Group C*-algebra A.K. Recall 1 ha if G i a nonempy e, he linear pace c oo (G) = {f : G C : upp f finie } ha a Hamel bai coniing of he funcion {δ : G} where { 1, = δ () = 0, Thu every f c oo (G) i a finie um f = G f()δ. The Hilber pace l 2 (G) i he compleion of c oo (G) wih repec o he calar produc f, g = f()g() G and hen {δ : G} become an orhonormal bai of l 2 (G). In cae G i a group, he group operaion (, ) and 1 exend linearly o make c oo (G) ino a *-algebra: we define δ δ = δ and (δ ) = δ 1, o ha ( ) ( ) f g = f()δ g()δ = f()g()δ, and 1 gpar, 16/1/07 ( ) f = f()δ = f()δ 1 1
in oher word (eing r = ) f g = ( ) f()g( 1 r) δ r = r r ( ) f(r 1 )g() δ r and (changing o r = 1 ) f = r f(r 1 )δ r. Thu (f g)(r) = f()g( 1 r) = f(r 1 )g() (r G) and (f )(r) = f(r 1 ) (r G). We may alo complee c oo (G) wih repec o he l 1 norm f 1 = f() o obain he Banach pace l 1 (G). Noe ha becaue of he relaion f g 1 f 1 g 1 and f 1 = f 1 (he proof of he fir one i eay 2 and he econd one i obviou) he muliplicaion and he involuion exend coninuouly o l 1 (G), which become a Banach algebra wih iomeric involuion, alhough rarely a C*-algebra. For example if e, and 2 are differen elemen of G and f = δ 1 +δ e δ hen f 1 = 1 + 1 + 1 and f f = (δ + δ e δ 1)(δ 1 + δ e δ ) = δ 2 + 3δ e δ 2 hence f f 1 = 1 + 3 + 1. In order o equip c oo (G) wih a uiable C*-norm, we udy i *-repreenaion on Hilber pace. 2 r f()g( 1 r) r f(). g( 1 r) = r f(). g( 1 r) = f(). r g( 1 r) = f() g() 2
The lef regular repreenaion Le H = l 2 (G). Each G define a uniary operaor λ on H by he formula λ ( ξ()δ ) = ξ()δ (ξ = ξ()δ l 2 (G)). For example, if G = Z hen λ n = U n where U i he bilaeral hif, U(δ n ) = δ n+1, on l 2 (Z). Making he change of variable r =, we find (λ ξ)(r) = ξ( 1 r) (r G). Noe ha λ i a well-defined linear iomery, becaue λ (ξ) 2 2 = r ξ( 1 r) 2 = ξ() 2 = ξ 2 2. Alo λ e = I (he ideniy operaor) and becaue λ λ = λ λ (λ δ r ) = λ (δ r ) = δ r = δ ()r = λ (δ r ) for each r G. Since he operaor involved are bounded and linear and he {δ } pan l 2 (G) he claim follow. In paricular i follow ha each λ i inverible wih invere (λ ) 1 = λ 1 and o i i an ono iomery, i.e. a uniary, wih (λ ) = λ 1. Thu we have a group homomorphim G U(B(H)) ino he group of uniary operaor on H = l 2 (G). Thi i called a uniary repreenaion of G on H. The uniary repreenaion λ immediaely exend o a *-repreenaion, alo denoed by λ, of he *-algebra c oo (G) on l 2 (G). More preciely, given f = f()δ c oo (G) we define λ(f) = i.e. (λ(f)ξ)(r) = f()λ f()ξ( 1 r) (ξ l 2 (G)). 3
Thi i a bounded operaor becaue λ(f) = f()λ f() λ = f() = f 1 ince each λ i uniary. In fac hi inequaliy how ha λ exend o a (conracive) map l 1 (G) B(l 2 (G)). The fac ha λ i a *-repreenaion immediaely follow from he properie of i rericion o G: (( ) ( )) ( ) λ f()δ g()δ = λ f()g()δ = f()g()λ,, = ( ) ( ) f()g()λ λ = f()λ g()λ = λ(f)λ(g), (( ) ) ( ) and λ f()δ = λ f()δ 1 = f()λ 1 = ( ) f()λ = f()λ. The above calculaion can be carried ou for any uniary repreenaion of G. The deail are lef a an exercie. Propoiion 1 There i a bijecive correpondence beween uniary repreenaion of G and *-repreenaion of c oo (G): If π : G U(B(H)) i any uniary repreenaion of he group G, he formula ( ) π f()δ = f()π() define a unial *-repreenaion of c oo (G) (and of l 1 (G)) on he ame Hilber pace H which i 1 -conracive. Converely, every unial 1 -conracive *-repreenaion ρ of c oo (G) (or of l 1 (G)) define a uniary repreenaion π by rericion : π() = ρ(δ ) aifying π = ρ. We uually ue he ymbol π for π. 4
Definiion 1 Le Σ be he e of all 1 -conracive *-repreenaion (π, H π ) of c oo (G) (equivalenly, of l 1 (G)). The C*-norm on c oo (G) (or l 1 (G)) i defined by he formula f = up{ π(f) : π Σ}. The group C*-algebra C (G) i defined o be he compleion of c oo (G) (or equivalenly of l 1 (G)) wih repec o hi norm. Remark 2 Fir of all, he e Σ i non-empy: i conain he lef regular repreenaion. Clearly i a eminorm on c oo (G), being he upremum of eminorm, all of which are (by definiion) bounded by 1, hence o i. Alo, aifie he C*-ideniy, becaue all he eminorm f π(f) aify i. 3 Bu why i a norm? In oher word, why i i rue ha f > 0 whenever f c oo (G) i nonzero? The reaon i ha he lef regular repreenaion i faihful on c oo (G) and l 1 (G); hu if f l 1 (G) i nonzero hen λ(f) 0 and o f λ(f) > 0. Indeed if f = f()δ l 1 (G) i nonzero hen here exi G wih f() 0 and hen 4 λ(f)δ e, δ l 2 (G) = f()λ (δ e ), δ = f() δ, δ = f() becaue he δ are orhonormal in l 2 (G). Thu λ(f) 0. The uefulne of C (G) come from he following propery, whoe proof i an immediae conequence of he previou propoiion and he fac ha c oo (G) i a dene *-ubalgebra of C (G). Propoiion 3 There i a bijecive correpondence beween uniary repreenaion of G and unial *-repreenaion of he group C*-algebra C (G). In paricular, he lef regular repreenaion λ exend o a *-repreenaion of C (G) on l 2 (G). However, he fac ha λ i faihful on c oo (G) doe NOT mean ha i exenion remain faihful on C (G)! 3 π(f f) = π(f) π(f) = π(f) 2. 4 ince f() <, he um f()λ converge (aboluely) in he norm of B(l 2 (G)). 5
The image λ(c (G)) in B(l 2 (G)) i a C*-algebra; i equal he cloure of λ(c oo (G)) in he norm of B(l 2 (G)) and i called he reduced C*-algebra Cr (G) of G. In many cae, for example when G i abelian, λ i faihful on C (G), o ha Cr (G) C (G) (iomerically and *-iomorphically). In general, however, Cr (G) i a quoien of C (G) and doe no conain all uniary repreenaion of G. Example 4 Le G = F 2 be he free group in wo generaor a and b; ha i, any elemen of G i a (finie) word of he form a n b m a k b j where n, m, k, j Z and here are no relaion beween a and b. I i known ha he reduced C*-algebra C r (F 2 ) i imple, i.e. i ha no nonrivial cloed wo-ided ideal. Thu all of i repreenaion are iomorphim; Since C r (F 2 ) i obviouly infinie-dimenional, i canno have finie dimenional repreenaion. On he oher hand, he group F 2 doe have uniary repreenaion on finiedimenional pace: Ju ake any wo uniary n n marice U and V and define π(a) = U and π(b) = V. Since here are no relaion beween a and b, hi exend o a uniary repreenaion of F 2 on C n ; for example π(a n b m a k b j ) = U n V m U k V j. Hence C (F 2 ) doe have nonrivial finiedimenional repreenaion: herefore i canno be iomorphic o C (F 2 ). Thu, C r (F 2 ) i a proper quoien of C (F 2 ). Example 5 Le G = Z. If we repreen each n Z by he funcion ζ n (z) = z n, z T, he convoluion produc ζ n ζ m = ζ n+m become poinwie produc, involuion become complex conjugaion, and he elemen of c oo (G) become rigonomeric polynomial. Hence if P C(T) i he e of rigonomeric polynomial we have a *-iomorphim c oo (Z) P : n f(n)δ n p f, where p f (z) n f(n)z n. Noe ha, a oberved earlier, he lef regular repreenaion λ i generaed by λ(1) = U, he bilaeral hif on l 2 (Z), which i uniarily equivalen o muliplicaion by ζ on L 2 (T). Therefore, for each f c oo (Z), λ(f) = p f (U) i uniarily equivalen o he muliplicaion operaor M pf acing on L 2 (T) and o λ(f) = M pf = p f = up{ p f (z) : z T}. I follow ha he cloure C r (Z) of λ(c oo (Z) i iomerically iomorphic o he up-norm cloure of he rigonomeric polynomial, namely C(T). 6
We will how ha C (Z) i iomerically *-iomorphic wih C(T). Since c oo (Z) i -dene in C (Z) and P i -dene in C(T) (Sone- Weierra) i uffice o how ha he norm f on c oo (Z) coincide wih he up norm p f of C(T). For hi, ince we ju proved ha p f = λ(f) f, i i enough o prove he revere inequaliy, namely ha if π i any uniary repreenaion of Z on ome Hilber pace, hen π(f) p f for any f = n f(n)δ n c oo (Z). Indeed le V = π(1); hi i a uniary operaor and π(f) = n f(n)π(n) = n f(n)v n = p f (V ). Now p f (V ) i a normal operaor and hence i norm equal i pecral radiu. By he pecral mapping heorem, σ(p f (V )) = {p f (z) : z σ(v )} {p f (z) : z T} becaue V i uniary and o σ(v ) T. Thu π(f) = p f (V ) up{ p f (z) : z T} = p f. Abelian group The iuaion of hi la example generalize o arbirary abelian group. Briefly, if G i an abelian group, hen of coure c oo (G) i abelian, and hence o i C (G). Thu C (G) C(K), where K i he compac pace of muliplicaive linear funcional on C (G) wih he weak* opology. We idenify he pace K: Define he e of characer of G Ĝ = = {γ : G T : homomorphim}. Wih he opology of poinwie convergence, i i no hard o ee ha hi i a compac pace (a cloed ubpace of he Careian produc T G ) and i i a group wih poinwie operaion. In fac i can be hown o be a opological group (he group operaion are coninuou). I i called he dual group of G. Any γ i a *-repreenaion of G on he Hilber pace C (ince γ()γ() = γ() and γ( 1 ) = (γ()) 1 = γ()) and hu exend (Propoiion 7
3) o a *-repreenaion γ of C (G) on C, i.e. a muliplicaive linear funcional on C (G). Converely, any muliplicaive linear funcional on C (G) reric o a characer on G. Thu here i a bijecion beween he e of characer of G and he e K of muliplicaive linear funcional on C (G). We claim ha hi bijecion i a homeomorphim; ince boh pace are compac and Haudorff, i uffice o prove ha i i coninuou. Le γ i γ in ; hi mean γ i () γ() for each G. To prove ha γ i γ in K, we need o prove ha γ i (a) γ(a) for all a C (G). Fix uch an a. Since c oo (G) i dene in C (G), given ɛ > 0 here exi f c oo (G) wih a f < ɛ. Now each γ i and γ ha norm 1, and o γ i (a f) γ(a f) 2 a f < 2ɛ. On he oher hand, if f i a finie um f()δ, we have γ i (f) γ(f) = f()(γ i () γ()) f(). γ i () γ(). Now ince γ i () γ() for each G, here i i o uch ha γ i () γ() < ɛ for each i i o and each in he finie uppor of f. Combining wih he previou inequaliy we conclude ha γ i (a) γ(a) < (2 + f 1 )ɛ whenever i i o ; hu γ i γ in he weak*-opology. Thi conclude he proof ha and K are homeomorphic; we henceforh idenify K wih and now we can conclude by Gelfand heory ha C (G) C(). In fac he *-iomorphim i given by a â, where â(γ) = γ(a), a C (G) and in paricular ˆf(γ) = f()γ(), f c oo (G). Haar meaure on We now wih o equip wih a uiable probabiliy meaure µ and form L 2 (, µ). We fir define a ae: ω : c oo (G) C : f f(e) 8
Clearly hi i linear 5 and ω(1) = ω(δ e ) = δ e (e) = 1. We check poiiviy: ω(f f) =(f f)(e) = f( 1 ) 2 = f ()f( 1 e) = f( 1 )f( 1 ) = f() 2 0 (1) for all f = f()δ c oo (G). Noe alo ha ω i coninuou in he norm of C (G) C(): Indeed ω(f) = f(e) = λ(f)δ e, δ e λ(f) δ e 2 2 = λ(f) f = ˆf when f c oo (G). Therefore ω exend o a coninuou linear form on he compleion C() and he exenion i a ae. By he Riez repreenaion heorem, here exi a unique Borel probabiliy meaure µ on he compac pace uch ha ω(a) = â(γ)dµ(γ) for all a C (G). (2) Lemma 6 The meaure µ (i) i lef invarian, i.e. µ(γe) = µ(e) for every Borel ube of and any g (where γe = {γγ : γ E}), and (ii) ha full uppor, i.e. µ(u) > 0 for every nonempy open e U. Proof (i) Fix γ. We claim ha ˆf(γ 1 γ )dµ(γ ) = ˆf(γ )dµ(γ ) for all f c oo (G). Indeed, eing g() = γ()f() we eaily find ha ĝ(γ ) = ˆf(γ 1 γ ˆf(γ ) and o 1 γ )dµ(γ ) = ĝ(γ )dµ(γ ) = g(e) = γ(e)f(e) = f(e). Since c oo (G) i dene in C() i follow ha a(γ 1 γ )dµ(γ ) = a(γ )dµ(γ ) for all a C(). 5 Thi i no muliplicaive: he produc on c oo (G) i no poinwie muliplicaion, i i convoluion 9
By uniquene of µ hi implie χ E (γ 1 γ )dµ(γ ) = χ E (γ )dµ(γ ) for every Borel e E. Bu ince χ E (γ 1 γ ) = χ γe (γ ), claim (i) follow. (ii) Le U be a nonempy open e. Oberve ha {γu : γ } i an open cover of (he map γ γγ i a homeomorphim) and o here i a finie ubcover {γ i U : i = 1,..., n}. Now µ(γ i U) = µ(u) by lef invariance, hence ( n ) n µ() = µ γ i U µ(γ i U) = nµ(u). i=1 Since µ() > 0 i follow ha µ(u) > 0. i=1 The Fourier ranform I follow from (2) ha for f c oo (G) (remembering ha he Gelfand ranform i a *-morphim, o ha ĝ f = ĝ ˆf) we have ω(f f) = f fdµ = f ˆfdµ = ˆf ˆfdµ = ˆf(γ) 2 dµ(γ). (3) Combine hi wih (1) o conclude ha ˆf(γ) 2 dµ(γ) = f() 2 for all f c oo (G). Thi equaliy how (if we wrie L 2 () for L 2 (, µ)) ha he linear map (c oo (G), l 2 (G) ) (C(), L 2 () ) : f ˆf i iomeric and ha dene range, and hu exend o a uniary bijecion F : l 2 (G) L 2 () which i called he Fourier ranform. Finally, if f c oo (G) and ξ c oo (G) l 2 (G) we have F (λ(f)ξ) = F (f ξ) = f ξ = ˆf ˆξ = M ˆf ˆξ = M ˆfF ξ 10
where M g denoe muliplicaion by g on L 2 (). The operaor F λ(f) and M ˆfF are boh bounded operaor on l 2 (G) and coincide on he dene ubpace c oo (G); herefore hey are equal: F λ(f) = M ˆfF or F λ(f)f = M ˆf (F i uniary). I follow ha λ(f) = M ˆf. Bu, ince µ ha full uppor, 6 M ˆf = ˆf. Thu finally λ(f) = ˆf for all f c oo (G) o ha he lef regular repreenaion i iomeric on c oo (G). Therefore i exenion o C (G) C() i alo iomeric, hence injecive, and implemen a *-iomorphim beween C (G) and C r (G). We ummarize: Theorem 7 If G i an abelian group and = Ĝ, hen C (G) C() and he Fourier ranform F : l 2 (G) L 2 () implemen a uniary equivalence beween he lef regular repreenaion λ of C (G) on l 2 (G) and he muliplicaion repreenaion g M g of C() on L 2 (). Hence λ i iomeric and o C (G) C r (G). 6 If U i an open e on which ˆf ˆf ɛ, hen ξ = χ U i a nonzero elemen of L 2 () and M ˆf ξ 2 M ˆf ξ 2 ( ˆf ɛ) ξ 2. 11