Harmonic Analysis of Covariant Functions of Characters of Normal Subgroups
aa r X i v : . [ m a t h . F A ] F e b HARMONIC ANALYSIS OF COVARIANT FUNCTIONS OFCHARACTERS OF NORMAL SUBGROUPS
ARASH GHAANI FARASHAHI
Abstract.
Let G be a locally compact group with the group algebra L ( G ) and N be a closed normal subgroup of G . Suppose that ξ : N → T is a continuous characterand L ξ ( G, N ) is the L -space of all covariant functions of ξ on G . We showed that L ξ ( G, N ) is isometrically isomorphic to a quotient space of L ( G ). It is also provedthat the dual space L ξ ( G, N ) ∗ is isometrically isomorphic to L ∞ ξ ( G, N ). Introduction
The covariant functions of characters (one-dimensional continuous irreducible unitaryrepresentations) of closed subgroups arise as building blocks with different applicationsin variant areas such as number theory (automorphic forms), induced representations,homogeneous spaces, complex (hypercomplex) analysis, coherent states, and covariantanalysis, see [1, 7, 10, 11, 12, 13, 14, 15].In general, classical harmonic analysis methods cannot be applied for covariant func-tions of characters of arbitrary closed subgroups. In the case of a compact subgroup,harmonic analysis on covariant functions studied in [8]. The following paper presentsa unified operator theoretic approach to study abstract harmonic analysis on covariantfunctions of characters of closed normal subgroups. We consider L -spaces of covariantfunctions of characters of normal subgroups in locally compact groups. It is discussedthat the introduced approach extends techniques used in [8] when the subgroup isnormal and compact. The presented approach also generalizes classical methods ofabstract harmonic analysis and functional analysis of L -spaces on quotient (factor)groups by considering the character as the trivial character of the normal subgroup,see [2, 3, 5, 16].This article contains 5 sections and organized as follows. Section 2 is devoted to fixnotations and provides a summary of classical harmonic analysis on locally compactcompact groups and factor (quotient) groups of locally compact groups. Let G bea locally compact group with the group algebra L ( G ) and N be a closed normalsubgroup of G . Suppose that ξ : N → T is a fixed character of N . In section 3 we studyabstract harmonic analysis of covariant functions of the character ξ using a classicaloperator theoretic approach. Next section considers the Banach space L ξ ( G, N ), the L -space of covariant functions of ξ on G , and investigates abstract harmonic analysis on L ξ ( G, N ). We then study some fundamental properties of the Banach space L ξ ( G, N ).Section 5 is devoted to show that L ξ ( G, N ) is isometrically isomorphic to a quotientspace of L ( G ). We conclude the paper by showing that the dual space L ξ ( G, N ) ∗ isisometrically isomorphic to L ∞ ξ ( G, N ). Mathematics Subject Classification.
Primary 43A15, 43A20, 43A85.
Key words and phrases. covariance property, normal subgroup, covariant function, character.E-mail addresses: [email protected] (Arash Ghaani Farashahi) . Preliminaries and Notations
Let X be a locally compact Hausdorff space. Then C c ( X ) denotes the space of allcontinuous complex valued functions on X with compact support. If λ is a positiveRadon measure on X , for each 1 ≤ p < ∞ the Banach space of equivalence classes of λ -measurable complex valued functions f : X → C such that k f k L p ( X,λ ) := (cid:18)Z X | f ( x ) | p d λ ( x ) (cid:19) /p < ∞ , is denoted by L p ( X, λ ) which contains C c ( X ) as a k · k L p ( X,λ ) -dense subspace.Let G be a locally compact group. A left Haar measure on G is a non-zero positiveRadon measure λ G on G which satisfies λ G ( xE ) = λ G ( E ) for every Borel subset E ⊆ G and every x ∈ G . It is well-known that every locally compact group G possessesa left Haar measure (Theorem 2.10 of [6]) and left Haar measures on G are uniqueup to scaling, see Theorem 2.20 of [6]. If λ G is a fixed left Haar measure on G and1 ≤ p < ∞ , then L p ( G ) stands for the Banach function space L p ( G, λ G ). For a function f : G → C and x ∈ G , the functions L x f, R x f : G → C are given by L x f ( y ) := f ( x − y )and R x f ( y ) := f ( yx ) for y ∈ G . The modular function of G , denoted by ∆ G is thecontinuous homomorphism ∆ G : G → (0 , ∞ ) which satisfies Z G f ( y )d λ G ( y ) = ∆ G ( x ) Z G R x f ( y )d λ G ( y ) , for every x ∈ G , left Haar measure λ G on G and f ∈ L ( G ).Suppose that Aut ( G ) is the group of all bicontinuous group automorphisms of G and α ∈ Aut ( G ). The Haar modulus σ G ( α ) ∈ (0 , ∞ ) is given by(2.1) Z G f ( y )d λ G ( y ) = σ G ( α ) Z G f ( α ( y ))d λ G ( y ) , for every left Haar measure λ G on G and f ∈ L ( G ), see [9]. Then σ G : Aut ( G ) → (0 , ∞ )given by α σ G ( α ) is a group homomorphism. It should be mentioned that existenceof the positive real number σ G ( α ) satisfying (2.1) guaranteed by the uniqueness of leftHaar measures on the locally compact group G .Let N be a closed normal subgroup of G with the left Haar measure λ N . Theneach left coset of N is a right coset of N and hence the coset space G/N is a locallycompact group called as factor (quotient) group of N in G . Then C c ( G/N ) consists ofall functions T N ( f ), where f ∈ C c ( G ) and(2.2) T N ( f )( xN ) = Z N f ( xs )d λ N ( s ) ( xN ∈ G/N ) . The factor (quotient) group
G/N has a left Haar measure λ G/N , which is normalizedwith respect to the following Weil’s formula(2.3) Z G/N T N ( f )( xN )d λ G/N ( xN ) = Z G f ( x )d λ G ( x ) , for all f ∈ L ( G ), see Thereom 3.4.6 of [16].A character ξ of N , is a continuous group homomorphism ξ : N → T , where T := { z ∈ C : | z | = 1 } is the circle group. In terms of group representation theory, eachcharacter of N is a 1-dimensional irreducible continuous unitary representation of N .We then denote the set of all characters of N by χ ( N ). A function ψ : G → C satisfiescovariant property associated to the character ξ ∈ χ ( N ), if ψ ( xs ) = ξ ( s ) ψ ( x ), for x ∈ G and s ∈ N . In this case, ψ is called as a covariant function of ξ . OVARIANT FUNCTIONS OF CHARACTERS OF NORMAL SUBGROUPS 3 Covariant Functions of Characters
Throughout this section, we shall present some of the theoretical aspects of contin-uous covariant functions of characters of closed normal subgroup. In this direction,we first review some fundamental properties of continuous covariant functions. Thecovariant functions developed in abstract harmonic analysis and group representationtheory in the construction of induced representations, see [6, 10]. We here discuss oneof the classical approaches to produce covariant functions.Suppose that λ N is a left Haar measure on N . For ξ ∈ χ ( N ) and f ∈ C c ( G ), definethe function T ξ ( f ) : G → C via T ξ ( f )( x ) := Z N f ( xs ) ξ ( s )d λ N ( s ) , for every x ∈ G .Suppose C ξ ( G, N ) is the linear subspace of C ( G ) given by C ξ ( G, N ) := { ψ ∈ C c ( G | N ) : ψ ( xk ) = ξ ( k ) ψ ( x ) , for all x ∈ G, k ∈ N } , where C c ( G | N ) := { ψ ∈ C ( G ) : q(supp( ψ )) is compact in G/N } , and q : G → G/N is the canonical map given by q( x ) := xN for x ∈ G . It is proventhat the linear operator T ξ maps C c ( G ) onto C ξ ( G, N ), see Proposition 6.1 of [6].
Remark . If ξ = 1 is the trivial character of N , we then have T ( f ) = T N ( f ). In thiscase, C ( G, N ) consists of functions on G which are constant on cosets of N . Therefore, C ( G, N ) can be canonically identified with C c ( G/N ) via ψ e ψ , where e ψ : G/N → C is given by e ψ ( xN ) := ψ ( x ) for every x ∈ G .We then have the following observations. Proposition 3.2.
Let G be a locally compact group, N be a closed subgroup of G , and ξ ∈ χ ( N ) . Suppose k ∈ N and y ∈ G . Then, (1) T ξ ◦ R k = ∆ N ( k − ) ξ ( k ) T ξ on C c ( G ) . (2) T ξ ◦ L y = L y ◦ T ξ on C c ( G ) .Proof. (1) Suppose that f ∈ C c ( G ), and k ∈ N . Let x ∈ G be given. We then have T ξ ( R k f )( x ) = Z N f ( xsk ) ξ ( s )d λ N ( s ) = Z N f ( xs ) ξ ( sk − )d λ N ( sk − )= ∆ N ( k − ) ξ ( k ) Z N f ( xs ) ξ ( s )d λ N ( s ) = ∆ N ( k − ) ξ ( k ) T ξ ( f )( x ) , which implies that T ξ ( R k f ) = ∆ N ( k − ) ξ ( k ) T ξ ( f ).(2) Suppose that f ∈ C c ( G ) and y ∈ G . Let x ∈ G be given. We then have T ξ ( L y f )( x ) = Z N L y f ( xs ) ξ ( s )d λ N ( s ) = Z N f ( y − xs ) ξ ( s )d λ N ( s ) = L y ( T ξ ( f ))( x ) , implying that T ξ ( L y f ) = L y ( T ξ ( f )). (cid:3) A. GHAANI FARASHAHI
Covariant Functions of Characters of Normal Subgroups.
We then con-tinue by investigation of some aspects of covariant functions of characters of normalsubgroups on locally compact groups. Throughout, let G be a locally compact groupand N be a closed normal subgroup of G . Then there exists a unique homomorphism σ N : G → (0 , ∞ ) such that(3.1) Z N v ( s )d λ N ( x − sx ) = σ N ( x ) Z N v ( s )d λ N ( s ) , and Z N v ( s )d µ N ( x − sx ) = σ N ( x ) Z N v ( s )d µ N ( s ) , for all left Haar measure λ N of N , right Haar measure µ N of N , v ∈ C c ( N ), and x ∈ G .It is worthwile to mention that existance of the homomorphism σ N : G → (0 , ∞ )guaranteed by the uniqueness of left Haar measures on the locally compact group N ,Theorem 2.20 of [6]. We then have σ N ( t ) = ∆ N ( t ) for t ∈ N and hence σ G ( x ) = ∆ G ( x )for x ∈ G . Also, the modular function of G/N satisfies ∆ G ( x ) = σ N ( x )∆ G/N ( xN ) for x ∈ G , see Proposition 11 of [2, Chap. VII, § N is central we thenhave σ N = 1.We then study some properties of T ξ , when the subgroup N is normal in G . Proposition 3.3.
Let G be a locally compact group and N be a closed normal subgroupof G . Suppose ξ ∈ χ ( N ) , x ∈ G , and f ∈ C c ( G ) . We then have T ξ ( f )( x ) = σ N ( x ) Z N f ( sx ) ξ ( x − sx )d λ N ( s ) . In particular, if N is a central subgroup of G then T ξ ( f )( x ) = Z N f ( sx ) ξ ( s )d λ N ( s ) . Proof.
Let f ∈ C c ( G ) be given. Then, for x ∈ G , we have T ξ ( f )( x ) = Z N f ( xs ) ξ ( s )d λ N ( s ) = Z N f ( xsx − x ) ξ ( s )d λ N ( s )= Z N f ( sx ) ξ ( x − sx )d λ N ( x − sx ) = σ N ( x ) Z N f ( sx ) ξ ( x − sx )d λ N ( s ) . If N is central in G then σ N ( x ) = 1 and ξ ( x − sx ) = ξ ( s ) for all x ∈ G and s ∈ N .Thus, we get T ξ ( f )( x ) = σ N ( x ) Z N f ( sx ) ξ ( x − sx )d λ N ( s ) = Z N f ( sx ) ξ ( s )d λ N ( s ) . (cid:3) Proposition 3.4.
Let G be a locally compact group, N be a closed normal subgroup of G , and ξ ∈ χ ( N ) . Suppose f ∈ C c ( G ) , x ∈ G , and k ∈ N . We then have T ξ ( L k f )( x ) = ξ ( x − kx ) T ξ ( f )( x ) . Proof.
Let τ x ( k ) := x − kx . Then, using Proposition 3.2(2), we have T ξ ( L k f )( x ) = L k T ξ ( f )( x ) = T ξ ( f )( k − x )= T ξ ( f )( xτ x ( k − )) = ξ ( τ x ( k − ) T ξ ( f )( x ) = ξ ( τ x ( k )) T ξ ( f )( x ) . (cid:3) OVARIANT FUNCTIONS OF CHARACTERS OF NORMAL SUBGROUPS 5
For functions f ∈ C c ( G ) and ψ ∈ C ξ ( G, N ), the functions f ψ and ψf are continuouswith compact supports. Therefore, they are integrable over G with respect to the leftHaar measure λ G . We shall denote the later integrals by h f, ψ i and h ψ, f i respectively. Theorem 3.5.
Let G be a locally compact group and N be a closed normal subgroupof G . Suppose ξ ∈ χ ( N ) and f, g ∈ C c ( G ) . We then have (3.2) h T ξ ( f ) , g i = h f, T ξ ( g ) i . Proof.
Let f, g ∈ C c ( G ) be given. We then have h T ξ ( f ) , g i = Z G T ξ ( f )( x ) g ( x )d λ G ( x )= Z G (cid:18)Z N f ( xs ) ξ ( s )d λ N ( s ) (cid:19) g ( x )d λ G ( x )= Z N (cid:18)Z G f ( xs ) g ( x )d λ G ( x ) (cid:19) ξ ( s )d λ N ( s )= Z N (cid:18)Z G f ( x ) g ( xs − )d λ G ( xs − ) (cid:19) ξ ( s )d λ N ( s )= Z N ∆ G ( s − ) (cid:18)Z G f ( x ) g ( xs − )d λ G ( x ) (cid:19) ξ ( s )d λ N ( s ) . Using normality of N in G and Proposition 3.3.17 of [16], we get ∆ G ( s ) = ∆ N ( s ) forevery s ∈ N . Therefore, we obtain h T ξ ( f ) , g i = Z N ∆ N ( s − ) (cid:18)Z G f ( x ) g ( xs − )d λ G ( x ) (cid:19) ξ ( s )d λ N ( s )= Z G f ( x ) (cid:18)Z N ∆ N ( s − ) g ( xs − ) ξ ( s )d λ N ( s ) (cid:19) d λ G ( x )= Z G f ( x ) (cid:18)Z N ∆ N ( s ) g ( xs ) ξ ( s )d λ N ( s − ) (cid:19) d λ G ( x )= Z G f ( x ) (cid:18)Z N g ( xs ) ξ ( s )d λ N ( s ) (cid:19) d λ G ( x ) = h f, T ξ ( g ) i . (cid:3) Harmonic Analysis on Covariant Functions of Characters of NormalSubgroups
In this section, we consider L -spaces of covariant functions of characters of normalsubgroups. We then study some of the basic properties of these classical Banach spacesof covariant functions of characters of normal subgroups. Throughout, suppose that G is a locally compact group and N is a closed normal subgroup of G . Let λ G be a leftHaar measure on G and λ N be a left Haar measure on N . Also, we assume that λ G/N is the left Haar measure on the factor (quotient) group
G/N normalized with respectto Weil’s formula (2.3). Let ξ ∈ χ ( N ) be given. For ψ ∈ C ξ ( G, N ), define the norm(4.1) k ψ k (1) := k| ψ |k L ( G/N,λ
G/N ) = Z G/N | ψ ( y ) | d λ G/N ( yN ) . If ψ ∈ C ξ ( G, N ), the function y
7→ | ψ ( y ) | reduces to constant on N and hence it dependsonly on the coset yN . Therefore, yN
7→ | ψ ( y ) | defines a function in C c ( G/N ) whichcan be integrated with respect to λ G/N . A. GHAANI FARASHAHI
Remark . Let ξ = 1 be the trivial character of N . Suppose that ψ ∈ C ( G, N ) isidentified with e ψ ∈ C c ( G/N ), due to Remark 3.1. Then k ψ k (1) = k e ψ k L ( G/N ) . Proposition 4.2.
Let G be a locally compact group and N be a compact normal sub-group of G . Suppose ξ ∈ χ ( N ) and ψ ∈ C ξ ( G, N ) . We then have (4.2) k ψ k (1) = λ N ( N ) k ψ k L ( G ) . Proof.
Since N is compact in G , we get C ξ ( G, N ) ⊆ C c ( G ), see Proposition 3.1(1) of[8]. Also, using compactness of N in G , each Haar measure of N is finite. Then usingWeil’s formula (2.3), for every ψ ∈ C ξ ( G, N ), we get k ψ k L ( G ) = Z G/N T N ( | ψ | )( xN )d λ G/N ( xN )= Z G/N Z N | ψ ( x ) | d λ N ( s )d λ G/N ( xN )= Z G/N (cid:18)Z N d λ N ( s ) (cid:19) | ψ ( x ) | d λ G/N ( xN ) = λ N ( N ) k ψ k (1) , (cid:3) Remark . Invoking Proposition 4.2, if N is compact in G and the left Haar measure λ G/N is normalized with respect to Weil’s formula and the probability measure λ N of N , we get k ψ k (1) = k ψ k L ( G ) , for every ψ ∈ C ξ ( G, N ). For more details on covariantfunctions of characters of compact subgroups, we refer the reader to [8].Next we show that T ξ : ( C c ( G ) , k · k L ( G ) ) → ( C ξ ( G, N ) , k · k (1) ) is a contraction. Theorem 4.4.
Let G be a locally compact group, N be a closed normal subgroup of G ,and ξ ∈ χ ( N ) . The linear operator T ξ : ( C c ( G ) , k · k L ( G ) ) → ( C ξ ( G, N ) , k · k (1) ) is acontraction.Proof. Let f ∈ C c ( G ) be given. Then, using Weil’s formula (2.3), we get k T ξ ( f ) k (1) = Z G/N | T ξ ( f )( y ) | d λ G/N ( yN )= Z G/N (cid:12)(cid:12)(cid:12)(cid:12)Z N f ( ys ) ξ ( s )d λ N ( s ) (cid:12)(cid:12)(cid:12)(cid:12) d λ G/N ( yN ) ≤ Z G/N Z N | f ( ys ) | d λ N ( s )d λ G/N ( yN )= Z G/N T N ( | f | )( xN )d λ G/N ( xN ) = k f k L ( G ) . (cid:3) We then continue by proving the following fundamental property of k · k (1) . Proposition 4.5.
Let G be a locally compact group, N be a closed normal subgroup of G , and ξ ∈ χ ( N ) . Then, for each ψ ∈ C ξ ( G, N ) , we have k ψ k (1) = inf (cid:8) k f k L ( G ) : f ∈ C c ( G ) , T ξ ( f ) = ψ (cid:9) . Proof.
Let ψ ∈ C ξ ( G, N ) be given. Suppose V ψ := { f ∈ C c ( G ) : T ξ ( f ) = ψ } . We thendefine γ ψ := inf (cid:8) k f k L ( G ) : f ∈ V ψ (cid:9) . Using Theorem 4.4, for every f ∈ V ψ , we have k ψ k (1) ≤ k f k L ( G ) . So we obtain k ψ k (1) ≤ γ ψ . We then also show that k ψ k (1) ≥ γ ψ . To OVARIANT FUNCTIONS OF CHARACTERS OF NORMAL SUBGROUPS 7 prove this, using Lemma 2.47 of [6], let h ∈ C c ( G ) be a positive function with T N ( h ) = 1on q(supp( ψ )). Put g ( x ) := ψ ( x ) h ( x ) for x ∈ G . Then T ξ ( g ) = ψ and hence g ∈ V ψ .Also, we get T N ( | g | )( xN ) = | ψ ( x ) | for x ∈ G . Then, using Weils’ formula (2.3), weachieve k g k L ( G ) = Z G/N T N ( | g | )( xN )d λ G/N ( xN ) = k ψ k (1) , implying that k ψ k (1) = k g k L ( G ) ≥ γ ψ . (cid:3) Suppose that N ξ ( G, N ) is the kernel of the linear operator T ξ : C c ( G ) → C ξ ( G, N ),which is the linear subspace given by N ξ ( G, N ) := { f ∈ C c ( G ) : T ξ ( f ) = 0 } . Then, N ξ ( G, N ) is a closed linear subspace of ( C c ( G ) , k · k L ( G ) ). Also, by applyingProposition 3.2(1), we achieve(4.3) span { R k f − ∆ N ( k − ) ξ ( k ) f : k ∈ N, f ∈ C c ( G ) } ⊆ N ξ ( G, N ) . Let X ξ ( G, N ) := C c ( G ) / N ξ ( G, N ) be the quotient normed space of N ξ ( G, N ) in C c ( G ),that is C c ( G ) / N ξ ( G, N ) = { f + N ξ ( G, N ) : f ∈ C c ( G ) } , equipped with the quotient norm given by(4.4) k f + N ξ k X ξ := inf (cid:8) k f + g k L ( G ) : g ∈ N ξ ( G, N ) (cid:9) . Suppose that X ξ ( G, N ) is the Banach completion of the normed space X ξ ( G, N ) withrespect to the quotient norm k · k X ξ given by (4.4).The following result characterizes the covariant function space C ξ ( G, N ) as a quotientspace of C c ( G ). Theorem 4.6.
Let G be a locally compact group, N be a closed normal subgroup of G , and ξ ∈ χ ( N ) . Then, ( C ξ ( G, N ) , k · k (1) ) is isometrically isomorphic to the quotientspace X ξ ( G, N ) .Proof. Invoking Proposition 6.1 of [6], the linear operator T ξ : C c ( G ) → C ξ ( G, N ) issurjective. Therefore, we deduce that the linear space C ξ ( G, N ) is isomorphic to thequotient linear space X ξ ( G, N ) via the linear operator U ξ : X ξ ( G, N ) → C ξ ( G, N )defined by U ξ ( f + N ξ ) := T ξ ( f ) for every f ∈ C c ( G ). The algebraic isomorphism U ξ isisometric as well, if the quotient linear space X ξ ( G, N ) is equipped with the quotientnorm (4.4). To show this, using Proposition 4.5, for every f ∈ C c ( G ), we achieve k U ξ ( f + N ξ ) k (1) = k T ξ ( f ) k (1) = inf {k h k L ( G ) : T ξ ( h ) = T ξ ( f ) } = inf {k h k L ( G ) : h − f ∈ N ξ } = inf {k f + g k L ( G ) : g ∈ N ξ } = k f + N ξ k X ξ . (cid:3) A. GHAANI FARASHAHI The Banach Space L ξ ( G, N )In this section, we study further properties of the L -spaces of covariant functionsof characters of normal subgroups. Throughout, we still suppose that G is a locallycompact group and N is a closed normal subgroup of G . Let λ G be a left Haar measureon G and λ N be a left Haar measure on N . We also assume that λ G/N is the left Haarmeasure on the factor (quotient) group
G/N normalized with respect to Weil’s formula(2.3). Let ξ ∈ χ ( N ) and L ξ ( G, N ) be the Banach completion of the normed linearspace C ξ ( G, N ) with respect to k · k (1) given by (4.1). We shall use the completion normby k · k (1) as well.The following characterization of L ξ ( G, N ) is also a canonical consequence of Theo-rem 4.6 and construction of the Banach space X ξ ( G, N ). Proposition 5.1.
Let G be a locally compact group, N be a closed normal subgroupof G , and ξ ∈ χ ( N ) . Then, X ξ ( G, N ) is isometrically isomorphic to the Banach space L ξ ( G, N ) .Remark . The abstract space L ξ ( G, N ) can be identified with a linear space ofcomplex-valued functions on G , where two functions are identified when they are equallocally almost everywhere (l.a.e). Let A ∈ L ξ ( G, N ) be an equivalent class of k · k (1) -Cauchy sequences in C ξ ( G, N ). Suppose ( ψ n ) ∞ n =1 is a representative for A . So ( ψ n ) ∞ n =1 is a k · k (1) -Cauchy sequence in C ξ ( G, N ) and k A k (1) = lim n k ψ n k (1) . Then choose asubsequence ( ψ ǫ ( k ) ) such that k ψ ǫ ( k +1) − ψ ǫ ( k ) k (1) ≤ / k , for k ∈ N . This impliesthat P ∞ k =1 k ψ ǫ ( k +1) − ψ ǫ ( k ) k (1) < ∞ . Hence, P ∞ k =1 k Ψ k k L ( G/N ) < ∞ with Ψ k := | ψ ǫ ( k +1) − ψ ǫ ( k ) | for k ∈ N . Since L ( G/N ) is a Banach space, there exists Ψ ∈ L ( G/N )with Ψ = P ∞ k =1 Ψ k in L ( G/N ), where the sum converges pointwise almost everywhereas well. Thus, ∞ X k =1 | ψ ǫ ( k +1) ( x ) − ψ ǫ ( k ) ( x ) | = ∞ X k =1 Ψ k ( xN ) < ∞ , for xN ∈ G/N except those in a subset set E of G/N with λ G/N ( E ) = 0. So, thecomplex series P ∞ k =1 ψ ǫ ( k +1) ( x ) − ψ ǫ ( k ) ( x ) , converges to a complex number, denotedby ψ ( x ), for x ∈ G with xN E . For x ∈ G with xN / ∈ E , let ψ ( x ) := ψ ǫ (1) ( x ) + ψ ( x ). Further, ψ ( x ) is independent of the choice of the Cauchy sequence ( ψ n ) as arepresentative of the class A , since locally null sets in G are precisely the images underthe canonical projection of the locally null sets in G/N , Theorem 3.3.28 of [16]. Theclass A can be uniquely determined with the locally almost everywhere defined function x ψ ( x ), as a complex valued function on G . Hence, we getlim n ψ ǫ ( n ) ( x ) = lim n ψ ǫ (1) ( x ) + n − X k =1 ψ ǫ ( k +1) ( x ) − ψ ǫ ( k ) ( x ) ! = ψ ( x ) . Under this identification, for each n , we have | ψ ǫ ( n ) ( x ) | ≤ | ψ ǫ (1) ( x ) | + n − X k =1 Ψ k ( x ) ≤ | ψ ǫ (1) ( x ) | + Ψ( x ) . Then Lebesgue’s Dominated Convergence Theorem, implies that the function xN ψ ( x ) | belongs to L ( G/N ) and(5.1) k A k (1) = lim n k ψ ǫ ( n ) k (1) = Z G/N | ψ ( x ) | d λ G/N ( xN ) . OVARIANT FUNCTIONS OF CHARACTERS OF NORMAL SUBGROUPS 9
Invoking the structure of the Banach space L ξ ( G, N ) and Theorem 4.4, we concludethat the linear operator T ξ : ( C c ( G ) , k · k L ( G ) ) → ( C ξ ( G, N ) , k · k (1) ) has a uniqueextension to a contraction from L ( G ) into L ξ ( G, N ), which we still denote this uniquebounded linear operator by T ξ : L ( G ) → L ξ ( G, N ).We then demonstrate the following explicit formula for T ξ : L ( G ) → L ξ ( G, N ). Theorem 5.3.
Let G be a locally compact group, N be a closed normal subgroup of G , and ξ ∈ χ ( N ) . The extended linear operator T ξ : L ( G ) → L ξ ( G, N ) is given by f T ξ ( f ) , where (5.2) T ξ ( f )( x ) = Z N f ( xs ) ξ ( s )d λ N ( s ) , for l . a . e . x ∈ G. Proof.
Suppose that f ∈ L ( G ) is given. For x ∈ G , let f x : N → C be given by f x ( s ) := f ( xs ) for s ∈ N . Invoking Theorem 3.4.6 of [16], there exists a λ G/N -nullset A in G/N such that f x ∈ L ( N ) for every x ∈ G with xN A . Since any λ G/N -null set is a locally λ G/N -null set, using Theorem 3.3.28 of [16], we conclude thatq − ( A ) is locally λ G -null in G . So f x ∈ L ( N ) for every x ∈ G with x q − ( A ).This implies that the right hand side of (5.2) is well-defined as a function on G , forl.a.e x ∈ G . Let ( f n ) ⊂ C c ( G ) with k f n − f k L ( G ) < − ( n +1) for every n ∈ N . Then k f n +1 − f n k L ( G ) < − n for n ∈ N . We also have T ξ ( f ) = lim n T ξ ( f n ) in L ξ ( G, N ).Let ψ n = T ξ ( f n ) for n ∈ N . Invoking Theorem 4.4, we get k ψ n +1 − ψ n k (1) < − n for n ∈ N . This implies that lim n ψ n ( x ) = T ξ ( f )( x ) for l.c.a x ∈ G , according to Remark5.2. Using Weil’s formula (2.3), for n ∈ N , we obtain Z G/N Z N | f n ( xs ) − f ( xs ) | d λ N ( s )d λ G/N ( xN ) = k f n − f k L ( G ) , which implies that lim n Z N | f n ( xs ) − f ( xs ) | d λ N ( s ) = 0 , for a.e. xN ∈ G/N . Since for n ∈ N and x ∈ G , we have (cid:12)(cid:12)(cid:12)(cid:12)Z N f n ( xs ) ξ ( s )d λ N ( s ) − Z N f ( xs ) ξ ( s )d λ N ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z N | f n ( xs ) − f ( xs ) | d λ N ( s ) , we conclude that(5.3) lim n Z N f n ( xs ) ξ ( s )d λ N ( s ) = Z N f ( xs ) ξ ( s )d λ N ( s ) , for a.e. xN ∈ G/N . Using (5.3), for l.c.a x ∈ G , we achieve T ξ ( f )( x ) = lim n T ξ ( f n )( x ) = lim n Z N f n ( xs ) ξ ( s )d λ N ( s ) = Z N f ( xs ) ξ ( s )d λ N ( s ) . (cid:3) Assume that N ξ ( G, N ) is the kernel of the linear map T ξ : L ( G ) → L ξ ( G, N ) in L ( G ), which is the linear subspace N ξ ( G, N ) := (cid:8) f ∈ L ( G ) : T ξ ( f ) = 0 in L ξ ( G, N ) (cid:9) . We then show that N ξ ( G, N ) is the closure of N ξ ( G, N ) in L ( G ). Proposition 5.4.
Let G be a locally compact group, N be a closed normal subgroup of G , and ξ ∈ χ ( N ) . Then, N ξ ( G, N ) is the closure of N ξ ( G, N ) in L ( G ) . Proof.
Suppose that X denotes the closure of N ξ ( G, N ) in L ( G ). Let f ∈ X bearbitrary. Then, f ∈ L ( G ) and lim n f n = f for some sequence ( f n ) ⊂ N ξ ( G, N ). Thencontinuity of the linear operator T ξ : L ( G ) → L ξ ( G, N ) guarantees that T ξ ( f ) = 0in L ξ ( G, N ). Therefore, f ∈ N ξ ( G, N ). Since f was given, we get X ⊆ N ξ ( G, N ).Conversely, suppose f ∈ L ( G ) with T ξ ( f ) = 0 in L ξ ( G, N ). Let ε > h ∈ C c ( G ) such that k f − h k L ( G ) < ε/ φ := T ξ ( h ). Let u ∈ C c ( G ) be anon-negative function with T N ( u ) = 1 on q(supp( φ )), according to Lemma 2.47 of [6].We then consider the function g : G → C defined by g ( x ) := h ( x ) − u ( x ) φ ( x ), for every x ∈ G . Then, g ∈ C c ( G ) and T ξ ( g ) = 0. Because, for every x ∈ G , we obtain T ξ ( g )( x ) = φ ( x ) − Z N u ( xs ) φ ( xs ) ξ ( s ) dλ N ( s ) = φ ( x ) − φ ( x ) T N ( u )( xN ) = 0 . Hence, we conclude that g ∈ N ξ ( G, N ). Then, using Weil’s formula (2.3), we achieve k h − g k L ( G ) = Z G u ( x ) | φ ( x ) | d λ G ( x ) = Z G/N T N ( u | φ | )d λ G ( x )= Z G/N T N ( u )( xN ) | φ ( x ) | d λ G ( x ) = Z G/N | φ ( x ) | d λ G ( x )= k T ξ ( h ) k (1) ≤ k T ξ ( h − f ) k (1) + k T ξ ( f ) k (1) ≤ k h − f k L ( G ) < ε . Therefore, we have k f − g k L ( G ) ≤ k f − h k L ( G ) + k h − g k L ( G ) < ε, which implies f ∈ X . Since f ∈ N ξ ( G, N ) was arbitrary, we get N ξ ( G, N ) ⊆ X . (cid:3) Next we have the following characterization of L ξ ( G, N ) as a quotient subspace of L ( G ). Theorem 5.5.
Let G be a locally compact group, N be a closed normal subgroup of G ,and ξ ∈ χ ( N ) . The Banach space L ξ ( G, N ) is isometrically isomorphic to the quotientBanach space L ( G ) / N ξ ( G, N ) .Proof. Invoking Proposition 5.4 and also Lemma 3.4.4 of [16], one can conclude thatthe Banach space X ξ ( G, N ) is isometrically isomorphic to the quotient Banach space L ( G ) / N ξ ( G, N ). Then, Proposition 5.1 guarantees that the Banach space L ξ ( G, N )is isometrically isomorphic to the quotient Banach space L ( G ) / N ξ ( G, N ). (cid:3) Corollary 5.6.
Let G be a locally compact group and N be a closed normal subgroupof G . Suppose ξ ∈ χ ( N ) and ψ ∈ L ξ ( G, N ) . We then have k ψ k (1) = inf (cid:8) k f k L ( G ) : f ∈ L ( G ) , T ξ ( f ) = ψ (cid:9) . Corollary 5.7.
Let G be a locally compact group, N be a closed normal subgroup of G , and ξ ∈ χ ( N ) . The extended contraction T ξ maps L ( G ) onto L ξ ( G, N ) . Let L ∞ ( G ) be the Banach space of all locally λ G -measurable functions f : G → C which are bounded except on a locally λ G -null set, modulo functions which are zerolocally a.e. on G , equipped with the norm k f k ∞ := inf { t : | f ( x ) | ≤ t l . a . e . x ∈ G } . OVARIANT FUNCTIONS OF CHARACTERS OF NORMAL SUBGROUPS 11
Then C ξ ( G, N ) ⊂ L ∞ ( G ). Suppose that L ∞ ξ ( G, N ) is the closed subspace of L ∞ ( G )defined by L ∞ ξ ( G, N ) := { ψ ∈ L ∞ ( G ) : R k ψ = ξ ( k ) ψ, for k ∈ N } . Then C ξ ( G, N ) ⊆ L ∞ ξ ( G, N ) and hence the closure of C ξ ( G, N ) in L ∞ ( G ) is included in L ∞ ( G, N ).We conclude the paper by the following characterization for the dual space L ξ ( G, N ) ∗ . Theorem 5.8.
Let G be a locally compact group, N be a closed normal subgroup of G ,and ξ ∈ χ ( N ) . Then, L ξ ( G, N ) ∗ is isometrically isomorphic to L ∞ ξ ( G, N ) .Proof. Let g Λ g be the antilinear isometric isomorphism identification of L ∞ ( G )as L ( G ) ∗ , where Λ g : L ( G ) → C is given by Λ g ( f ) := h f, g i for every f ∈ L ( G ).Using Theorem 5.5, the dual space L ξ ( G, N ) ∗ is isometrically isomorphic to N ξ ( G, N ) ⊥ ,where N ξ ( G, N ) ⊥ = (cid:8) g ∈ L ∞ ( G ) : Λ g ( f ) = 0 , for all f ∈ N ξ ( G, N ) (cid:9) . We then show that N ξ ( G, N ) ⊥ = L ∞ ξ ( G, N ). To this end, suppose that g ∈ N ξ ( G, N ) ⊥ is arbitrary. Then, g ∈ L ∞ ( G ) and Λ g ( f ) = 0 for every f ∈ N ξ ( G, N ). Let k ∈ N be given. Then, using (4.3) and Proposition 5.4, for every f ∈ C c ( G ) we get R k − f − ∆ N ( k ) ξ ( k ) f ∈ N ξ ( G, N ). So Λ g ( R k − f ) = ∆ N ( k ) ξ ( k )Λ g ( f ) and hence∆ N ( k − )Λ g ( R k − f ) = ξ ( k )Λ g ( f ), for every f ∈ C c ( G ). Since N is normal, we have∆ G | N = ∆ N . Therefore, for every f ∈ C c ( G ), we achieveΛ R k g ( f ) = Z G f ( x ) g ( xk )d λ G ( x )= Z G f ( xk − ) g ( x )d λ G ( xk − )= ∆ G ( k − ) Z G f ( xk − ) g ( x )d λ G ( x )= ∆ N ( k − ) Z G f ( xk − ) g ( x )d λ G ( x )= ∆ N ( k − )Λ g ( R k − f ) = ξ ( k )Λ g ( f ) = Λ ξ ( k ) g ( f ) . Since f ∈ C c ( G ) is given, we obtain R k g = ξ ( k ) g in L ∞ ( G ). Because k ∈ N was alsoarbitrary, we get g ∈ L ∞ ξ ( G, N ). Conversely, let g ∈ L ∞ ξ ( G, N ) be given. We thenclaim that Λ g ( f ) = 0 for every f ∈ N ξ ( G, N ). To this end, suppose that f ∈ N ξ ( G, N )is given. Then Λ g ( f ) = Z G f ( x ) g ( x )d λ G ( x )= Z G/N (cid:18)Z N f ( xs ) g ( xs )d λ N ( s ) (cid:19) d λ G/N ( xN )= Z G/N (cid:18)Z N f ( xs ) ξ ( s )d λ N ( s ) (cid:19) g ( x )d λ G/N ( xN )= Z G/N T ξ ( f )( x ) g ( x )d λ G/N ( xN ) = 0 , implying that g ∈ N ξ ( G, N ) ⊥ . (cid:3) Acknowledgement.
This project has received funding from the European Union’sHorizon 2020 research and innovation programme under the Marie Sklodowska-Curiegrant agreement No. 794305. The author gratefully acknowledges the supportingagency. The findings and opinions expressed here are only those of the author, andnot of the funding agency.The author would like to express his deepest gratitude to Vladimir V. Kisil for sug-gesting the problem that motivated the results in this article, stimulating discussionsand pointing out various references.
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Department of Pure Mathematics, School of Mathematics, University of Leeds, LeedsLS2 9JT, United Kingdom
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