aa r X i v : . [ m a t h . F A ] S e p Admissible vectors and Radon-Nikodym theorems
F. G´omez-Cubillo
Dpto de An´alisis Matem´atico, IMUVa, Universidad de Valladolid, Facultad de Ciencias, 47011Valladolid, Spain. e-mail: [email protected]
Abstract
Admissible vectors lead to frames or coherent states under the action of a group by meansof square integrable representations. This work shows that admissible vectors can be seen asweights with central support on the (left) group von Neumann algebra. The analysis involvesspatial and cocycle derivatives, noncommutative L p -Fourier transforms and Radon-Nikodymtheorems. Square integrability confine the weights in the predual of the algebra and everythingmay be written in terms of a (right selfdual) bounded element. Keywords: locally compact group, unitary representation, admissible vector, von Neu-mann algebra, weight, Radon-Nikodym theorem, frame, coherent state
Admissible vectors lead to frames or coherent states under the action of a group. They fall intothe fields of square integrable representations and reproducing kernel Hilbert spaces. Exhaustivecriteria for the existence and characterization of admissible vectors have been established in thecase the group von Neumann algebra is type I, using the Plancherel formula of the group [12]. Lieand discrete groups have also received special attention; see [1, 4] and references therein. Techniquesin von Neumann algebras, using central decompositions with respect to a trace in the semifinitecase, have been used in the study of admissible vectors for unimodular separable groups [11, 3]and countable discrete groups [15, 3, 2]. The analysis in the general case (with type III summand)requires the use of weights (or Hilbert algebras), which generalize the notions of positive functionaland trace [24, 26]. A study for arbitrary groups by means of convolution Hilbert algebras havebeen done in [13].This work characterizes admissible vectors in terms of weights on the group von Neumannalgebras. For each admissible vector there exists a weight with central support that satisfies certaininequalities with respect to the canonical weight associated to the convolution Hilbert algebra.These inequalities move directly to Connes’s spatial derivatives and are also equivalent to certainholomorphic extension properties of cocycle derivatives. The result is a noncommutative Radon-Nikodym theorem for the involved weights. See Theorem 1 below.Square integrability is reflected in the fact that weights corresponding to admissible vectors arein the predual of the (left) group von Neumann algebra and, then, they are associated with positiveelements of the Fourier algebra of the group. This implies that the spatial and cocycle derivativescan be expressed in terms of a (right selfdual) bounded element and its noncommutative L -Fouriertransform. The Radon-Nikodym derivatives are then bounded operators (Theorem 2).The involved weights may commute, that is, each one may be invariant under the modular auto-morphism group of the other. This is the case when the cocycle derivatives form an one-parametergroup of unitary elements in the reduced von Neumann algebra, whose Stone’s generator is just theRadon-Nikodym derivative (Theorem 4). The reduced algebra is then semifinite (Corollary 6). If,moreover, the weights satisfy the KMS condition, the Radon-Nikodym derivative is in the centreof the algebra (Corollary 7). 1he unimodular groups are those whose canonical weight is a trace, with trivial modular group.Any weight is then invariant with respect to the canonical weight and Theorem 4 is applicable.Furthermore, for unimodular groups even the spatial derivatives are bounded operators (Theorem8) and KMS condition lead to admissible vectors described by traces (Corollary 9). For commutativegroups, admissible vectors are associated with classical (Abelian) Radon-Nikodym derivatives atthe side of the dual group (Theorem 10).The content is organized as follows: Section 2 introduces the terminology and notation usedalong the work. Section 3 deals with admissible vectors for arbitrary locally compact groups. Section4 restricts attention to commuting weights. Sections 5 and 6 treat the cases of unimodular andcommutative groups, respectively. Section 7 remarks some connections of this work with previousones on tracial conditions [11], dual integrable representations and brackets [2] and formal degrees[9]. Let us begin by introducing some terminology and notation mainly borrowed from Folland [10],Str˘atil˘a [23] and Takesaki [26].
Let G be a locally compact group (lc group, for brevity) with fixed left Haar measure ds . C c ( G )denotes the space of complex-valued continuous functions on G with compact support and L p ( G ),1 ≤ p ≤ ∞ , the usual L p -spaces with respect to ds . ( ·|· ) and || · || are, respectively, the innerproduct and norm in L ( G ). The modular function of G is denoted by δ G . Recall that δ G is acontinuous homomorphism from G into the multiplicative group R + that satisfies d ( st ) = δ G ( t ) ds , d ( s − ) = δ G ( s − ) ds , s, t ∈ G . (1)The left and right regular representations λ and ρ of G on L ( G ) are defined by( λ ( s ) f )( t ) := f ( s − t ) , s ∈ G, f ∈ L ( G ) , (2)( ρ ( s ) f )( t ) := δ / G ( s ) f ( ts ) , s ∈ G, f ∈ L ( G ) . (3)The left and right group von Neumann algebras of G are denoted by L ( G ) and R ( G ), i.e., L ( G ) := { λ ( s ) : s ∈ G } ′′ , R ( G ) := { ρ ( s ) : s ∈ G } ′′ (4)(double commutant). One has L ( G ) ′ = R ( G ). See e.g. [26, Sect.VII.3].The convolution product f ∗ g and involutions f f ♯ and f f ♭ are defined at first in C c ( G )by [ f ∗ g ]( s ) := R G f ( t ) g ( t − s ) dt, s ∈ G ,f ♯ ( s ) := δ G ( s − ) f ( s − ) , s ∈ G ,f ♭ ( s ) := f ( s − ) , s ∈ G , (5)where the bar denotes complex conjugation.The extension of the convolution product ∗ and involution ♯ to L ( G ) leads to a structure ofBanach *-algebra for L ( G ). The Banach dual space of L ( G ) is L ∞ ( G ). The symbol h· , ·i representsthis duality: h f, ϕ i := Z G f ( s ) ϕ ( s ) ds, f ∈ L ( G ) , ϕ ∈ L ∞ ( G ) . A function of positive type on G is a function ϕ ∈ L ∞ ( G ) that defines a positive linearfunctional on the C ∗ -algebra L ( G ), i.e., that satisfies h f ♯ ∗ f, ϕ i ≥ f ∈ L ( G ). In whatfollows P ( G ) denotes the set of continuous functions of positive type on G .Let A ( G ) be the Fourier algebra of G . It is well known [17, Sect.2.3] [26, Sect.VII.3] that thefollowing conditions are equivalent: 2i) ϕ ∈ A ( G ) + := P ( G ) ∩ A ( G ).(ii) ϕ = g ∗ g ♭ , g ∈ L ( G ).(iii) There exists a positive normal functional ω on L ( G ) such that ω ( λ ( s )) = ϕ ( s ), for all s ∈ G .Moreover, ϕ ω is a bijection of A ( G ) + onto L ( G ) + ∗ , the positive part of the predual of L ( G ).A vector f ∈ L ( G ) is left (right) bounded if there exists a bounded operator L f ( R f ) on L ( G ) such that L f g := f ∗ g ( R f g := g ∗ f ) , g ∈ C c ( G ) . A weight on a von Neumann algebra M is a map ω : M + → [0 , ∞ ], defined at first on the positivecone M + of M , satisfying the following conditions: ω ( A + B ) = ω ( A ) + ω ( B ) , ω ( αA ) = α ω ( A ) , A, B ∈ M + , α ≥ , where the convention 0(+ ∞ ) = 0 is used. This map can be extended uniquely to M . The weight ω is said to be semifinite if { A ∈ M + : ω ( A ) < ∞} generates M ; faithful if ω ( A ) = 0 for nonzero A ∈ M + ; normal if ω (sup A i ) = sup ω ( A i ) for bounded increasing nets { A i } in M + ; a trace if ω ( A ∗ A ) = ω ( AA ∗ ) for A ∈ M . See, e.g., [24, Sect.10.14] and [26, Sect.VII.1] for details. The canonical weight Ω l on L ( G ) is the normal semifinite faithful ( n.s.f. ) weight associatedto the left convolution Hilbert algebra C c ( G ) [26, Sect.VII.3]:Ω l ( A ) := (cid:26) || f || , if A = L ∗ f L f , f left bounded , + ∞ , otherwise . (6)Ω l is a trace iff G is unimodular.Let ∆ be the modular operator on L ( G ), multiplication by the modular function δ G , and J the modular conjugation , J : L ( G ) → L ( G ) , ( Jf )( s ) := δ − / G ( s ) f ( s − ) . The modular automorphism group { σ Ω l t } t ∈ R on L ( G ) associated with the canonical n.s.f.weight Ω l (see (6)) is of the form σ Ω l t ( A ) := ∆ it A ∆ − it , A ∈ L ( G ) . The centralizer of Ω l is the fixed-point set of the modular group { σ Ω l t } t ∈ R and is denoted by L ( G ) Ω l , i.e., A ∈ L ( G ) belongs to L ( G ) Ω l iff σ Ω l t ( A ) = A , t ∈ R . L ( G ) Ω l is a von Neumannsubalgebra of L ( G ). See [26, VIII.3].The canonical n.s.f. weight Ω r on R ( G ) is given by Ω r ( A ) := Ω l ( JAJ ), for A ∈ R ( G ), andthe corresponding modular automorphism group { σ Ω r t } t ∈ R on R ( G ) by σ Ω r t ( A ) := ∆ − it A ∆ it , A ∈ R ( G ) . Connes [7] introduced the notion of spatial derivative of a n.s. weight defined on a von Neu-mann algebra
M ⊂ B ( H ) with respect to a n.s.f weight defined on the commutant M ′ ⊂ B ( H ).In particular, for a n.s. weight ω on L ( G ), the spatial derivative dω/d Ω r is the unique (notnecessarily bounded) positive selfadjoint operator T such that ω ( L ∗ f L f ) = (cid:26) || T / f || , if f left bounded, f ∈ D ( T / ) , + ∞ , otherwise . For the definition and properties of the cocycle derivative ( Dω : D Ω) t of a n.s weight ω relative to a n.s.f weight Ω defined on a von Neumann algebra M see, e.g., [26, VIII.3] [23, Sect.3]. The support s ( ω ) of a normal weight defined on a von Neumann algebra M ⊂ B ( H ) is the (orthogonal)projection P ∈ M such that I − P is the lower upper bound of the family of projections P ′ ∈ M satisfying ω ( P ′ ) = 0, where I denotes the identity operator on H ; see [24, 5.15]. .3 Noncommutative L p -Fourier transforms We introduce the theory of noncommutative L p -spaces associated with a weight on the basis ofspatial derivatives, as proposed by Connes [7] and Hilsum [16]. These spatial L p -spaces are thebasis for the Fourier analysis on l.c. groups given by Terp [28]. For a survey of the theory ofnoncommutative L p -spaces see, e.g., [21] and references therein.In what follows, if T is a (not necessarily bounded) positive selfadjoint operator and P theprojection onto ker( T ) ⊥ , by definition, T it , t ∈ R , is the partial isometry coinciding with theunitary operator ( T P ) it on ker( T ) ⊥ and 0 on ker( T ).Let α ∈ R and let T be a closed densely defined operator on L ( G ) with polar decomposition T = V | T | . T is called α -homogeneous if it satisfies the following equivalent conditions [28,Lem.2.7]:(i) ρ ( s ) T = δ − αG ( s ) T ρ ( s ), for all s ∈ G .(ii) V ∈ L ( G ) and σ Ω r αt ( A ) | T | it = | T | it A , for all A ∈ R ( G ) and t ∈ R .A positive selfadjoint operator T on L ( G ) is ( − T = dω/d Ω r for a (necessarilyunique) n.s. weight ω on L ( G ) [7, Th.13]. In such case, the integral of T with respect to Ω r isdefined by Z T d Ω r := ω ( I ) ∈ [0 , ∞ ] , where I denotes the identity operator on L ( G ).For each p ∈ [1 , ∞ ), the noncommutative L p -space L p (Ω r ) is the set of ( − /p )-homogeneousoperators T on L ( G ) satisfying || T || pp := R | T | p d Ω r < ∞ . And L ∞ (Ω r ) is identified with L ( G )with the usual operator norm || · || ∞ := || · || B ( L ( G )) .In the sequel, for a preclosed operator T , we denote by [ T ] the closure of T . With the obviousscalar multiplication, sum ( S, T ) [ S + T ] and norm || · || p , L p (Ω r ) is a Banach space. Theoperation T T ∗ is an isometry of L p (Ω r ). By linearity, the integral R T d Ω r defined on L (Ω r ) + extends to a linear form on L p (Ω r ).For p ∈ [1 ,
2] and 1 /p + 1 /q = 1, the L p -Fourier transform of f ∈ L p ( G ) is the operator F p ( f ) on L ( G ) given by F p ( f ) g := f ∗ ∆ /q g, g ∈ D ( F p ( f )) . F ( f ) g = f ∗ g with domain D ( F ( f )) = L ( G ). F is a unitary transformation from L ( G ) onto L (Ω r ) [28, Th.3.2].Let p ∈ [1 ,
2] and 1 /p + 1 /q = 1. For each T ∈ L p (Ω r ), F p ( T ) denotes the unique function in L q ( G ) such that Z h ( t ) F p ( T )( t ) dt = Z [ F p ( h ) T ] dψ , h ∈ C c ( G ) . For p ∈ (1 , F p is simply the transpose of F p . For p = 1, the mapping F takes an element T ∈ L (Ω r ) into the unique function ϕ ∈ A ( G ) that defines the same element of L ( G ) ∗ as T does;in particular, F (cid:16) dωd Ω r (cid:17) = ϕ, ϕ ∈ L ( G ) + ∗ ≃ A ( G ) + . The mapping F is an isometry from L (Ω r ) onto A ( G ) [28, Th.5.2]. For p = 2, the contransfor-mation F is not exactly the inverse of F ; they are related by the formula F ( T ) = F − ( T ∗ ), T ∈ L (Ω r ). It follows that F : L (Ω r ) → L ( G ) is unitary.In the sequel we shall say that a bounded operator A on a Hilbert space H is nonsingular ona closed subspace H of H if the support and range of A are included in H and the restriction A |H admits a (bounded or not) inverse in H . In such case, the inverse ( A |H ) − shall be denotedsimply by A − . 4 Admissible vectors. General case.
Let G be a lc group with fixed left Haar measure ds and let π be a (continuous) unitary represen-tation of G on a Hilbert space H π with inner product ( ·|· ) π and norm || · || π . Given an element η ∈ H π , the orbit { π ( s ) η } s ∈ G is called a covariant frame with (frame) bounds < α ≤ β < ∞ when(F1) G ∋ s ( ξ | π ( s ) η ) π ∈ C is a measurable function for all ξ ∈ H π ;(F2) α || ξ || π ≤ Z G | ( ξ | π ( s ) η ) π | ds ≤ β || ξ || π for all ξ ∈ H π .In such case there exists ψ ∈ H π giving rise to a dual covariant frame { π ( s ) ψ } s ∈ G with the same frame bounds and every ξ ∈ H π has the representations ξ = Z G ( ξ | π ( s ) η ) π ( s ) ψ ds = Z G ( ξ | π ( s ) ψ ) π ( s ) η ds . (7)These representations must be interpreted in weak sense. Relations of this type are known as covariant frame expansions in harmonic analysis [4, Section 5.8] or covariant coherent state expan-sions in mathematical physics [1, Chapters 7 and 8]. The pair of vectors { η, ψ } is then called an admissible pair for ( π, H π ) with bounds α, β . If η = ψ , it is said that η is an admissible vector .The inequalities in (F2) have their images in terms of positive elements of the Fourier algebra A ( G ), weights on the left group von Neumann algebra L ( G ), the corresponding spatial derivativesand holomorphic extensions of the cocycle derivatives: Theorem 1
Let G be a lc group, ( π, H π ) a unitary representation of G , η ∈ H π and ϕ η ( s ) := ( π ( s ) η | η ) π , s ∈ G .
The following statements are equivalent:(i) { π ( s ) η } s ∈ G is a covariant frame for ( π, H π ) with bounds α, β .(ii) ϕ η ∈ A ( G ) + and α ( P f | f ) ≤ h f ♯ ∗ f, ϕ η i ≤ β ( P f | f ) , f ∈ C c ( G ) , (8) with P an (orthogonal) projection in the centre Z ( G ) = L ( G ) ∩ R ( G ) .(iii) The normal finite functional ω η ∈ L ( G ) + ∗ , corresponding to ϕ η ∈ A ( G ) + , has support s ( ω η ) ∈Z ( G ) and α Ω l ( A ) ≤ ω η ( A ) ≤ β Ω l ( A ) , A ∈ ( L ( G ) s ( ω η )) + . (9) In particular, ω η is faithful on L ( G ) s ( ω η ) . Moreover, s ( ω η ) = P .(iv) The spatial derivative dω η /d Ω r satisfies α ∆ s ( ω η ) ≤ dω η d Ω r ≤ β ∆ s ( ω η ) . (10) (v) ω η ∈ L ( G ) + ∗ is faithful on L ( G ) s ( ω η ) and the cocycle derivatives ( Dω η : D Ω l ) t and ( D Ω l : Dω η ) t , t ∈ R , can be extended to L ( G ) s ( ω η ) -valued σ -weakly continuous bounded functionson the horizontal strip D / := { z ∈ C : − / ≤ Im ( z ) ≤ } which are holomorphic in theinterior of the strip and such that || ( Dω η : D Ω l ) − i/ || ≤ p β and || ( D Ω l : Dω η ) − i/ || ≤ / √ α . f this is the case, ( Dω η : D Ω l ) − i/ is nonsingular on s ( ω η ) L ( G ) with bounded inverse, onehas ( D Ω l : Dω η ) − i/ = ( Dω η : D Ω l ) − − i/ and ω η ( A ) = Ω l (( Dω η : D Ω l ) ∗− i/ A ( Dω η : D Ω l ) − i/ ) , A ∈ L ( G ) , (11)Ω l ( A ) = ω η ((( Dω η : D Ω l ) − − i/ ) ∗ A ( Dω η : D Ω l ) − − i/ ) , A ∈ L ( G ) s ( ω η ) . (12) Proof: (i) ⇔ (ii) ⇔ (iii): If { π ( s ) η } s ∈ G is a covariant frame with frame bounds α, β , for a fixed ξ ∈ H π , Cauchy-Schwarz inequality implies that the map H π ∋ ζ R G ( ξ | π ( s ) η ) π ( π ( s ) | ζ ) π ds ∈ C is conjugated linear and bounded. By Riesz representation theorem, there exists a unique element S η ξ ∈ H π such that ( S η ξ | ζ ) π = Z G ( ξ | π ( s ) η ) π ( π ( s ) η | ζ ) π ds, ζ ∈ H π . (13)The mapping S η : H π → H π so defined is linear and, by (F2), satisfies α || ξ || π ≤ ( S η ξ | ξ ) π ≤ β || ξ || π , ξ ∈ H π , (14)i.e., αI H π ≤ S η ≤ βI H π , where I H π denotes the identity operator on H π . Thus, S η is a boundedpositive operator with bounded inverse. Put ψ := S − η η . Then the equalities ξ = S − η S η ξ = S η S − η ξ coincide with the representation formulas given in (7). In other words, { π ( s ) η } s ∈ G is a covariantframe with frame bounds α, β iff the operator V η : H π → L ( G ) , ( V η ξ )( s ) := ( ξ | π ( s ) η ) π , (15)is a bounded map and V ∗ η V η = S η satisfies (14). Furthermore, (13) implies that S η commutes with π ( s ) for all s ∈ G . Hence V S − η η = V η S − η is also bounded and the representation formulas (7) areequivalent to V ∗ S − η η V η = S − η V ∗ η V η = I H π , V ∗ η V S − η η = V ∗ η V η S − η = I H π . (16)From (7) it is clear that η and ψ are cyclic vectors for ( π, H π ). A similar calculation shows that V ∗ S − / η η V S − / η η = I H π , (17)that is, S − / η η is an admissible vector for ( π, H π ).On the other hand, ϕ η ∈ P ( G ) by its own definition [10, Prop.3.15]. Let ( π ′ , H π ′ ) be the cyclicrepresentation of G , with cyclic vector η ′ , induced by ϕ η [10, Sect.3.3]. Then, for any f ∈ L ( G ), h f, ϕ η i = Z G f ( s ) ϕ η ( s ) ds = ( π ′ ( f ) η ′ | η ′ ) π ′ = Z G f ( s )( π ′ ( s ) η ′ | η ′ ) π ′ ds . Hence, being ϕ η continuous, ϕ η ( s ) = ( π ( s ) η | η ) π = ( π ′ ( s ) η ′ | η ′ ) π ′ , s ∈ G . (18)By (18), the cyclic representations ( π ′ , H π ′ , η ′ ) and ( π, H π , η ) of G are equivalent and the unitaryoperator U : H π ′ → H π such that U ( π ′ ( s ) η ′ ) = π ( s ) η resolves the equivalence [10, Prop.3.23].Thus, { π ′ ( s ) η ′ } s ∈ G and { π ( s ) η } s ∈ G are simultaneously covariant frames with frame bounds α, β or not.Now, let us assume that h f ♯ ∗ f, ϕ η i ≤ β ( f | f ) , f ∈ C c ( G ) . (19)Then, for f ∈ C c ( G ), || π ′ ( f ) η ′ || π ′ = ( π ′ ( f ) η ′ | π ′ ( f ) η ′ ) π ′ = ( π ′ ( f ) ∗ π ′ ( f ) η ′ | η ′ ) π ′ = h f ♯ ∗ f, ϕ η i ≤ β || f || . T : L ( G ) → H π ′ such that T f = π ′ ( f ) η ′ for f ∈ C c ( G ). || T || ≤ √ β and T has dense range since η ′ is cyclic. Moreover, by the definition of π ′ ,for f, g ∈ C c ( G ), T L f g = T ( f ∗ g ) = π ′ ( f ∗ g ) η ′ = π ′ ( f ) π ′ ( g ) η ′ = π ′ ( f ) T g , that is, T intertwines π ′ and the left regular representation: T L f = π ′ ( f ) T, f ∈ C c ( G ) . Then one has, for f ∈ C c ( G ), T ∗ T L f = T ∗ π ′ ( f ) T = ( π ′ ( f ) ∗ T ) ∗ T = ( T L ∗ f ) ∗ T = L f T ∗ T .
Thus, T ∗ T belongs to the commutant R ( G ) of L ( G ).Let T = V | T | be the polar decomposition of T , where | T | = ( T ∗ T ) / . Then V V ∗ = I H π ′ and V ∗ V = P , where P is the projection from L ( G ) onto the support of | T | , (Ker | T | ) ⊥ =: H . For f ∈ C c ( G ) and g ∈ L ( G ), V L f | T | g = V | T | L f g = T L f g = π ′ ( f ) T g = π ′ ( f ) V | T | g . Being | T | L ( G ) dense in H , V puts into equivalence { π ′ , H π ′ } and the subrepresentation { λ |H , H } of the left regular representation. Let η := V ∗ η ′ ∈ L ( G ). Then η is a cyclic vector for { λ |H , H } and, for s ∈ G , ϕ η ( s ) = ( π ( s ) η | η ) π = ( π ′ ( s ) η ′ | η ′ ) π ′ = ( λ ( s ) η | η ) . (20)Note that ( λ ( s ) η | η ) = R G η ( s − t ) η ( t ) dt = ( η ∗ η ♭ )( s ). This proves that ϕ η ∈ A ( G ) + . Moreover,for f ∈ C c ( G ), f ∗ η = L f η = L f V ∗ η ′ = V ∗ π ′ ( f ) η ′ = V ∗ T f = | T | f . Hence, η is right bounded and R η = | T | . In particular, since || T || ≤ √ β ,0 ≤ R η ≤ p β P . (21)Moreover, being R η positive, η = η ♭ [20, Prop.2.5].Since, for f ∈ C c ( G ), h f, ϕ η i = Z G f ( s ) ϕ η ( s ) ds = Z G f ( s )( λ ( s ) η | η ) ds = ( L f η | η ) , the positive normal functional ω η ∈ L ( G ) + ∗ is of the form ω η ( A ) = ( Aη | η ) , A ∈ L ( G ) . (22)Then, for f ∈ L ( G ), f left bounded, ω η ( L ∗ f L f ) = || L f η || = || R η f || , (23)and (21) implies that ω η ( A ) ≤ β Ω l ( A ) for A ∈ ( P L ( G ) P ) + and ω η is null on (( I − P ) L ( G )( I − P )) + .In the opposite direction, assume now that ϕ η ∈ A ( G ) + and ω η ( A ) ≤ β Ω l ( A ) , A ∈ L ( G ) + . (24)Recall that, since ω η is normal and ω η ( λ ( · )) = ϕ η ( · ), for f ∈ C c ( G ), ω η ( L ∗ f L f ) = ω η (cid:16) Z G f ♯ ( s ) λ ( s ) ds Z G f ( t ) λ ( t ) dt (cid:17) = Z G Z G f ♯ ( s ) f ( t ) ω η ( λ ( st )) ds dt = Z G Z G f ♯ ( s ) f ( s − t ) ϕ η ( t ) ds dt = h f ♯ ∗ f, ϕ η i . (25)7hus, (24) implies (19) and then, repeating the above arguments, for f ∈ C c ( G ), ω η ( L ∗ f L f ) = ( L f η | L f η ) = ( R η f | R η f ) = h f ♯ ∗ f, ϕ η i , (26)so that, by (21), condition (24) really implies h f ♯ ∗ f, ϕ η i ≤ β ( P f | f ) , f ∈ C c ( G ) . Similar arguments can be found in Combes [5, Lemma 2.3] and Haagerup [14, Prop.2.4].Reasoning as before with (18), due to (20), the cyclic representations ( λ |H , H , η ), ( π ′ , H π ′ , η ′ )and ( π, H π , η ) of G are unitarily equivalent and the orbits { λ |H ( s ) η } s ∈ G , { π ′ ( s ) η ′ } s ∈ G and { π ( s ) η } s ∈ G are simultaneously covariant frames with frame bounds α, β or not. Thus, it onlyremains to deal with the first inequalities in (27), (8) and (9).For ( λ |H , H , η ) the corresponding operator V η given in (15) is of the form( V η f )( s ) = ( f | λ ( s ) η ) = Z G f ( t ) η ( s − t ) dt = Z G f ( t ) η ♭ ( t − s ) dt = ( f ∗ η ♭ )( s ) = ( f ∗ η )( s ) , because η = η ♭ . That is, V η = R η |H = | T | |H and S η = | T | |H . Recall also that H is just thesupport of | T | . Using (26) and reasoning as above, the first inequality in (14), α || f || ≤ ( S η f | f ) = ( | T | f | f ) = ( R η f | R η f ) , f ∈ H , is equivalent to √ α P ≤ R η and also to the first inequalities in (8) and (9) on ( P L ( G ) P ) + , being ω η null on (( I − P ) L ( G )( I − P )) + . By the first inequality in (9), ω η is faithful on ( P L ( G ) P ) + since Ω l is. Thus, the support s ( ω η ) coincides with P .To complete the proof of (i) ⇔ (ii) ⇔ (iii) we must see that P belongs to the centre Z ( G ) = L ( G ) ∩ R ( G ). Since ( λ |H , H ) is a subrepresentation of the left regular representation of G , H is left invariant, so that P ∈ R ( G ). On the other hand, according to [23, Th.7.4], the supports of ω η and dω η /d Ω r coincide. Since dω η /d Ω r is ( − dω η /d Ω r ) is invariant underall ρ ( s ), s ∈ G ; thus, P ∈ L ( G ).(iii) ⇔ (iv): Let us recall that d Ω l /d Ω r = ∆ on L ( G ) and that the centralizer L ( G ) Ω l can alsobe defined as the set of elements in L ( G ) that commute with ∆ [19, page 61]. In particular, thecentre Z ( G ) of L ( G ) is contained in L ( G ) Ω l [26, VI.1.23]. Since P ∈ Z ( G ), P commutes with ∆.This implies that d Ω l /d Ω r = ∆ P on L ( G ) P an the result follows from [7, Prop.8.(a)]: For n.s.weights ω and ω on a von Neumann algebra M ⊆ B ( H ) and a n.s.f. weight Ω ′ on M ′ , ω ≤ ω iff dω /d Ω ′ ≤ dω /d Ω ′ .(iii) ⇔ (v): By (9), ω η ≤ β Ω l as n.f.s weights on L ( G ) P . According to [26, VIII.3.17] (a resultdue to Connes [6]), this is equivalent to the fact that the cocycle derivative ( Dω η : D Ω l ) t canbe extended to an L ( G ) P -valued σ -weakly continuous bounded function on the horizontal strip D / which is holomorphic in the interior of the strip and such that || ( Dω η : D Ω l ) − i/ || ≤ √ β .Furthermore, in such case, (11) is satisfied. Now, by the uniqueness of the cocycle derivative andthe chain rule for cocycle derivatives of n.f.s weights [26, VIII.3.7],( D Ω l : Dω η ) t = ( Dω η : D Ω l ) − t , t ∈ R , and, using [26, VIII.3.17] again, the inequality α Ω l ≤ ω η as n.f.s weights on L ( G ) P is equivalentto the extension of ( D Ω l : Dω η ) t on D / such that the inequality || ∆ / R − η || ≤ / √ α and (12)hold. (cid:3) The proof of Theorem 1 uses a right selfdual bounded element η of L ( G ). Furthermore, η is also left bounded and all the ingredients in statements (i)-(v) of Theorem 1 can be written interms of η and its L -Fourier transform F ( η ):8 heorem 2 Under the equivalent conditions (i)-(v) of Theorem 1, the representation ( π, H π ) isunitarily equivalent to a cyclic subrepresentation ( λ |H , H ) of the left regular representation of G with a cyclic vector η such that:(a) P is the projection of L ( G ) onto H .(b) η is right bounded, η = η ♭ and √ α P ≤ R η ≤ p β P , (27) Thus, R η is positive and nonsingular on H with bounded inverse.(c) { λ |H ( s ) η } s ∈ G is a covariant frame for ( λ |H , H ) with bounds α, β .(d) ϕ η ( s ) = ( λ ( s ) η | η ) , for s ∈ G .(e) ω η ( A ) = ( Aη | η ) , for A ∈ L ( G ) .(f ) η is left bounded, L η is nonsingular on H with bounded inverse and ( Dω η : D Ω l ) − i/ = L η , ( D Ω l : Dω η ) − i/ = L − η . Therefore, ω η ( A ) = Ω l ( L ∗ η AL η ) , A ∈ L ( G ) , (28)Ω l ( A ) = ω η (( L − η ) ∗ AL − η ) , A ∈ L ( G ) P . (29) (g) ϕ η ( s ) = Ω l ( L ∗ η λ ( s ) L η ) , for s ∈ G .(h) The spatial derivative dω η /d Ω r is given by dω η d Ω r = F ( η ) F ( η ) ∗ = F ( η ) F ( Jη ) = L η ∆ L ∗ η . (i) The cocycle derivative ( Dω η : D Ω l ) t ∈ L ( G ) P is ( Dω η : D Ω l ) t = ( F ( η ) F ( η ) ∗ ) it ∆ − it = ( L η ∆ L ∗ η ) it ∆ − it , t ∈ R . (j) { η , R − η η } is an admissible pair for ( λ |H , H ) with bounds α, β .(k) R − η η = L − η η is an admissible vector for ( λ |H , H ) . Proof: (a)-(e) are implicit in the proof (i) ⇔ (ii) ⇔ (iii) of Theorem 1.(f): According to [26, VIII.3.18.(ii)], for f ∈ C c ( G ), the element in L ( G ) corresponding to( Dω η : D Ω l ) − i/ (we are using the GNS representation of L ( G ) associated with Ω l ) is given by Jσ Ω l − i/ (( Dω η : D Ω l ) ∗− i/ ) Jf = J ∆ / ( Dω η : D Ω l ) ∗− i/ ∆ − / Jf .
On the other hand, (23) says that ω η ( L ∗ f L f ) = ( R η f | R η f ), for f ∈ C c ( G ). These facts togetherwith (11) imply that J ∆ / ( Dω η : D Ω l ) ∗− i/ ∆ − / J = R η and, then,( Dω η : D Ω l ) ∗− i/ = ∆ − / JR η J ∆ / = L η ♯ = L ∗ η . Now, (28) and (29) are just (11) and (12).(g): The result follows from (28) and ϕ η ( s ) = ω η ( λ ( s )) for s ∈ G .9h): According to (20), for s ∈ G , ϕ η ( s ) = ( λ ( s ) η | η ) = Z G η ( s − t ) η ( t ) dt = ( η ∗ η ♭ )( s ) . Since f ∗ g ♭ = F ([ F ( g ) F ( f ) ∗ ]) for all f, g ∈ L ( G ) [28, Cor.5.7], we have ϕ η = F ( F ( η ) F ( η ) ∗ ) , (30)that is, dω η /d Ω r = F ( η ) F ( η ) ∗ . Now, the second equality follows from the fact that F ( Jf ) = F ( f ) ∗ for all f ∈ L ( G ) [28, Prop.3.3]. Also, by (f), L η is nonsingular on H with bounded inverseand the n.s.f. weights ω η and Ω l on L ( G ) P satisfy ω η ( · ) = Ω l ( L ∗ η · L η ). Under these conditions,according to [23, Prop.7.13], dω η d Ω r = d Ω l ( L ∗ η · L η ) d Ω r = L η d Ω l d Ω r L ∗ η = L η ∆ L ∗ η . (i): According to [7, Th.9.(2)] (see also [23, Th.7.4]),( dω η /d Ω r ) it = ( Dω η : D Ω l ) t ( d Ω l /d Ω r ) it , t ∈ R . Then, (i) follows from (h) and d Ω l /d Ω r = ∆ [28, page 551].(j) and (k): At the end of the proof proof (i) ⇔ (ii) ⇔ (iii) of Theorem 1 we got V η = R η |H .Then, the arguments preceding (16) and (17) show that { η , R − η η } is an admissible pair for( λ |H , H ) with bounds α, β and R − η η is an admissible vector for ( λ |H , H ). Let us note that,since L η and R η commute, L − η η = R − η η ⇔ L η η = R η η ⇔ η ∗ η = η ∗ η . (cid:3) Remark 3
It is possible to dualize the above discussion entirely using the modular conjugation J in such a way that the roles of the left and right regular representations and the correspondingvon Neumann algebras are interchanged. See [13] for details. σ Ω -invariance Note that the result in Theorem 1.(v), concerning cocycle derivatives, is a Radon-Nikodym theo-rem for the weights ω η and Ω l ; see Equations (11) and (12). In this section we study particularcases of Theorems 1 and 2 closely related to the Pedersen-Takesaki work [19] on noncommutativeRadon-Nikodym theorems. They turn out when the weight ω η is invariant under the action ofthe automorphism group σ Ω l . This is the case when the reduced von Neumann algebra L ( G ) P issemifinite and { ( Dω η : D Ω l ) t } t ∈ R is an one-parameter group of unitary elements of L ( G ) P . Theorem 4
Assume that the equivalent statements (i)-(v) of Theorem 1 are satisfied. Put H := ( Dω η : D Ω l ) − i/ ( Dω η : D Ω l ) ∗− i/ = L η L ∗ η . Then the following additional conditions are equivalent:(PT.1) ( Dω η : D Ω l ) − i/ = L η commutes with ∆ , that is, it belongs to the centralizer L ( G ) Ω l .(PT.2) ω η is invariant with respect to { σ Ω l t } t ∈ R on L ( G ) , that is, ω η ( A ) = ω η (cid:0) σ Ω l t ( A ) (cid:1) , for A ∈L ( G ) and t ∈ R .(PT.3) Ω l is invariant with respect to { σ ω η t } t ∈ R on L ( G ) P , that is, Ω l ( A ) = Ω l (cid:0) σ ω η t ( A ) (cid:1) , for A ∈ L ( G ) P and t ∈ R . PT.4) ( Dω η : D Ω l ) t ∈ L ( G ) Ω l , for t ∈ R .(PT.5) ( Dω η : D Ω l ) t ∈ ( L ( G ) P ) ω η , for t ∈ R .(PT.6) ( D Ω l : Dω η ) t ∈ L ( G ) Ω l , for t ∈ R .(PT.7) ( D Ω l : Dω η ) t ∈ ( L ( G ) P ) ω η , for t ∈ R .(PT.8) { ( Dω η : D Ω l ) t } t ∈ R is an one-parameter group of unitary elements of L ( G ) P generated by H , i.e., ( Dω η : D Ω l ) t = H it , for t ∈ R .(PT.9) { ( D Ω l : Dω η ) t } t ∈ R is an one-parameter group of unitary elements of L ( G ) P generated by H − , i.e., ( D Ω l : Dω η ) t = H − it , for t ∈ R .(PT.10) ω η ( A ) = Ω l ( H / AH / ) , for A ∈ L ( G ) .(PT.11) Ω l ( A ) = ω η ( H − / AH − / ) , for A ∈ L ( G ) P .If this is the case, αP ≤ H ≤ βP . Proof:
Let us begin by recalling some results of the Pedersen-Takesaki work [19]. Let M be avon Neumann algebra and Ω a n.s.f weight on M . For each H ∈ M Ω+ ,Ω H ( A ) := Ω( H / AH / ) , A ∈ M + , (31)is a s.n. weight on M . If H is a positive selfadjoint operator affiliated with M Ω and H ε := H (1 + εH ) − for ε >
0, Ω H ( A ) := lim ε ↓ Ω( H / ε AH / ε ) , A ∈ M + , (32)also defines a n.s. weight on M ; this weight is faithful iff H is nonsingular. If ω is a n.s. weight on M , the following conditions are equivalent:(i) ω = ω ◦ σ Ω t , for t ∈ R .(ii) ( Dω : D Ω) t ∈ M ω , for t ∈ R .(iii) ( Dω : D Ω) t ∈ M Ω , for t ∈ R .(iv) { ( Dω : D Ω) t } t ∈ R is a strong(operator)-continuous group of unitary elements of s ( ω ) M s ( ω ),, where s ( ω ) denotes the support of ω .(v) ω = Ω H for some nonsingular positive selfadjoint operator H affiliated with M Ω .If moreover ω is faithful, then also the following statement is equivalent to those above:(vi) Ω = Ω ◦ σ ωt , t ∈ R .See also [26, Lem.VIII.2.7], [26, Lem.VIII.2.8], [26, Cor.VIII.3.6] and [23, Th.4.10].In our context, under the equivalent equivalent conditions (i)-(v) of Theorem 1, by Theorem2.(i), when L η commutes with ∆ one has( Dω η : D Ω l ) t = ( L η ∆ L ∗ η ) it ∆ − it = ( L η L ∗ η ) it = H it , t ∈ R , and, using [23, Cor.3.4] on L ( G ) P ,( D Ω l : Dω η ) t = ( Dω η : D Ω l ) − t = H − it , t ∈ R .
11n this case it is clear that the extensions of these cocycle derivatives on the horizontal strip D / given in Theorem 1.(v) lead to ( Dω η : D Ω l ) − i/ = H / and ( D Ω l : Dω η ) − i/ = H − / , so that || H / || ≤ p β and || H − / || ≤ / √ α or, in other words, αP ≤ H ≤ βP . Now, (PT.10) and (PT.11) follows, respectively, from (11)and (12).Note that, since u t = ( Dω η : D Ω l ) t ∈ L ( G ) Ω l for t ∈ R , one has u s + t = u s σ Ω l s ( u t ) = u s u t , for s, t ∈ R , that is, { ( Dω η : D Ω l ) t } t ∈ R is a group. (cid:3) Remark 5
The modular automorphism group { σ ω η t } t ∈ R on L ( G ) P is given by [7, Th.9.(1)]: σ ω η t ( A ) = (cid:16) dω η d Ω r (cid:17) it A (cid:16) dω η d Ω r (cid:17) − it , A ∈ L ( G ) P , t ∈ R . It is well known that every von Neumann algebra M is uniquely decomposable into the directsum of those of type I, II , II ∞ and III. If there is not summand of type III, then M is said to be semifinite . See, e.g., [25, Sect.V.1] for details.Pedersen-Takesaki work includes the following characterization of semifinite von Neumann alge-bras [19, Th.7.4]: There exists a n.s.f. weight ω on M such that { σ ωt } t ∈ R is inner in the sense thatthere exists a strong-continuous one parameter unitary group { u t } t ∈ R in M such that σ ωt ( · ) = u t · u ∗ t , t ∈ R . If this is the case, then with a fixed n.s.f. trace τ on M , every s.n. weight on M is writtenuniquely in the form τ H , with H a positive selfadjoint operator affiliated with M , where τ H isgiven by (32). See also [26, Th.VIII.3.14].The above comments and Theorem 4.(4) imply the following result. Corollary 6
Assume that the equivalent statements (i)-(v) of Theorem 1 and the additional equiv-alent conditions (PT.1)-(PT.11) of Theorem 4 are satisfied. Then the von Neumann algebra L ( G ) P is semifinite. Now, let M be a von Neumann algebra equipped with a one parameter automorphism group { σ t } t ∈ R . Let ω be an s.n. weight on M and put N ω := { A ∈ M : ω ( A ∗ A ) < ∞} . It is said that ω satisfies the (Kubo-Martin-Schwinger) KMS condition for { σ t } t ∈ R if the following two conditionshold:(i) ω = ω ◦ σ t , for t ∈ R .(ii) For every pair A, B ∈ N ω ∩ N ∗ ω there exists a function f = f A,B defined, continuous andbounded on the closed horizontal strip D − := { z ∈ C : 0 ≤ Im( z ) ≤ } , holomorphic on theopen strip D − and such that f ( t ) = ω ( σ t ( A ) B ) , f ( t + i ) = ω ( Bσ t ( A )) , t ∈ R . Corollary 7
Assume that the equivalent statements (i)-(v) of Theorem 1 are satisfied. Then thefollowing additional conditions are equivalent:(KMS.1) ω η satisfies the KMS condition with respect to { σ Ω l t } t ∈ R . (Here, N ω η = L ( G ) , since ω η ∈ L ( G ) + ∗ .)(KMS.2) σ ω η t = σ Ω l t |L ( G ) P for all t ∈ R . Really, these cocycle derivatives admit holomorphic extensions on C , since H ∈ L ( G ) Ω l is an entire analyticalelement. See [23, Sect.2.15]. KMS.3) H := L η L ∗ η belongs to the centre Z ( G ) . Proof:
Again Pedersen-Takesaki work [19] implies the following result (see also [23, Cor.4.11]):For a pair ω, Ω of n.s.f. weights on a von Neumann algebra M with centre Z , such that ω is finite,the following conditions are equivalent:(i) ω satisfies the KMS condition with respect to { σ Ω t } t ∈ R .(ii) s ( ω ) ∈ Z and σ ωt = σ Ω t |M s ( ω ) for all t ∈ R .(iii) ω ( · ) = Ω( H / · H / ) for some nonsingular positive selfadjoint operator H ∈ Z .The result is a straightforward consequence of Theorem 4 and these equivalences. (cid:3) Assume now that G is a lc unimodular group , i.e., the modular function is δ G = 1. Then:(U1) The modular operator is ∆ = I .(U2) Jf = f ♯ = f ♭ , for all f ∈ L ( G ). The sets of left bounded and right bounded elements of L ( G ) coincide, since L f = JR Jf J and R g = JL Jg J on them. In the sequel, left and rightbounded elements shall be called simply bounded .(U3) The canonical weight Ω l is a trace on L ( G ) and the corresponding modular automorphismgroup { σ Ω l t } t ∈ R is trivial, that is, σ Ω l t ( A ) = A , for A ∈ L ( G ) and t ∈ R .(U4) The von Neumann algebra L ( G ) is semifinite.(U5) The centralizer L ( G ) Ω l coincides with L ( G ).(U6) The α -homogeneous operators for any α ∈ R are simply the operators affiliated with L ( G ).The spaces L p (Ω r ) reduce to the noncommutative L p ( L ( G ) , Ω l ) spaces associated with atrace on a von Neumann algebra. This theory was laid out in the early 50’s by Segal [22] andDixmier [8]. See also [18], [27], [21] and [26, Sect.IX.2].(U7) For p ∈ [1 , L p -Fourier transform of f ∈ L p ( G ) is the left-convolution by f operatoron L ( G ): F p ( f ) g := f ∗ g, g ∈ D ( F p ( f )) . (U8) Fixed the n.s.f. trace Ω l on L ( G ), every s.n. weight on L ( G ) is written uniquely in the formΩ Hl , with H a positive selfadjoint operator affiliated with L ( G ), where Ω Hl is given by (32).See [26, Th.VIII.3.14].(U9) For all positive selfadjoint operators H affiliated with L ( G ), Z H d Ω r = Ω Hl ( I ) , d Ω Hl d Ω r = H , ( D Ω Hl : D Ω l ) t = (cid:16) d Ω Hl d Ω r (cid:17) it = H it , t ∈ R . See [28, Rem.3.2] and [7, Th.9.(2)].Due to these facts, Theorems 1 and 2 together with Theorem 4 lead to the following result forunimodular lc groups: 13 heorem 8
Let G be an unimodular lc group, ( π, H π ) a unitary representation of G , η ∈ H π and ϕ η ( s ) := ( π ( s ) η | η ) π , s ∈ G .
The following statements are equivalent:(u.i) { π ( s ) η } s ∈ G is a covariant frame for ( π, H π ) with bounds α, β .(u.ii) ϕ η ∈ A ( G ) + and α ( P f | f ) ≤ h f ♯ ∗ f, ϕ η i ≤ β ( P f | f ) , f ∈ C c ( G ) , (33) with P a projection in Z ( G ) .(u.iii) The normal finite functional ω η ∈ L ( G ) + ∗ , corresponding to ϕ η ∈ A ( G ) + , has support s ( ω η ) = P and satisfies α Ω l ( A ) ≤ ω η ( A ) ≤ β Ω l ( A ) , A ∈ ( L ( G ) P ) + . (34) (u.iv) ( Dω η : D Ω l ) − i/ = F ( η ) = L η ∈ L ( G ) + , dω η /d Ω r = L η and αP ≤ L η ≤ βP . (35) (u.v) The cocycle derivatives are of the form ( Dω η : D Ω l ) t = L itη , ( D Ω l : Dω η ) t = L − itη , t ∈ R , (36) where L η satisfies (35), and one has ω η = Ω L η l on L ( G ) and Ω l ( A ) = ω L − η η on L ( G ) P ,that is, ω η ( A ) = Ω l ( L η AL η ) , A ∈ L ( G ) , (37)Ω l ( A ) = ω η ( L − η AL − η ) , A ∈ L ( G ) P . (38) Proof:
Theorem 4 is of applicability thanks to (U5). L η is positive in this case, since η = η ♭ = Jη and L ∗ η = L η ♯ = L Jη . (cid:3) Corollary 7 for unimodular lc groups reads as follows:
Corollary 9
Assume that the equivalent statements (u.i)-(u.v) of Theorem 8 are satisfied. Thenthe following additional conditions are equivalent:(KMSu.1) For every pair
A, B ∈ L ( G ) there exists a function f = f A,B defined, continuous andbounded on the closed horizontal strip D − := { z ∈ C : 0 ≤ Im ( z ) ≤ } , holomorphicon the open strip D − and such that f ( t ) = ω η ( AB ) , f ( t + i ) = ω η ( BA ) , t ∈ R . (KMSu.2) ω η is a trace.(KMSu.3) L η ∈ Z ( G ) . Proof:
Note that (KMSu.1) is just the KMS condition for ω η with respect to the trivial au-tomorphism group { σ Ω l t } t ∈ R . With respect to (KMSu.2), ω η is a trace iff the corresponding auto-morphism group { σ ω η t } t ∈ R is trivial. (cid:3) Commutative case
Let G be a commutative lc group. Then:(C1) λ = ρ , L ( G ) = R ( G ) and, obviously, G is unimodular.(C2) The irreducible unitary representations of G are all one-dimensional [10, Cor.3.6]. They arecontinuous homomorphisms γ from G into the multiplicative group T of complex numberswith modulus 1, the (unitary) characters of G . With the topology of compact convergenceon G , the set ˆ G of all characters of G is a commutative lc group called the dual group of G .(C3) ˆ G can be identified with the spectrum of L ( G ) [10, Th.4.2]. The Gelfand transform on L ( G )then becomes the map from L ( G ) to C ( ˆ G ) defined by F f ( γ ) := ˆ f ( γ ) := Z G γ ( s ) f ( s ) ds . The map F is the Fourier transform on G and F ( f ∗ g ) = ˆ f · ˆ g , for f, g ∈ L ( G ).(C4) Let M ( ˆ G ) denote the set of bounded Radon complex-valued measures on ˆ G . If µ ∈ M ( ˆ G )we define the bounded continuous function ϕ µ on G by ϕ µ ( s ) := Z ˆ G γ ( s ) dµ ( γ ) . (39) Bochner theorem [10, Th.4.18] asserts that if ϕ ∈ P ( G ), there is a unique positive µ ∈ M ( ˆ G ) such that ϕ = ϕ µ as in (39).(C5) The Fourier-Stieltjes algebra B ( G ) is given by B ( G ) := { ϕ µ : µ ∈ M ( ˆ G ) } . The cor-respondence µ ϕ µ is a bijection from M ( ˆ G ) to B ( G ). We shall denote its inverse by ϕ µ ϕ . By Bochner’s theorem, B ( G ) is the linear span of P ( G ). The Fourier algebra is A ( G ) = B ( G ) ∩ L ( G ).(C6) One of the Fourier inversion theorems in this context reads as follows [10, Th.4.21]: If f ∈ A ( G ) then ˆ f ∈ L ( ˆ G ), and if Haar measure dγ on ˆ G is suitably normalized relative tothe given Haar measure ds on G , we have dµ f ( γ ) = ˆ f ( γ ) dγ ; that is, f ( s ) = Z ˆ G γ ( s ) ˆ f ( γ ) dγ . From now on, it will always be tacitly assumed that the Haar measures of G and ˆ G are soadjusted that the inversion theorem holds.(C7) In this case, Plancherel theorem [10, Th.4.25] says that the Fourier transform F on L ( G ) ∩ L ( G ) extends uniquely to a unitary isomorphism from L ( G ) onto L ( ˆ G ).Let us come back to the context of Theorem 8. Since H is invariant under convolution and F transform convolution into usual product, there exists a measurable closed subset S of ˆ G suchthat ˆ H := FH = { ˆ g ∈ L ( ˆ G ) : supp(ˆ g ) ⊆ S } . Let χ S denote the characteristic function of S .The following result shows that, when G is commutative, covariant frames are characterized bymeans of the classical Radon-Nikodym theorem on the dual group ˆ G . Theorem 10 If G is a commutative lc group, the equivalent statements (u.i)-(u.v) of Theorem 8are equivalent to the following ones: c.vii) ˆ η ≥ ; η ∗ η ∈ A ( G ) + or, equivalently, ˆ η ∈ L ( ˆ G ) + ; and αχ S ≤ ˆ η ≤ βχ S . (40) Note that this implies R ˆ G χ S dγ < ∞ .(c.viii) One has α Z ˆ G ˆ f dγ ≤ Z ˆ G ˆ f dµ η ∗ η ≤ β Z ˆ G ˆ f dγ, ˆ f ∈ ˆ H +0 . (41) (c.ix) The positive measures dµ η ∗ η = ˆ η dγ, dγ | S = ˆ η − dµ η ∗ η (42) are absolutely continuous one with respect to the other on S and, moreover, the Radon-Nikodym derivative ˆ η satisfies (40). Proof: ˆ η ≥ L η ∈ L ( G ) + in (u.v). ϕ η ∈ A ( G ) + in (u.iv) means that η ∗ η ∈ A ( G ) + . The equivalence η ∗ η ∈ A ( G ) + ⇔ ˆ η ∈ L ( ˆ G ) + follows from the Fourierinversion theorem in (C6). Equation (40) is just (35) moved to the dual group by means of theFourier transform given in (C3). The equivalence of Equations (41) and (34) is again a consequenceof the Fourier inversion theorem in (C6). Finally, (37) and (38) move to (42). (cid:3) We finish by including some brief remarks about connections of this work with the tracial condition in [11], dual integrable representations and brackets in [2] and the formal degree in [9].
Remark 11
For unimodular lc groups G and subrepresentations ( λ |H , H ) of the left regularrepresentation of G , F¨uhr [11, Th.2.2] gives a characterization of admissible vectors η as tracial ones on the reduced right von Neumann algebra P R ( G ) P , i.e., such thatΩ r ( B ) = ( Bη | η ) , B ∈ P R ( G ) P . (43)In this work, for arbitrary lc groups G , according to the definition of Ω l , (12) and Theorem 2.(e)and (k), for left bounded f ,( P f | f ) = Ω l ( L ∗ f L f P ) = ω η (( L − η ) ∗ L ∗ f L f P L − η )= ( L − η ) ∗ L ∗ f L f P L − η η | η ) = ( P L f L − η η | L f L − η η ) . Since the set of left bounded vectors is dense in L ( G ), this means that the weight Ω l on the reducedleft von Neumann algebra L ( G ) P is recovered by means of the admissible vector L − η η = R − η η .In other words, R L − η η = P , as required in [13, Th.5]. These results generalize the tracial condition(43); see Remark 3. Remark 12
Barbieri, Hern´andez and Parcet [2] analyze the properties of principal shift invariantspaces in a Hilbert space given by the action of an arbitrary countable discrete group. To be precise,let G be a countable discrete group and { δ s } s ∈ G the standard unit vector basis of l ( G ). Considerthe usual normalized trace τ on L ( G ): τ ( A ) := ( Aδ e | δ e ) , A ∈ L ( G ) , where e denotes the identity element of G . A unitary representation ( π, H π ) of G is called dualintegrable whenever there exists a map [ · , · ] : H × H → L ( L ( G ) , τ ) such that( φ | π ( s ) ψ ) π = τ (cid:0) [ φ, ψ ] λ ( s ) ∗ (cid:1) , φ, ψ ∈ H π , s ∈ G . η ∈ H π , let us consider the principal invariant subspace generated by η : h η i := span { π ( s ) η : s ∈ G } . Let P T denote the orthogonal projection in l ( G ) onto (Ker T ) ⊥ for any densely defined operator T on l ( G ). One of the main results is [2, Th.A]: Given any η ∈ H π , the system { π ( s ) η } s ∈ G isi) An orthonormal basis for h η i iff [ η, η ] = I .ii) A Riesz basis for h η i with frame bounds 0 < α ≤ β < ∞ iff α I ≤ [ η, η ] ≤ β I . iii) A frame for h η i with frame bounds 0 < α ≤ β < ∞ iff α P [ η,η ] ≤ [ η, η ] ≤ β P [ η,η ] . A central tool in the proof of this result is the Hilbert space L ( L ( G ) , [ η, η ]) defined as follows:The functional || A || ,η := (cid:0) τ ( | A | [ η, η ]) (cid:1) / = (cid:12)(cid:12)(cid:12)(cid:12) A [ η, η ] / (cid:12)(cid:12)(cid:12)(cid:12) , A ∈ L ( G ) , is a seminorm on L ( G ). Let N η be the null space associated with the seminorm || · || ,η . Thespace L ( L ( G ) , [ η, η ]) is the closure of L ( G ) /N η in the || · || ,η norm. To each 0 = η ∈ H π therecorresponds an isometric isomorphism S η : h η i → L ( L ( G ) , [ η, η ]) satisfying S η [ π ( s ) η ] = λ ( s ), for s ∈ G ; see [2, Prop.3.4].Here, clearly, we can identify L ( L ( G ) , [ η, η ]) with the GNS representation associated with theweight ω η (see, e.g., [23, Sect.1.2]) and the bracket [ η, η ] itself with the spatial derivative dω η /d Ω r .In [2] the authors apply these results in the concrete framework of unitary representations givenby measurable actions of G on σ -finite measure spaces ( X, µ ). In this scenario, a noncommutativeZak transform , defined as a measurable field of operators over X , and a tiling property play thecentral roles. The generalization of these ideas to arbitrary groups deserves further study. Remark 13
Given a representation ( π, H π ) of a lc group G , the coefficients of π are the functions c φ,ψ , φ, ψ ∈ H π , where c φψ ( s ) := ( φ | π ( s ) ψ ) π , s ∈ G .
Duflo and Moore [9, Th.2] prove that an irreducible representation π of G is equivalent to asubrepresentation of the left regular representation λ iff it has a nonzero square integrable co-efficient. Representations satisfying these conditions are usually called square integrable . Todetermine which coefficients are square integrable and certain relations of orthogonality, they ex-tend to nonunimodular groups the notion of formal degree , which is a positive number in theunimodular case.For it, recall that a character γ of G is a continuous homomorphism of G in C ∗ := C \{ } . Let( π, H ) and ( π ′ , H ′ ) be representations of G . A densely defined closed operator T from H to H ′ iscalled semi-invariant with weight γ if π ′ ( s ) T = γ ( s ) T π ( s ) , s ∈ G . (44)Given an arbitrary unit vector η of H π , let us consider T η : D ( T η ) ⊆ H π → L ( G ) : ψ c ηψ = ( π ( s ) ψ | η ) , where the domain D ( T η ) is the set of ψ ∈ H π such that c ηψ ∈ L ( G ). T η is a closed operatorsemi-invariant relative to π and ρ with weight δ / G , that is, ρ ( s ) T φ = δ / G ( s ) T φ π ( s ) , s ∈ G . formal degree of the representation ( π, H π ) is the operator K defined by K := ( T ∗ η T η ) − and the result reads as follows [9, Th.3]: If ( π, H π ) is a square integrable representation of G , thenthere exists a unique operator K in H π , selfadjoint positive, semi-invariant with weight δ − G , andsatisfying the following conditions:(a) If φ, ψ ∈ H π , f = 0, then c φψ is square integrable iff ψ ∈ dom K − / .(b) If φ, φ ′ ∈ H π and ψ, ψ ′ ∈ dom K − / ,( c φψ | c φ ′ ψ ′ ) = ( φ | φ ′ ) π ( K − / ψ | K − / ψ ′ ) π . The uniqueness of K in the case of square integrable representations is due to an extension ofSchur lemma [9, Sect.2]: If η ′ ∈ H π , η ′ = 0, then T η ′ = c η ′ U η ′ K − / , where c η ′ ∈ (0 , ∞ ) and U η ′ isan isometry that intertwines π and ρ .Here, for an arbitrary representation ( π, H π ) of G and η ∈ H π , we consider the operator T η T ∗ η on L ( G ), which is selfadjoint positive and semi-invariant with respect to ρ with weight δ G . That is,using the terminology of noncommutative L p -spaces, T η T ∗ η is ( − ω η on L ( G ) such that T η T ∗ η = dω η d Ω r . Theorem 1 says that { π ( s ) η } s ∈ G is a covariant frame for ( π, H π ) with bounds α, β iff ω η satisfiesany of the equivalent statements (iii)-(v) of the theorem. In this context, orthogonality relationshave a clear description dealing with standard forms ; see [13, Sect.4] for details. Acknowledgments
I would like to thank Ulpiano Ossorio and Margot Men´endez for stimulating discussions and kindhospitality at Madrid and San Juan de la Mata.
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