), ⊗ s s −1 Z −1 G x G1x y G2y ∩

x,y with the quasi invariant measure µx,y given by Def. 5. Then U N is determined 1 to within Krein-unitary equivalence by the double coset D = G1xy− G2 to which 1 N x,y D L M xy− belongs and we may write U = U . Finally U U is Krein-unitary D ⊗ equivalent to the direct integral of U with respect to the measure ν12 (Def. 5) on G1 : G2. By the above remarks the Subgroup Theorem 7 is applicable to the re- µ µ L M striction of the representation 1× 2 U × of G G to the subgroup G. By this ×

86 µ µ L M theorem, 1× 2 U × restricted to G is a direct integral over the space of double cosets (G1 G2)(x, y)G with exactly one representant (x, y) for each double × (L M)(x,y) coset, of the representations U × of the subgroup G (i. e. with (x, y) ranging over B1 B2 – the corresponding section of G G with respect to double × (L×M)(x,y) cosets (G1 G2): G). Each of the representations U × of G is induced by the Krein-unitary× representation (L M)(x,y) : (s,s) (L M) = × 7→ × (x,y)(s,s)(x,y)−1 1 Lxsy−1 Mxsy−1 of the subgroup G(x,y) = G (x, y)− (G1 G2)(x, y) G in ⊗ ∩ (L M)(x,y×) ⊂ the Krein space ( L M , JL JM ). Moreover U × acts in the Krein (L M)(x,yH) ⊗ H ⊗ (L M)(x,y) space × , J where for each function w × we have H (x,y) ∈ H −1 J(x,y) w = (L M)h((x,y) (s,s))JL M (L M)h((x,y) (s,s)) w (s,s) × · × × · (s,s) =L M)(h (xs),h (ys))JL M (L M)(h (xs)−1,h (ys)−1) w × 1 2 × × 1 2 (s,s) = Lh (xs) JL Lh (xs)−1 Mh (ys) JM Mh (xs)−1 w . 1 1 ⊗ 2 2 (s,s) (L M)(x,y) (L M)(x,y) The inner product in × and Krein-inner product in × , J(x,y) are deﬁned by H H

(x,y) (w,g)(x,y) = JL M J(x,y) w , g d(µ1 µ2) ([(s,s)]) × (s,s) (s,s) × Z G/G(x,y)

(x,y) = JL JM J(x,y) w , g d(µ1 µ2) ([(s,s)]) ⊗ (s,s) (s,s) × Z G/G(x,y) and

(w,g)J = (J(x,y) w,g)(x,y) (x,y)

(x,y) = JL M w , g d(µ1 µ2) ([(s,s)]) × (s,s) (s,s) × Z G/G(x,y)

(x,y) = JL JM w , g d(µ1 µ2) ([(s,s)]); ⊗ (s,s) (s,s) × Z G/G(x,y) with the quasi invariant measure (µ µ )(x,y) on G/G given by Def. 4. 1 × 2 (x,y) Now under the natural isomorphism (x, x) x transferring G onto G the 1 7→ group G(x,y) = G (x, y)− (G1 G2)(x, y) is transferred onto the subgroup 1 1 ∩ × x− G1x y− G2y of G and the homogeneous space G/G(x,y) with the quasi ∩ (x,y) invariant measure (µ1 µ2) is transferred over to the homogeneous space 1 1 × G x− G1x y− G2y with the quasi invariant measure, which we denote by µx,y. ∩

87 11 Krein-isometric representations induced by decomposable Krein-unitary representations

We say a family S of operators in a Hilbert space is reducible by an idempotent P (i. e. a bounded operator P which satisfiesH the identity P 2 = P ), or equivalently by a closed subspace equal to the range P of P , in case P UP = UP for all U S. We say the family S is decomposableH in case PU = UP for all U S∈. In this case the Hilbert space is the direct sum of ∈ H closed subspaces 1 = P and 2 = (I P ) and every operator in S is a direct sum of operatorsH UH and UH with U− actingH in , i = 1, 2. The closed 1 2 i Hi subspaces i, i = 1, 2, are orthogonal iff P is self adjoint. Moreover if ( , J) is a KreinH space, the closed subspaces , i =1, 2, are Krein-orthogonal iffH the Hi idempotent P is Krein-self-adjoint: P † = P . Now the Krein-isometric representations U L inherit decomposability from decomposability of L. Namely for each idempotent PL acting in the Krein space of the representation L we may define a natural idempotent P L by the formula (P Lf) = P f for f L provided P commutes with the representation x L x ∈ H L L. Checking that P L is well defined (with measurable x P Lf , υ for 7→ x L L L each υ L and P f = Lh P f ) and that P is a bounded idempotent hx x ∈ H L L is immediate. Moreover P likewise commutes with U and is self-adjoint whenever PL is. Thus in particular for the standard Krein-isometric representation we have the following THEOREM 9. Let H be a closed subgroup of the separable locally compact group G. Let U L be the Krein-isometric representation of G acting in the Krein space ( L, JL), induced by the Krein-unitary representation L of the subgroup H H, acting in the Krein space ( , J ). Let for a measure space (R, RR,m) the HL L operators of the representation L and the fundamental symmetry JL be decomposable:

L = λ L dm(λ) and JL = J L dm(λ) λ ZR ZR with respect to a direct integral decomposition

L = L dm(λ), (40) H Hλ ZR of the Hilbert space . Then HL

L L = λ dm(λ) (41) H H ZR and all operators of the representation U L and the fundamental symmetry JL

88 are decomposable with respect to (41), i. e.

L L U L = U λ dm(λ) and JL = Jλ dm(λ); ZR ZR

L L L where U λ is the Krein-isometric representation in the Krein space λ , Jλ induced by the Krein-unitary representation L of the subgroup H, actingH in the λ Krein space L, J L . Hλ λ

[Outline of the proof.] Let λ E(λ)L be the spectral measure associated with the decomposition (40). Consider7→ the direct integral decompositions

L = L(λ) dm(λ), H H ZR

JL = JL(λ) dm(λ) and U L = U L(λ) dm(λ), ZR ZR of L, JL and U L, associated with the corresponding spectral measure λ E(λH)L and the same measure m. Using the vector-valued version of (28) and7→ the Fubini theorem one shows that L(λ) = and the equalities of the λ L Radon-Nikodym derivatives H H

L d E(λ)LJLf , E(λ)Lg d E(λ)LJλ f , E(λ)Lg = , dm(λ) dm(λ) L d E(λ)LU Lf , E(λ)Lg d E(λ)LU λ f , E(λ)Lg = , dm(λ) dm(λ)

L for all f,g L in the domain of U L, which means that JL(λ) = Jλ and ∈L H U L(λ)= U λ . Using the Dunford-Gelfand-Mackey [8, 16] (or more general [15, 28]) spectral measures and corresponding decompositions, we could generalize the last theorem keeping Krein self adjointness of the idempotents of the decomposition of L just using Dunford or more general spectral measures), but abandoning their commutativity with JL, and thus discarding their self-adjointness.

But in decomposition of the Krein-isometric induced representation restricted to a closed subgroup as in the Subgroup Theorem (or respectively in decomposition of the tensor product of Krein-isometric induced representations as in the Kronecker Product Theorem) we have encountered Krein-isometric induced Lx Lx representations U in the Krein space ( , Jx ) with the non-standard funda- H Lx mental symmetry Jx instead of the standard one J (respectively their tensor N x,y N x,y Lx product U acting in the tensor product Krein space , Jx,y = y H H ⊗ M , J J ). In this case for each idempotent P acting in the representation H x ⊗ y L 89 x space ( L, JL) and commuting with L we could similarly define the correspond- x,y H L L Lx N ing operator P : (P f)x = PLfx for f . (Similarly we can define P ∈ H x,y for each idempotent PNx,y commuting with N .) However in this case with −1 non-standard fundamental symmetry Jx g = Lh (xs) JL Lh (xs) g , g s 1 1 s Lx ∈ −1 −1 (resp. Jx,y w = Lh (xs) JL Lh (xs) Mh (ys) JM Mh (ys) w , s 1 1 2 2 s H x,y ⊗ N x,y L N L w ) the operator P (or P ) is in general unbounded. Moreover P ∈ H Nx,y (resp. P ) is non self-adjoint in this case even if PL (resp. PNx,y ) is self- x,y adjoint. We hope the slightly misleading (unjustified) notation U N will cause no serious troubles. Thus in particular the Theorem 9 (and its generalizations with Dunford- Gelfand-Mackey spectral measure decompositions) cannot in general be imme- x,y diately applied to the representations U N standing in the Kronecker Product Theorem for the tensor product ofLopuszański representations of the double N x,y covering G = T4sSL(2, C) of the Poincarégroup. But U as a Krein- isometric representation of the semi-direct product G = T4sSL(2, C) defines an imprimitivity system in Krein space (in the sense of Sect 5) which is concentrated on a single orbit. We then restore the form of the ordinary induced x,y representation to U N by applying Theorem 5 of Sect. 5 but we need the generalized version of this theorem with the finite multiplicity condition 3) discarded. We hope to present in a subsequent paper the full analysis of the component x,y representations U N in the decomposition of tensor product ofLopuszański representations. The necessity of restoring the standard form of induced Krein-isometric representation to the component Krein-isometric representations in the decomposition of tensor product of standard induced Krein-isometric representations is the main difference in comparison to the Mackey theory of unitary induced representations. In case of the double covering G of the Poincarégroup this “restoring” is quite elaborate, but effectively computable. The case of tensor products of ordinary unitary induced representations may be rather effectively reduced to C R the harmonic analysis on “small groups” Gχp = SU(2, ) or SL(2, ) (see Sect. 5; for the harmonic analysis on SL(2, C) compare [23, 10, 11, 12].) and to the tensor products (computed in [42, 43, 44]) of Gelfand-Neumark representations of SL(2, C), with the help of the original Mackey’s Subgroup and Kronecker Product Theorems for unitary induced representations. Indeed for tensor products of integer spin representations (for both versions of the energy sign) these decompositions have indeed been computed by Tatsuuma [53]. Unfortunately the paper [53] presents only the results without proofs, and some of the results presented there are not correct, namely those under X). REMARK 8. Because the representation of the translation subgroup T 4 ⊂ T4sSL(2, C) inLopuszański-type representation is equivalent to the representation of T4 in direct sum of several (four in case of theLopuszański representation) representations of, say helicity zero, ordinary unitary induced representations of T4sSL(2, C), corresponding to the “light-cone orbit” in the momentum space, and the representation of T4sSL(2, C) in the Fock space is the direct

90 sum of symmetrized/anitisymmetrized tensor products of one-particle representations, then investigation of the multiplicity of the representation of T4 in the Fock space is reduced to the decomposition of tensor products of ordinary unitary induced representations of T4sSL(2, C).

ACKNOWLEDGEMENTS

The author is indebted for helpful discussions to prof. A. Staruszkiewicz. The author would especially like to thank prof. A. K. Wr´oblewski, prof. A. Staruszkiewicz, and prof. M. Je˙zabek for the warm encouragement. I would like to thank prof. M. Je˙zabek for the excellent conditions for work at INP PAS where the most part of this paper has come into being.

References

[1] Bleuler, K: Helv. Phys. Acta 23, 567 (1950). [2] Bognar, J.: Indeﬁnite Inner Product Spaces. Springer, Berlin (1974). [3] Bourbaki, N.: Elements of Mathematics. Integration I. Chapters 1-6, Springer-Verlag, Berlin, Heidelberg (2004). [4] Bourbaki, N.: Elements of Mathematics. Integration II. Chapters 7 - 9, Springer-Verlag, Berlin, Heidelberg (2004) (reed.). [5] Bourbaki, N.: Elements of Mathematics. General Topology. Chapters 5-10, Springer-Verlag, Berlin, Heidelberg (1989). [6] Connes, A.: Noncommutative Geometry, Acad. Press, San Diego (1994). [7] Dirac, P. A. M.: Comm. Dublin Inst. Advenced Studies A, No 1 (1943). [8] Dunford, N.: Bull. Amer. Math. Soc. 64, 217 (1958). [9] D¨utsch, M., Fredenhagen, K.: Commun. Math. Phys. 203, 71 (1999). [10] Ehrenpreis, L., Mautner, F. I.: Ann. of Math. 61, 406 (1955). [11] Ehrenpreis, L., Mautner, F. I.: Trans. Amer. Math. Soc. 84, 1 (1957). [12] Ehrenpreis, L., Mautner, F. I.: Trans. Amer. Math. Soc. 90, 431 (1959). [13] Engelking, R., Sieklucki, K.: Wst¸ep do topologii, BM 62, PWN, Warszawa 1986. [14] Federer, H. and Morse, A. P.: Bull. Amer. Math. Soc. 49, 270 (1943). [15] Foias, C.: Bull. Sci. Math. 84, 147 (1960).

91 [16] Gelfand, I. M.: Notes of the Moscow State University, vol. IV, No 148, p. 224 (1951). [17] Gelfand, I. M., Silov,ˇ C. R. Acad. Sci. SSSR 23 (No 5), (1939). [18] Gelfand, I. M., Graev, M. I. and Vilenkin, N. Ya.: Integral Geometry and Representation Theory: Generalized functions. Vol. 5., Acad. Press, New York, 1966. [19] Gelfand, I. M. and Vilenkin, N. Ya.: Applications of Harmonic Analysis: Generalized functions. Vol. 4., Acad. Press, New York, 1964. [20] Gupta, S. N.: Proc. Phys. Soc. 63, 681 (1950). [21] Haag, R.: Local Quantum Physics. Springer Verlag, 1996. [22] Halpern, F. R.: Special Relativity and Quantum Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, 1968. [23] Harish-Chandra: Proc. Nat. Acad. Sci. U. S. A. 38, 337 (1952). [24] Jech, T.: Set Theory (third ed.), Springer Verlag, berlin, Heidelberg, New York (2003). [25] Krein, M. G., Krein, S. G.: Doklady SSSR 27 (No 5), (1940). [26] Kupersztych, J.: Nuovo Cimento 31B, 1 (1976). [27] Kuratowski, K.: Topologie, Vol. I, Warsaw, 1933 (1st ed.). [28] Lanze, W. E.: Mat. Sborn. 61, 80 (1963). [29]Lopuszański, J.: Rachunek spinorów. PWN, Warszawa 1985. [30]Lopuszański, J.: Fortschritte der Physik 26, 261 (1978). [31] Mackey, G. W.: Ann. Math. 55, 101 (1952). [32] Mackey, G. W.: Proc. Nat. Acad. Sci. U.S.A. 35, 537 (1949). [33] Mackey, G. W.: Acta Math. 99, 265 (1958). [34] Mackey, G. W.: The Theory of Unitary Group Representations, The Uni- veristy of Chicago Press, Chicago, London, 1976. [35] Murray, F. J., von Neumann, J.: Ann. of Math. 37, 116 (1936). [36] von Neumann, J.: Ann. Math. 50, 401 (1949). [37] Neumark, M. A.: Math. Ann. 162, 147 (1965). [38] Neumark, M. A.: Acta. Sci. Szeged 26, 3 and 201 (1965). [39] Neumark, M. A.: Izv. AN SSSR (Math. Series) 30, 1111 (1966).

92 [40] Neumark, M. A.: Izv. AN SSSR (Math. Series) 30, 1229 (1966). [41] Naimark(Neumark) M. A.: Normed Rings, P.Nordhoff N. V.-Groningen- The Netherlands (1964). [42] Neumark, M. A.: Trudy Matem. O-va 8, 121 (1959). [43] Neumark, M. A.: Trudy Matem. O-va 9, 237 (1960). [44] Neumark, M. A.: Trudy Matem. O-va 10, 181 (1961). [45] Pauli, W.: Rev. Mod. Phys. 15, 175 (1945). [46] Pontrjagin, L. S.: Izv. Akad. Nauk SSSRSer. Mat. 8,243 (1944). [47] Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Func- tional Analysis. Acad. Press, New York 1980. [48] Rohlin, V. A.: Rec. Math. (Mat. Sbornik) (NS) 25, 107 (1949). [49] Segal, I. E.: Ann. Math. 57, 401 (1953). [50] Segal, I. E.: Decomposition of Operator Algebras. I. Memoirs of the Amer- ican Mathematical Society. No. 9, 1951. [51] Segal, I. E., Kunze, R. A.: Integrals and Operators, Springer-Verlag, Berlin (1978). [52] Strohmaier, A.: J. Geom. Phys. 56, 175 (2006). [53] Tatsuuma, N.: Proc. Japan Acad. 38, 156 (1962). [54] Wawrzycki, J.: Int. J. Theor. Phys 52, 2910 (2014). [55] Wawrzycki, J.: math-ph/180206719v3. [56] Weil, A.: L’intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles, Paris 1940. [57] Weinberg, S.: Phys. Rev. 134, B882 (1964). [58] Weinberg, S.: Phys. Rev. 135, B1049 (1964). [59] Wigner, E. P.: Ann. Math. 40, 149 (1939). [60] Wigner, E. P.: Z. Phys. 124, 665 (1948).