Covariant Functions of Characters of Compact Subgroups
aa r X i v : . [ m a t h . F A ] F e b COVARIANT FUNCTIONS OF CHARACTERS OF COMPACTSUBGROUPS
ARASH GHAANI FARASHAHI
Abstract.
This paper presents a systematic study for abstract harmonic analysis on clas-sical Banach spaces of covariant functions of characters of compact subgroups. Let G be alocally compact group and H be a compact subgroup of G . Suppose that ξ : H → T is acontinuous character, 1 ≤ p < ∞ and L pξ ( G, H ) is the set of all covariant functions of ξ in L p ( G ). It is shown that L pξ ( G, H ) is isometrically isomorphic to a quotient space of L p ( G ). Itis also proven that L qξ ( G, H ) is isometrically isomorphic to the dual space L pξ ( G, H ) ∗ , where q is the conjugate exponent of p . The paper is concluded by some results for the case that G is compact. Introduction
The Banach spaces consist of covariant functions on locally compact groups associatedto continuous characters (one-dimensional continuous irreducible unitary representations) ofclosed subgroups appear in variant mathematical areas and their applications including calcu-lus of pseudo-differential operators, number theory (automorphic forms), induced representa-tions, homogenuos spaces, complex (hypercomplex) analysis, theoretical aspects of mathemat-ical physics including coherent states and covariant analysis, see [1, 5, 14, 15, 16, 17, 18, 19, 20].The following paper presents a unified operator theoretic approach to study abstract har-monic analysis of L p -spaces of covariant functions of characters of compact subgroups inlocally compact groups. The introduced approach canonically extends classical methods ofabstract harmonic analysis and functional analysis on coset spaces (homogenouse spaces) ofcompact subgroups by assuming the character to be the identity character of the compactsubgroup, see [3, 7, 9, 21].This article contains 4 sections and organized as follows. Section 2 is devoted to fix notationsand provides a summary of classical harmonic analysis on locally compact compact groupsand covariant functions of characters of closed subgroups. Let G be a locally compact group, H be a compact subgroup of G , and 1 ≤ p < ∞ . Suppose that ξ : H → T is a fixed characterof H . In section 3, we study some of the operator theoretic aspects of covariant functionson G associated to the character ξ of the subgroup H . Next section investigates harmonicanalysis foundations on the Banach space L pξ ( G, H ), the space of covariant functions of thecharacter ξ in L p ( G ). In this direction, we study fundamental properties of classical Banachspaces of covariant functions of characters of compact subgroups. It is shown that L pξ ( G, H )is isometrically isomorphic to a quotient space of L p ( G ). We then proved that L qξ ( G, H ) isisometrically isomorphic to the dual space L pξ ( G, H ) ∗ , where q is the conjugate exponent of p . Finally, we conclude the paper by some related results for the case that G is compact. Mathematics Subject Classification.
Primary 43A15, 43A20, 43A85.
Key words and phrases. covariance property, compact 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 all continuouscomplex valued functions on X with compact support. If λ is a positive Radon measure on X ,for each 1 ≤ p < ∞ the Banach space of equivalence classes of λ -measurable complex valuedfunctions 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 with the modular function ∆ G and a fixed left Haarmeasure λ G . For a function f : G → C and x ∈ G , the functions L x f, R x f : G → C are givenby L x f ( y ) := f ( x − y ) and R x f ( y ) := f ( yx ) for y ∈ G . For 1 ≤ p < ∞ , L p ( G ) stands for theBanach space L p ( G, λ G ). The convolution for f, g ∈ L ( G ), is defined via(2.1) f ∗ G g ( x ) := Z G f ( y ) g ( y − x )d λ G ( y ) ( x ∈ G ) . We then have f ∗ G g ∈ L p ( G ) with k f ∗ G g k L p ( G ) ≤ k f k L ( G ) k g k L p ( G ) , if f ∈ L ( G ) and g ∈ L p ( G ). It is well known as a classical result in abstract harmonic analysis that theBanach function space L ( G, λ G ) is a Banach ∗ -algebra with respect to the bilinear product ∗ G : L ( G ) × L ( G ) → L ( G ) given by ( f, g ) f ∗ G g , with f ∗ G g defined by (2.1) andinvolution given by f f ∗ G where f ∗ G ( x ) := ∆ G ( x − ) f ( x − ) for x ∈ G . Also, for each p >
1, the Banach function space L p ( G ) is a Banach left L ( G )-module equipped with theleft module action ∗ G : L ( G ) × L p ( G ) → L p ( G ) given by ( f, g ) f ∗ G g , with f ∗ g is definedby (2.1), see [2, 4, 12, 13, 22] and the classical list of references therein.Suppose that H is a closed subgroup of G . A character ξ of H , is a continuous grouphomomorphism ξ : H → T , where T := { z ∈ C : | z | = 1 } is the circle group. In terms of grouprepresentation theory, each character of H is a 1-dimensional irreducible continuous unitaryrepresentation of H . We then denote the set of all characters of H by χ ( H ).Let G be a locally compact group, H be a closed subgroup of G , and ξ ∈ χ ( H ). A function ψ : G → C satisfies covariant property associated to the character ξ , if(2.2) ψ ( xs ) = ξ ( s ) ψ ( x ) , for every x ∈ G and s ∈ H . The covariant functions appear in abstract harmonic analysis inthe construction of induced representations, see [4, 14]. We here employ some of the classicaltools in this direction. Suppose that λ H is a left Haar measure on H . For each character ξ ∈ χ ( H ) and a function f ∈ C c ( G ), define the function T ξ ( f ) : G → C via T ξ ( f )( x ) := Z H f ( xs ) ξ ( s )d λ H ( s ) , for every x ∈ G . Suppose M ξ ( G, H ) is the linear subspace of C ( G ) given by M ξ ( G, H ) := { ψ ∈ C c ( G | H ) : ψ ( xh ) = ξ ( h ) ψ ( x ) , for all x ∈ G, h ∈ H } , where C c ( G | H ) := { ψ ∈ C ( G ) : q(supp( ψ )) is compact in G/H } , and q : G → G/H is the canonical map given by q( x ) := xH for x ∈ G . Using continuity ofthe canonical map q : G → G/H , one can deduce that C c ( G ) ⊆ C c ( G | H ). It is shown that thelinear operator T ξ maps C c ( G ) onto M ξ ( G, H ), see Proposition 6.1 of [4].
OVARIANT FUNCTIONS OF CHARACTERS OF COMPACT SUBGROUPS 3 Covariant Functions of Characters of Compact Subgroups
In this section, we shall study some of the fundamental theoretical aspects of covariantfunctions of characters of compact subgroup in locally compact groups. Throughout, let G bea locally compact group, H be a compact subgroup of G , and ξ ∈ χ ( H ) be a fixed character. Proposition 3.1.
Let G be a locally compact group and H be a compact subgroup of G .Suppose that ξ ∈ χ ( H ) is a character and λ H is the probability Haar measure of H . Then, (1) M ξ ( G, H ) ⊆ C c ( G ) . (2) T ξ ◦ T ξ = T ξ on C c ( G ) .Proof. (1) Let H be a compact subgroup of G . We then have C c ( G ) = C c ( G | H ). To seethis, let ψ ∈ C c ( G | H ) be given. Then q(supp( ψ )) is compact in G/H . Using Lemma 2.46 of[4], there exists a compact subset F of G such that q( F ) = q(supp( ψ )). This implies thatsupp( ψ ) ⊆ F H . Since H is compact, we conclude that supp( ψ ) is compact in G and so ψ ∈ C c ( G ). Therefore, we deduce that M ξ ( G, H ) ⊆ C c ( G ).(2) Let f ∈ C c ( G ) be given. Using (1), we have T ξ ( f ) ∈ C c ( G ). Then, for each x ∈ G andsince λ H is a probability measure, we get T ξ ( T ξ ( f ))( x ) = Z H T ξ ( f )( xs ) ξ ( s )d λ H ( s )= Z H T ξ ( f )( x ) ξ ( s ) ξ ( s )d λ H ( s )= T ξ ( f )( x ) Z H | ξ ( s ) | d λ H ( s ) = T ξ ( f )( x ) (cid:18)Z H d λ H ( s ) (cid:19) = T ξ ( f )( x ) . (cid:3) Invoking Proposition 3.1, if H is compact, one can regard the linear map T ξ : C c ( G ) → C c ( G )as a projection of the linear space C c ( G ) onto the subspace M ξ ( G, H ) ⊆ C c ( G ). Remark . If ξ = 1 is the trivial character of H , we then have T ( f ) = T H ( f ), see [8, 10].Then, M ( G, H ) consists of functions on G which are constant on cosets of N . Therefore, M ( G, H ) can be canonically identified with C c ( G/H ) via the isometric identification ψ e ψ ,where e ψ : G/H → C is given by e ψ ( xH ) := ψ ( x ) for every x ∈ G , see Corollary 3.4 of [11]. Inthis case, harmonic analysis on M ( G, H ) studied from different prespevtives in [6, 7, 9, 21].We then have the following observations.
Proposition 3.3.
Let G be a locally compact group, H be a compact subgroup of G , and ξ ∈ χ ( H ) be a character. Let y ∈ G and h ∈ H . Then, (1) T ξ ◦ R h = ξ ( h ) T ξ on C c ( G ) . (2) T ξ ◦ L y = L y ◦ T ξ on C c ( G ) .Proof. (1) Suppose f ∈ C c ( G ) and h ∈ H . Let x ∈ G be given. We then have T ξ ( R h f )( x ) = Z H R h f ( xs ) ξ ( s )d λ H ( s ) = Z H f ( xsh ) ξ ( s )d λ H ( s )= Z H f ( xs ) ξ ( sh − )d λ H ( sh − ) = ξ ( h ) Z H f ( xs ) ξ ( s )d λ H ( s ) = ξ ( h ) T ξ ( f )( x ) , A. GHAANI FARASHAHI which implies that T ξ ( R h f ) = ξ ( h ) T ξ ( f ).(2) Suppose f ∈ C c ( G ) and y ∈ G . Let x ∈ G be given. We then have T ξ ( L y f )( x ) = Z H L y f ( xs ) ξ ( s )d λ H ( s ) = Z H f ( y − xs ) ξ ( s )d λ H ( s ) = L y ( T ξ ( f ))( x ) , implying that T ξ ( L y f ) = L y ( T ξ ( f )). (cid:3) For functions f, g ∈ C c ( G ), the function f g is continuous with compact support. Therefore,it is integrable over G with respect to every left Haar measure λ G on G . We denote the laterintegral by h f, g i . Theorem 3.4.
Let G be a locally compact group and H be a compact subgroup of G . Suppose ξ ∈ χ ( H ) is a character and f, g ∈ C c ( G ) . We then have (3.1) 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 H f ( xs ) ξ ( s )d λ H ( s ) (cid:19) g ( x )d λ G ( x )= Z H (cid:18)Z G f ( xs ) g ( x )d λ G ( x ) (cid:19) ξ ( s )d λ H ( s )= Z H (cid:18)Z G f ( x ) g ( xs − )d λ G ( xs − ) (cid:19) ξ ( s )d λ H ( s )= Z H ∆ G ( s − ) (cid:18)Z G f ( x ) g ( xs − )d λ G ( x ) (cid:19) ξ ( s )d λ H ( s ) . Since H is compact in G , we have ∆ G | H = ∆ H = 1. Therefore, we get h T ξ ( f ) , g i = Z H ∆ H ( s − ) (cid:18)Z G f ( x ) g ( xs − )d λ G ( x ) (cid:19) ξ ( s )d λ H ( s )= Z G f ( x ) (cid:18)Z H g ( xs − ) ξ ( s )d λ H ( s ) (cid:19) d λ G ( x )= Z G f ( x ) (cid:18)Z H g ( xs ) ξ ( s )d λ H ( s − ) (cid:19) d λ G ( x )= Z G f ( x ) (cid:18)Z H g ( xs ) ξ ( s )d λ H ( s ) (cid:19) d λ G ( x ) = h f, T ξ ( g ) i . (cid:3) We then conclude this section by the following convolution property of the linear map T ξ . Theorem 3.5.
Let G be a locally compact group, H be a compact subgroup of G , and ξ ∈ χ ( H ) be a character. Suppose f, g ∈ C c ( G ) are given. We then have T ξ ( f ∗ G g ) = f ∗ G T ξ ( g ) . OVARIANT FUNCTIONS OF CHARACTERS OF COMPACT SUBGROUPS 5
Proof.
Let G be a locally compact group and H be a compact subgroup of G . Suppose that f, g ∈ C c ( G ) and x ∈ G are given. We then have T ξ ( f ∗ G g )( x ) = Z H f ∗ G g ( xs ) ξ ( s )d λ H ( s )= Z H (cid:18)Z G f ( y ) g ( y − xs )d λ G ( y ) (cid:19) ξ ( s )d λ H ( s )= Z G f ( y ) (cid:18)Z H g ( y − xs ) ξ ( s )d λ H ( s ) (cid:19) d λ G ( y )= Z G f ( y ) T ξ ( g )( y − x )d λ G ( y ) = f ∗ G T ξ ( g )( x ) . (cid:3) L p -Spaces of Covariant Functions of Characters of Compact Subgroups In this section, we study L p -spaces of covariant functions of characters of compact sub-groups. We then investigate some of the basic properties of these classical Banach spaces ofcovariant functions of characters of compact subgroups on locally compact groups. Through-out, suppose that G is a locally compact group, H is a compact subgroup of G , and ξ ∈ χ ( H ).Let λ G be a left Haar measure on G and λ H be a Haar measure on H .The following theorem proves the boundedness property of the linear map T ξ in the senseof L p ( G ). Theorem 4.1.
Let G be a locally compact group and ≤ p < ∞ . Suppose H is a compact sub-group of G and ξ ∈ χ ( H ) . The linear operator T ξ : ( C c ( G ) , k . k L p ( G ) ) → ( M ξ ( G, H ) , k . k L p ( G ) ) is bounded. In particular, if λ H is the probability Haar measure of H then the linear operator T ξ : ( C c ( G ) , k . k L p ( G ) ) → ( M ξ ( G, H ) , k . k L p ( G ) ) is a contraction.Proof. Let f ∈ C c ( G ) be given. Since H is compact and λ H ( H ) < ∞ , we have(4.1) (cid:18)Z H | f ( ys ) | d λ H ( s ) (cid:19) p ≤ λ H ( H ) p − Z H | f ( ys ) | p d λ H ( s ) , for each y ∈ G . Using compactness of H , we get ∆ G | H = ∆ H = 1. Then, compactness of H ,implies that ∆ G ( s ) = 1 for all s ∈ H . Thus, we get k T ξ ( f ) k pL p ( G ) = Z G (cid:12)(cid:12)(cid:12)(cid:12)Z H f ( xs ) ξ ( s )d λ H ( s ) (cid:12)(cid:12)(cid:12)(cid:12) p d λ G ( x ) ≤ Z G (cid:18)Z H | f ( xs ) | d λ H ( s ) (cid:19) p d λ G ( x ) ≤ λ H ( H ) p − Z G Z H | f ( xs ) | p d λ H ( s )d λ G ( x )= λ H ( H ) p − Z H Z G | f ( xs ) | p d λ G ( x )d λ H ( s )= λ H ( H ) p − Z H Z G | f ( x ) | p d λ G ( xs − )d λ H ( s )= λ H ( H ) p − k f k pL p ( G ) Z H ∆ G ( s − )d λ H ( s ) = λ H ( H ) p k f k pL p ( G ) , which guarantees that k T ξ ( f ) k L p ( G ) ≤ λ H ( H ) k f k L p ( G ) . Since f ∈ C c ( G ) was arbitrary, weconclude that the linear map T ξ : ( C c ( G ) , k . k L p ( G ) ) → ( M ξ ( G, H ) , k . k L p ( G ) ) is bounded with A. GHAANI FARASHAHI k T ξ k ≤ λ H ( H ). In particular, if λ H is the probability Haar measure of H then the oper-ator norm k T ξ k ≤
1. So the linear map T ξ : ( C c ( G ) , k . k L p ( G ) ) → ( M ξ ( G, H ) , k . k L p ( G ) ) is acontraction. (cid:3) Corollary 4.2.
Let G be a compact group and ≤ p < ∞ . Suppose H is a closed sub-group of G and ξ ∈ χ ( H ) . The linear map T ξ : ( C c ( G ) , k . k L p ( G ) ) → ( M ξ ( G, H ) , k . k L p ( G ) ) isbounded. In particular, if λ H is the probability Haar measure of H then the linear operator T ξ : ( C c ( G ) , k . k L p ( G ) ) → ( M ξ ( G, H ) , k . k L p ( G ) ) is a contraction.Remark . If ξ = 1 is the trivial character then Theorem 4.1 coincides with Proposition 3.4of [11], if M ( G, H ) is identified with C c ( G/H ).From now on, we assume that λ H is the probability Haar measure on H and hence thelinear operator T ξ : ( C c ( G ) , k . k L p ( G ) ) → ( M ξ ( G, H ) , k . k L p ( G ) ) is a contraction.We then conclude the following property of k . k L p ( G ) . Proposition 4.4.
Let G be a locally compact group and H be a compact subgroup of G .Suppose ≤ p < ∞ , ξ ∈ χ ( H ) and ψ ∈ M ξ ( G, H ) . Then, k ψ k L p ( G ) = inf (cid:8) k f k L p ( G ) : f ∈ C c ( G ) , T ξ ( f ) = ψ (cid:9) . Proof.
Let ψ ∈ M ξ ( G, H ) be given. Suppose that B ψ := { f ∈ C c ( G ) : T ξ ( f ) = ψ } and also γ ψ := inf (cid:8) k f k L p ( G ) : f ∈ B ψ (cid:9) . Using Theorem 4.1, for each f ∈ B ψ , we have k ψ k L p ( G ) ≤k f k L p ( G ) . Hence, we get k ψ k L p ( G ) ≤ γ ψ . Now, we claim that k ψ k L p ( G ) ≥ γ ψ as well. To thisend, since H is compact, using Proposition 3.1, we have ψ ∈ C c ( G ) and T ξ ( ψ ) = ψ . Therefore, ψ ∈ B ψ and hence we get k ψ k L ( G ) ≥ γ ψ , which completes the proof. (cid:3) Remark . Let ξ = 1 be the trivial character of H . Then Proposition 4.4 is a consequenceof Prospostion 3.4 and Corollary 3.4 of [11], if M ( G, H ) is identified with C c ( G/H ).Let N ξ = N ξ ( G, H ) be the kernel of the linear map T ξ in C c ( G ), that is the linear subspacegiven by N ξ ( G, H ) := { f ∈ C c ( G ) : T ξ ( f ) = 0 } . Then, N ξ ( G, H ) is a closed linear subspace of ( C c ( G ) , k . k L p ( G ) ) for every 1 ≤ p < ∞ . UsingProposition 3.3(1), we also have(4.2) span { R h f − ξ ( h ) f : h ∈ H, f ∈ C c ( G ) } ⊆ N ξ ( G, H ) . For every 1 ≤ p < ∞ , suppose that X ξ ( G, H ) := C c ( G ) / N ξ ( G, H ) is the quotient normedspace of N ξ ( G, H ) in C c ( G ), that is C c ( G ) / N ξ ( G, H ) = { f + N ξ : f ∈ C c ( G ) } , with the quotient norm given by(4.3) k f + N ξ k [ p ] := inf (cid:8) k f + g k L p ( G ) : g ∈ N ξ (cid:9) . Also, let X pξ ( G, H ) be the Banach completion of the normed linear space X ξ ( G, H ) with respectto the quotient norm k . k [ p ] . We may denote X ξ ( G, H ) by X ξ and X pξ ( G, H ) by X pξ at times.We also denote by L pξ ( G, H ) the Banach completion of the normed linear space M ξ ( G, H )with respect to k . k L p ( G ) . We shall use the completion norm by k . k L p ( G ) or just k . k p as well.It is then clear that L pξ ( G, H ) ⊆ L p ( G ). Also, one can see that L pξ ( G, H ) = { ψ ∈ L p ( G ) : R h ψ = ξ ( h ) ψ, for h ∈ H } Next we conclude the following characterisation of L pξ ( G, H ). OVARIANT FUNCTIONS OF CHARACTERS OF COMPACT SUBGROUPS 7
Theorem 4.6.
Let G be a locally compact group and H be a compact subgroup of G . Suppose ≤ p < ∞ and ξ ∈ χ ( H ) . Then, X pξ ( G, H ) is isometrically isomorphic to the Banach space L pξ ( G, H ) .Proof. Invoking the structure of the spaces X pξ ( G, H ) and L pξ ( G, H ), it is enough to show that( M ξ ( G, H ) , k . k L p ( G ) ) is isometrically isomorphic with the quotient normed space ( X ξ ( G, H ) , k·k [ p ] ). Since T ξ : C c ( G ) → M ξ ( G, H ) is a surjective linear map, we conclude that the linear space M ξ ( G, H ) is isomorphic with the quotient linear space X ξ ( G, H ) via the canonical linear map U ξ : X ξ ( G, H ) → M ξ ( G, H ) given by U ξ ( f + N ξ ) := T ξ ( f ) for every f ∈ C c ( G ). Further, theisomorphism U ξ is not only algebraic, but also isometric, if the quotient linear space X ξ ( G, H )is equipped with the classical quotient norm (4.3). Using Proposition 4.4, for f ∈ C c ( G ), wehave k U ξ ( f + N ξ ) k L p ( G ) = k T ξ ( f ) k L p ( G ) = inf (cid:8) k h k L p ( G ) : T ξ ( h ) = T ξ ( f ) (cid:9) = inf (cid:8) k h k L p ( G ) : h − f ∈ N ξ (cid:9) = inf (cid:8) k f + g k L p ( G ) : g ∈ N ξ (cid:9) = k f + N ξ k [ p ] . (cid:3) Invoking Theorem 4.1, one can conclude that if H is a compact subgroup of G then thebounded linear map T ξ : ( C c ( G ) , k . k L p ( G ) ) → ( M ξ ( G, H ) , k . k L p ( G ) ) has a unique extensionto a bounded linear map from L p ( G ) onto L pξ ( G, H ), which we still denote it by T ξ . Then T ξ ◦ R h = ξ ( h ) T ξ and T ξ ◦ L y = L y ◦ T ξ , for every y ∈ G and h ∈ H . It is easy to see that theextended map T ξ : L p ( G ) → L pξ ( G, H ) is given by f T ξ ( f ), where T ξ ( f )( x ) = Z H f ( xs ) ξ ( s )d λ H ( s ) , for x ∈ G. In particular, if λ H is the probability Haar measure of H , the extended linear operator T ξ : L p ( G ) → L pξ ( G, H ) is a contraction.Invoking structure of L pξ ( G, H ) and since L p ( G ) is a Banach L ( G )-module, we get that L pξ ( G, H ) is a Banach L ( G )-submodule of L p ( G ) as well. In particular, we conclude that L ξ ( G, H ) is a closed left ideal in L ( G ).We then prove the following the following multiplier property concerning convolution struc-ture of L pξ ( G, H ) in terms of the linear operator T ξ . Proposition 4.7.
Let G be a locally compact group and H be a compact subgroup of G .Suppose ξ ∈ χ ( H ) and ≤ p < ∞ . Then, T ξ : L p ( G ) → L p ( G ) is a L ( G ) -multiplier.Proof. Let f ∈ C c ( G ) and g ∈ L p ( G ). Suppose ( g n ) ⊂ C c ( G ) with g = lim n g n in L p ( G ). Usingboundedness of the linear operator T ξ : L p ( G ) → L p ( G ), continuity of the module action ineach argument, and Theorem 3.5, we get T ξ ( f ∗ G g ) = T ξ (lim n f ∗ G g n )= lim n T ξ ( f ∗ G g n )= lim n f ∗ G T ξ ( g n ) = f ∗ G T ξ ( g ) . Using a similar method, we get T ξ ( f ∗ G g ) = f ∗ G T ξ ( g ), if f ∈ L ( G ). (cid:3) A. GHAANI FARASHAHI
For 1 ≤ p < ∞ , let N pξ ( G, H ) be the kernel of extension of the linear map T ξ in L p ( G ),that is the linear subspace given by N pξ ( G, H ) := n f ∈ L p ( G ) : T ξ ( f ) = 0 in L pξ ( G, H ) o . Proposition 4.8.
Let G be a locally compact group and H be a compact subgroup of G .Suppose ≤ p < ∞ and ξ ∈ χ ( H ) . Then, N pξ ( G, H ) is the closure of N ξ ( G, H ) in L p ( G ) .Proof. Let X be the closure of N ξ ( G, H ) in L p ( G ). Suppose f ∈ X is given. Then, we have f ∈ L p ( G ) and lim n f n = f for some sequence ( f n ) ⊂ N ξ ( G, H ). Continuity of the extended T ξ : L p ( G ) → L pξ ( G, H ) implies that T ξ ( f ) = 0 in L pξ ( G, H ). Therefore, we get f ∈ N pξ ( G, H ).Since f was arbitrary, we deduce that X ⊆ N pξ ( G, H ). Conversely, let f ∈ N p ( G, H ) begiven. Then, f ∈ L p ( G ) with T ξ ( f ) = 0 in L pξ ( G, H ). Suppose that ε > h ∈ C c ( G ) with k f − h k L p ( G ) < ε/
2. Let φ := T ξ ( h ). We then define the function g : G → C by g ( x ) := h ( x ) − φ ( x ) for every x ∈ G . Then, we have g ∈ C c ( G ) and T ξ ( g ) = 0. Indeed, for x ∈ G , we have T ξ ( g )( x ) = φ ( x ) − Z H φ ( xs ) ξ ( s ) dλ H ( s ) = φ ( x ) − φ ( x ) = 0 . Hence, we get g ∈ N ξ ( G, H ). Also, we have k h − g k L p ( G ) = k φ k L p ( G ) = k T ξ ( h ) k L p ( G ) ≤ k T ξ ( h − f ) k L p ( G ) + k T ξ ( f ) k L p ( G ) ≤ k h − f k L p ( G ) < ε . Therefore, we achieve k f − g k L p ( G ) ≤ k f − h k L p ( G ) + k h − g k L p ( G ) < ε, which implies that f is in the closure of N ξ ( G, H ) in L p ( G ), that is f ∈ X . Since f ∈ N pξ ( G, H )was given, we conclude that N pξ ( G, H ) ⊆ X as well. (cid:3) We then have the following interesting characterization of the Banach space L pξ ( G, H ) as aquotient space of L p ( G ). Theorem 4.9.
Let G be a locally compact group and H be a compact subgroup of G . Suppose ≤ p < ∞ and ξ ∈ χ ( H ) . The Banach space L pξ ( G, H ) is isometrically isomorphic to thequotient Banach space L p ( G ) / N pξ ( G, H ) .Proof. Applying Propostion 4.8 and Lemma 3.4.4 of [22] we deduce that X pξ ( G, H ) is canon-ically isometric isomorphic to the quotient Banach space L p ( G ) / N pξ ( G, H ). Then, Theorem4.6 implies that the Banach space L pξ ( G, H ) is canonically isometric isomorphic to the quotientBanach space L p ( G ) / N pξ ( G, H ). (cid:3) We continue by some results concerning adjoint of the linear map T ξ : L p ( G ) → L p ( G ),when H is compact. Proposition 4.10.
Let G be a locally compact group and H be a compact subgroup of G .Suppose ξ ∈ χ ( H ) , and < p, q < ∞ with p − + q − = 1 . The adjoint of the bounded linearmap T ξ : L p ( G ) → L p ( G ) can be identified by T ξ : L q ( G ) → L q ( G ) . OVARIANT FUNCTIONS OF CHARACTERS OF COMPACT SUBGROUPS 9
Proof.
Let g Λ g be the canonical isometric isomorphism identification of L q ( G ) as L p ( G ) ∗ ,where the bounded linear functional Λ g : L p ( G ) → C is given by Λ g ( f ) := h f, g i for every f ∈ L p ( G ). Suppose that f, g ∈ C c ( G ) are given. Invoking the abstract structure of theadjoint linear map T ∗ ξ : L p ( G ) ∗ → L p ( G ) ∗ , and using (3.1), we get T ∗ ξ (Λ g )( f ) = Λ g ( T ξ ( f )) = h T ξ ( f ) , g i = h f, T ξ ( g ) i = Λ T ξ ( g ) ( f ) . Since f ∈ C c ( G ) was arbitrary, and using density of C c ( G ) in L p ( G ), we get T ∗ ξ (Λ g ) = Λ T ξ ( g ) .Then density of C c ( G ) in L q ( G ) implies that T ∗ ξ (Λ g ) = Λ T ξ ( g ) for every g ∈ L q ( G ). So, T ξ identifies T ∗ ξ . (cid:3) Corollary 4.11.
Let G be a locally compact group and H be a compact subgroup of G .Suppose ξ ∈ χ ( H ) , ≤ p < ∞ , and λ H be the probability Haar measure of H . Then, T ξ : L p ( G ) → L p ( G ) is the projection onto L pξ ( G, H ) . In particular, T ξ : L ( G ) → L ( G ) isthe orthogonal projection onto L ξ ( G, H ) . Next we obtain the following characterization for the dual space L pξ ( G, H ) ∗ , if p > Theorem 4.12.
Let G be a locally compact group and H be a compact subgroup of G . Suppose ξ ∈ χ ( H ) , and < p, q < ∞ with p − + q − = 1 . Then L qξ ( G, H ) is isometrically isomorphicto L pξ ( G, H ) ∗ .Proof. Let g Λ g be the canonical isometric isomorphism identification of L q ( G ) as L p ( G ) ∗ ,where Λ g : L p ( G ) → C is given by Λ g ( f ) := h f, g i for every f ∈ L p ( G ). Invoking Theorem 4.9,we conclude that the dual space L pξ ( G, H ) ∗ is isometrically isomorphic to N pξ ( G, H ) ⊥ , where N pξ ( G, H ) ⊥ = n g ∈ L q ( G ) : Λ g ( f ) = 0 , for all f ∈ N pξ ( G, H ) o . We then claim that N pξ ( G, H ) ⊥ = L qξ ( G, H ). To show this, let g ∈ N pξ ( G, H ) ⊥ be given.Then, g ∈ L q ( G ) and Λ g ( f ) = 0 for all f ∈ N pξ ( G, H ). Suppose that h ∈ H is arbitrary.Then, using (4.2) and Theorem 4.8, for every f ∈ C c ( G ) we have R h − f − ξ ( h ) f ∈ N pξ ( G, H ).This implies that Λ g ( R h − f ) = ξ ( h )Λ g ( f ). Therefore, using compactness of H , we getΛ R h g ( f ) = Z G f ( x ) g ( xh )d λ G ( x )= Z G f ( xh − ) g ( x )d λ G ( xh − )= ∆ G ( h − ) Z G f ( xh − ) g ( x )d λ G ( x )= ∆ H ( h − ) Z G f ( xh − ) g ( x )d λ G ( x )= Z G f ( xh − ) g ( x )d λ G ( x )= Λ g ( R h − f ) = ξ ( h )Λ g ( f ) = Λ ξ ( h ) g ( f ) . Since f ∈ C c ( G ) was arbitrary, we get R h g = ξ ( h ) g in L q ( G ). Since h ∈ H was arbitrary,we conclude that g ∈ L qξ ( G, H ). Conversely, suppose that g ∈ L qξ ( G, H ). We shall showthat Λ g ( f ) = 0 for every f ∈ N pξ ( G, H ). According to Theorem 2.49 of [4], let λ G/H be the G -invariant Radon measure on the left coset space G/H normalized with respect to the Weil’sformula (2.50) of [4]. Then, for every f ∈ N pξ ( G, H ), we haveΛ g ( f ) = Z G f ( x ) g ( x )d λ G ( x )= Z G/H (cid:18)Z H f ( xh ) g ( xh )d λ H ( h ) (cid:19) d λ G/H ( xH )= Z G/H (cid:18)Z H f ( xh ) ξ ( h )d λ H ( h ) (cid:19) g ( x )d λ G/H ( xH ) = Z G/H T ξ ( f )( x ) g ( x )d λ G/H ( xH ) = 0 , which implies that g ∈ N pξ ( G, H ) ⊥ . (cid:3) Suppose that L ∞ ( G ) is the Banach space of all locally λ G -measurable functions f : G → C that are bounded except on a locally λ G -null set, modulo functions which are zero locally a.e.on G given the norm k f k ∞ := inf { t : | f ( x ) | ≤ t l . a . e . x ∈ G } . We then have M ξ ( G, H ) ⊂ L ∞ ( G ). It is also routine to check that the linear operator T ξ : ( C c ( G ) , k · k L ∞ ( G ) ) → ( M ξ ( G, H ) , k · k L ∞ ( G ) ), is bounded with the operator norm k T ξ k ≤ λ H ( H ). In particular, if λ H is the probability Haar measure of H then the linear operator T ξ : ( C c ( G ) , k · k L ∞ ( G ) ) → ( M ξ ( G, H ) , k · k L ∞ ( G ) ) is a contraction.Let L ∞ ξ ( G, H ) be closed subspace of L ∞ ( G ) given by L ∞ ξ ( G, H ) := { ψ ∈ L ∞ ( G ) : R h ψ = ξ ( h ) ψ, for h ∈ H } . We then also obtain the following characterization for the dual space L ξ ( G, H ) ∗ . Theorem 4.13.
Let G be a locally compact group and H be a compact subgroup of G . Suppose ξ ∈ χ ( H ) . Then L ξ ( G, H ) ∗ is isometrically isomorphic to L ∞ ξ ( G, H ) .Proof. Let g Λ g be the canonical 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 f ∈ L ( G ), see Theorem 12.18 of [12].Invoking Theorem 4.9, the dual space L ξ ( G, H ) ∗ is isometric isomorphic to N ξ ( G, H ) ⊥ , where N ξ ( G, H ) ⊥ = (cid:8) g ∈ L ∞ ( G ) : Λ g ( f ) = 0 , for all f ∈ N ξ ( G, H ) (cid:9) . Then using a similar method used in Theorem 4.12, we get N ξ ( G, H ) ⊥ = L ∞ ξ ( G, H ). (cid:3) We then conclude the paper by the following inclusion property when G is compact. Proposition 4.14.
Let G be a compact group and H be a closed subgroup of G . Suppose ξ ∈ χ ( H ) and ≤ p < ∞ . Then, L pξ ( G, H ) ⊆ L ξ ( G, H ) .Proof. Since G is compact, we have k ψ k L ( G ) ≤ λ G ( G ) p − p k ψ k L p ( G ) , for ψ ∈ C c ( G ). Therefore, k ψ k L ( G ) ≤ λ G ( G ) p − p k ψ k L p ( G ) for ψ ∈ M ξ ( G, H ). This implies that L pξ ( G, H ) ⊆ L ξ ( G, H ). (cid:3) Corollary 4.15.
Let G be a compact group and H be a closed subgroup of G . Suppose ξ ∈ χ ( H ) and ≤ p < ∞ . Then, (1) T ξ : L p ( G ) → L p ( G ) is a L p ( G ) -multiplier. (2) L pξ ( G, H ) is a closed left ideal in L p ( G ) . OVARIANT FUNCTIONS OF CHARACTERS OF COMPACT SUBGROUPS 11
Acknowledgement.
This project has received funding from the European Union’s Hori-zon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agree-ment No. 794305. The author gratefully acknowledges the supporting agency. The findingsand opinions expressed here are only those of the author, and not of the funding agency.The author would like to express his deepest gratitude to Vladimir V. Kisil for suggesting theproblem that motivated the results in this article, stimulating discussions and pointing outvarious references.
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