Function representation of a noncommutative uniform algebra

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1 Fucti represetati f a cmmutative uifrm algebra Krzysztf Jarsz Abstract. We cstruct a Gelfad type represetati f a real cmmutative Baach algebra A satisfyig f 2 = kfk 2, fr all f 2 A:. Itrducti A uifrm algebra A is a Baach algebra such that (.) f 2 = kfk 2, fr all f 2 A: The cmmutative, cmplex uifrm algebras cstitute the mst classical class f Baach algebras, e which has bee itesely studied fr decades. It is well kw that ay such algebra is ismetrically ismrphic with a subalgebra f C C (X) - the algebra f all ctiuus cmplex valued fuctis de ed a cmpact set X ad equipped with the sup-rm tplgy. Hirschfeld ad Zelazk [5] prved that the assumpti that the algebra is cmmutative is super uus - cmmutativity already fllws frm (.). Algebras f aalytic fuctis serve as stadard examples f cmplex uifrm algebras. Cmmutative, real uifrm algebras have als bee studied fr years [9]. Ay such algebra A is ismetrically ismrphic with a real subalgebra f C C (X) fr sme cmpact set X; furthermre X ca be fte divided it three parts X ; X 2 ; ad X 3 such that A jx is a cmplex uifrm algebra, A jx2 csists f cmplex cjugates f the fuctis frm A jx, ad A jx3 is equal t C R (X 3 ). Ay cmmutative, real uifrm algebra ca be aturally ideti ed with a real subalgebra f C C (X; ) = f 2 C C (X) : f ( (x)) = f (x) fr x 2 X where : X! X is a surjective hmemrphism with 2 = id X. I case f real uifrm algebras the cditi (.) lger implies cmmutativity - the fur dimesial algebra f quateris serves as the simplest cuterexample. There has bee very little study f cmmutative real uifrm algebras. Oly recetly it was prved [8] that ay such algebra is ismrphic with a real subalgebra f C H (X) - the algebra f all ctiuus fuctis de ed a cmpact set X ad takig values i the eld H f quateris. There is simple thery f aalytic fuctis f quateri variables - the stadard de iti 2000 Mathematics Subject Classi cati. Primary 46H20, 46H05; Key wrds ad phrases. Uifrm algebra, fucti algebra, Baach algebra, quateris.

2 2 KRZYSZTOF JAROSZ f 0 (a) = lim q!0;q2h (f (a + q) f (a)) q is t useful sice ly liear fuctis are di eretiable (there is a iterestig alterative apprach t di eretiability f such fuctis - see fr example [0, 2], that apprach is hwever t applicable fr ur purpse). Hece there is bvius example f a cmmutative real uifrm algebra ther the the direct sum f the etire algebra C H (X) ad a cmmutative real uifrm algebra. We prve here that i fact there is trivial example ad ay such algebra is rughly equal t such direct sum; we als btai a Gelfad type represetati f such algebras aalgus t C C (X; ). All algebras are assumed t ctai a uit. 2. The Results We rst eed a easily veri able elemetary descripti f ismrphisms f the fur dimesial real algebra H. Prpsiti. A surjective map T : H! H is liear (ver the eld f real umbers) ad multiplicative if ad ly if a 0 (2.) T (a; b; c; d) = 6 0 M 4 4 b 3 7 c 5 5, fr a + bi + cj + dk 2 H d where M is a ismetry f 3 dimesial Euclidea space R 3 preservig the rietati f that space. The space f all such maps frms a 3 dimesial cmpact cected grup which we will dete by M. We shuld als tice that if A is a real algebra ad F : A! H is a multiplicative fuctial such that dim (F (A)) = 2 the F (A) is ismrphic with C ad the set ft F : T 2 Mg ca be ideti ed with the uit sphere i R 3 : Ideed t de e a ismetry T F (A) we just have t decide which uit vectr f the frm bi + cj + dk is t be the image f the imagiary uit f F (A) : Defiiti. We say that A C (X) separates the pits f X if fr ay x ; x 2 2 X there is f 2 A such that f (x ) 6= f (x 2 ). We say that A C (X) strgly separates the pits f X if fr ay x ; x 2 2 X there is f 2 A such that f (x ) 6= f (x 2 ) = 0. Defiiti 2. A real algebra A is fully cmmutative if ay zer liear ad multiplicative fuctial F : A! H is surjective. Ntice that A is t fully cmmutative i ff (a) : a 2 Ag is ismrphic with R r with C fr sme H-valued multiplicative fuctial F A. Defiiti 3. Let : M! Hm (X) : (T ) = T be a hmmrphism f the grup M t a grup f hmemrphisms f a cmpact set X. We de e C H (X; ) = ff 2 C H (X) : f T (x) = T f (x) fr x 2 X; T 2 Mg : Fr a cmmutative real r cmplex uifrm algebra A the stadard way t represet A as a subalgebra f C C (X) is t csider the space X = M (A) f all cmplex valued liear-multiplicative fuctials ad use the Gelfad trasfrmati: (2.2) A 3 a 7! ^a 2 C (X) : ^a (x) = x (a) :

3 3 Fr a real algebra A ad fr ay x 2 M (A) als the cmplex cjugate x = (x) f x is a elemet f M (A) hece the represetati f A as a subalgebra f C C (X; ). A imprtat feature f such represetati is that ^ maps the set f ivertible elemets f A exactly t the subset f ^A csistig f fuctis that d t vaish X. We will cstruct a similar represetati fr a cmmutative uifrm algebra. Therem. Assume A is a real uifrm algebra the there is a cmpact set X ad a ismrphism : M! Hm (X) such that A is ismetrically ismrphic with a subalgebra ^A f C H (X; ). Furthermre a 2 A is ivertible if ad ly if the crrespdig elemet ^a 2 ^A des t vaish X. If A is fully cmmutative the ^A = C H (X; ) : We have bvius cadidates fr X;, ad the map ^ : X = M H (A) = quateri valued real-liear ad multiplicative fuctials (2.3) T (x) = T x; ^a (x) = x (a) : It is clear that the map ^ is a hmmrphism f A it C H (X; ), we eed t shw that it is a ismetry, that it preserves the set f ivertible elemets, ad that ^A = C H (X; ) i the fully cmmutative case. S withut lss f geerality we may assume that A is separable, r equivaletly cutably geerated. Befre prvig the therem it will be useful t tice that fr x 2 X we may ecuter exactly three distict cases - the set ft x : T 2 Mg may be equal t the siglet fxg if x (A) = R, r hmemrphic with the uit sphere i R 3 if x (A) is a 2 dimesial cmmutative subalgebra f H ismrphic with C, r hmemrphic with M if x (A) = H. Csequetly the set M H (A) ca be divided it three parts X ; X 2 ; X 3 : X = fx 2 X : T x = x fr all T 2 Mg = fx 2 X : x (A) = Rg - that set is equal t the subset f X csistig f the pits where all the fuctis frm ^A are real valued; X 2 = fx 2 X : dim x (A) = 2g - that set is a ui f disjit cpies f the uit sphere i R 3 ; the algebra ^A restricted t X [ X 2 is cmmutative; X 3 = fx 2 X : x (A) = Hg - that set is a ui f disjit cpies f M ad the algebra ^A restricted t X 3 is fully cmmutative. That is aalgus t the cmmutative case whe a real-liear multiplicative fuctial x takes ly real values if (x) = x; r takes all cmplex values if (x) 6= x ad the set M (A) ca be divided it just tw parts. Put ad A R = a 2 A : ^a is cstat X [ X 2 ad ft x : T 2 Mg fr each x 2 X 3 ^A R = f^a 2 C R (X) : a 2 Ag : We prve the Therem i several shrt steps; the steps als prvide a much mre detail descripti f the represetati f A: () the map ^ de ed by (2.3) is ijective ad A is semisimple, ;

4 4 KRZYSZTOF JAROSZ (2) ^A f 2 CR (X) : f is cstat X R = [ X 2 ad ft x : T 2 Mg fr each x 2 X 3 k^ak fr ay a 2 A R,, furthermre kak = (3) X [ X 2 is a weak peak set fr ^A s ay fucti frm ^AjX[X 2 = f jx[x 2 2 C (X [ X 2 ) : f 2 ^A has a rm preservig extesi t a fucti i ^A; M ^AjX[X 2 = X [ X 2 ; (4) ff 2 C H (X; ) : f (x) = 0 fr x 2 X [ X 2 g ^A, (5) ^A = f 2 C H (X; ) : f jx[x 2 2 ^A jx[x 2 ; (6) the map ^ is a ismetry ad ^A jx[x 2 is a cmplete algebra, (7) ay ivertible elemet f A is ctaied i a kerel f a fuctial frm M H (A); ^A = CH (X; ) if A is fully cmmutative. The rst part fllws frm Th 3 f [8], we prvide a idepedet prf here fr cmpleteess. We will eed the fllwig special case f the mai result f []. Therem 2 (Aupetit-Zemaek). Let A be a real Baach algebra with uit. If lim p ka k = kak fr all a i A the fr every irreducible represetati : A! L (E), the algebra (A) is ismrphic with its cmmutat C i the algebra L (E) f all liear trasfrmati E. Prf. Part. Sice the cmmutat C is a rmed real divisi algebra ([2] p. 27) it is ismrphic with R; C; r H. Let X be the set f all irreducible represetatis f A; fr x 2 X ad a 2 A put ^a (x) = x (a) 2 H. The map ^ is a ismrphisms f A it the algebra f H-valued fuctis X. Let be the strgest tplgy X such that all the fuctis ^a; a 2 A, are ctiuus. Assume (X; ) is t cmpact ad let x 0 be a pit frm XX, the peratr A 3 a! ^a (x 0 ) 2 H is a irreducible represetati A s x 0 2 X, ctradicti. T shw that ^ is ijective we eed t shw that fr ay 0 6= a 2 A there is a irreducible represetati with (a) 6= 0. Fix 0 6= a 2 A ad let A a be the clsed subalgebra f A geerated by all the elemets f the frm q (a), where q is a ratial fucti with real ce ciets ad with ples utside the spectrum f a. Ntice that A a is a cmmutative uifrm algebra such that s rada a = f0g : If b 2 A \ A a the b as give by a ratial fucti is i A a, that meas that A \ A a = Aa. Hece by [] p. 476, A a \ rada rada a = f0g, s a =2 rada ad, sice a was arbitrary, A is semisimple. Part 2. Put R = (fg gf) 2 2 C H (X) : f; g 2 ^A : Fr arbitrary quateris w p = a p + b p i + c p j + d p k; p = ; 2 we have (w w 2 w 2 w ) 2 = (2 (b c 2 b 2 c ) ij + 2 (b d 2 b 2 d ) ik + 2 (c d 2 c 2 d ) jk) 2 = 4 ((b c 2 b 2 c ) k (b d 2 b 2 d ) j + (c d 2 c 2 d ) i) 2 = 4 (b c 2 b 2 c ) 2 + (b d 2 b 2 d ) 2 + (c d 2 c 2 d ) 2 2 C R (X) ; s R ctais ly real valued fuctis. Let be a equivalece relati X de ed by x x 2 i h (x ) = h (x 2 ) fr all h 2 R,

5 5 let X 0 = X = be the qutiet space, ad let : X! X 0 be the atural prjecti. Fr x 2 X there is a fucti i R t vaishig at x if ad ly if x 2 X 3, furthermre X 3 we have x x 2 if ad ly if ker x = ker x 2 ; r equivaletly if ad ly if ft x : T 2 Mg = ft x 2 : T 2 Mg. By the Ste-Weierstrass Therem ^A R ca be ideti ed with a dese subset f C R (X 0 ), we shall shw that i fact it is equal t the etire C R (X 0 ). A R is a cmmutative real uifrm algebra s it is ismetric with a clsed subalgebra f C R (M (A R )); X 0 ca be aturally ideti ed with a clsed subset f M (A R ) : Assume X 0 6= M (A R ), let y 0 2 M (A R ) X 0 : Sice M (A R ) is equipped with the weak * tplgy there are a ; :::; a 2 A R such that ^a (y 0 ) = ::: = ^a (y 0 ) = 0 ad max fj^a j (y)j : j = ; :::; g fr ay y 2 X 0 : Hece ^a 0 = ^a 2 + ::: + ^a 2 2 A R is strictly psitive X 0 = X =, ad sice A is semisimple a 0 is a ivertible elemet f A R A ctrary t the fact it is i the kerel f the fuctial y 0 : The ctradicti shws that X 0 = M (A R ) ad csequetly A R = C R (X 0 ). Assume there is a 2 A R such that kak = > k^ak. Sice k^a k! 0 while ka k = it fllws that the rage f ^ is t cmplete - that ctradicti shws that ^ is a ismetry A R. Part 3. We recall that a subset K f the maximal ideal space f a fucti algebra is called a weak peak set i fr ay pe eighbrhd U f K there is a fucti f i that algebra such that f (x) = = kfk > jf (x 0 )j fr ay x 2 K ad ay x 0 =2 U. Sice C R (X 0 ) ' A R A it fllws that X [ X 2 is a weak peak set. Assume a 2 A with sup fj^a (x)j : x 2 X [ X 2 g = : Sice sice M csists f ismetries maxf;j^ajg is cstat the sets ft x : T 2 Mg fr x 2 X 3 as well as the set X [X 2. Hece maxf;j^ajg 2 ^A s ^a maxf;j^ajg 2 ^A; that fucti cicides with ^a X [ X 2 ad has rm. It is imprtat t tice fr further referece that sice ^ is a ismetry A R we have a kak = kak maxf;jajg maxf;jajg s t ly a maxf;jajg is a rm preservig extesi f a with respect t the spectral rm but als with respect t the rigial rm f the algebra A. The maximal ideal space f A jx[x 2 is equal t X [ X 2 sice ay liear ad multiplicative fuctial A jx[x 2 gives a fuctial as well A. Part 4. Fix x 0 2 X 3 ad let f ; f 2 ; f 3 ; f 4 2 A be such that f (x 0 ) = ; f 2 (x 0 ) = i; f 3 (x 0 ) = j; f 4 (x 0 ) = k: Let U 0 be a pe eighbrhd f x 0 such that jf p (x) f p (x 0 )j < =4 fr x 2 U 0 ad p = ; 2; 3; 4: Let g 0 2 C H (X; ) ad assume the supprt f g 0 is ctaied i V 0 = ( (U 0 )). Fr ay x 2 U 0 ; the umbers f (x) ; f 2 (x) ; f 3 (x) ; f 4 (x) ca be see as liearly idepedet vectrs i H ' R 4 s there are uique real valued fuctis h p de ed U 0 such that (2.4) g 0 (x) = 4X h p (x) f p (x) fr x 2 U 0 : p= Let x 2 U 0 ad let x 0 = T x be ather pit f ( (x)), where T 2 M. By the de iti f C H (X; ) we have g 0 (x 0 ) = T g 0 (x) = 4X h p (x) T f p (x) = p= 4X h p (x) f p (x 0 ) : p=

6 6 KRZYSZTOF JAROSZ Hece the fuctis h p ; p = ; 2; 3; 4 are cstat the sets f the frm ( (x)) ad csequetly ca be aturally exteded t V 0 ; furthermre they ca be exteded t the etire set X by assigig value zer utside V 0. Sice the ce ciets with respect t a xed basis i R 4 are ctiuus fuctis f a vectr i R 4, the fuctis h p ; p = ; 2; 3; 4 belg t A R A. Hece g 0 2 A. We prved that fr ay pit x 0 i X there is a eighbrhd V 0 f ( (x)) such that A jv0 = C H (X; ) jv0. Let V s = ( (U s )) ; s 2 S be a pe cver f X csistig f such sets. Sice A is separable ad X 3 is a subset f its dual space equipped with the weak * tplgy, X 3 is metrizable ad csequetly paracmpact. Let P s2s s = be a lcally ite partiti f uity i C R (X 3 ) subrdiated t the cver (U s ) ; s 2 S f X 0 ([3], page 375). Fr ay g 2 C H (X; ) such that g (x) = 0 fr x 2 X [ X 2 we have g (x) = P s2s s (x) g (x) where supp s g V s s g 2 A. Part 5. Assume f 2 C H (X; ) is such that f jx[x 2 2 ^A jx[x 2. Let g 2 A be such that g jx[x 2 = f jx[x 2. Sice f g 2 ff 2 C H (X; ) : f (x) = 0 fr x 2 X [ X 2 g by the previus step f = g + (f g) 2 A. Part 6. Put I = fa 2 A : ^a (x) = 0 fr x 2 X [ X 2 g ad let B = A=I be the qutiet algebra. Ntice that B is a cmmutative Baach algebra with M (B) = X [ X 2 : Assume there is a + I 2 B such that ka + Ik = kak > 2 ad k^ak. By Part 3 we may assume that k^ak = ^a jx[x 2. Suppse that a 2 + I < s there is g 2 I with a 2 + g <, by Part 3 agai we may assume that ^a 2 + ^g is smaller tha X ad csequetly kgk = k^gk < 2. We have a 2 + g < ; kgk < 2; a 2 = 4; which is impssible. The abve prves that b 2 4 kbk2, fr all f 2 B: s B is ismrphic with a cmplete uifrm algebra which must be equal t ^A jx[x 2 ; by the same argumets as at the ed f Part 3 we cclude that ^ is i fact a ismetry. Part 7. A bvius csequece f Part 5 sice M ^AjX[X 2 = X [ X 2. Crllary. Assume A is a fully cmmutative clsed subalgebra f C H (Y ). The A = C H (Y ) if ad ly if A strgly separates the pits f Y. We d t assume here that A ctais all cstat fuctis; if we assume that the cstat fuctis i; j; k are i A the the abve versi f the Ste- Weierstrass Therem easily fllws frm fllwig fact: f 2 A implies that Re f = (f ifi jfj kfk)=4 2 A ([6]). Fr a fully cmmutative real uifrm algebra A its maximal ideal space M H (A) csists f disjit cpies f M, ad is lcally hmemrphic with a Cartesia prduct f M with ather set. Quite fte we als have a glbal decmpsiti: M H (A) = Y M. Hwever such glbal decmpsiti ca t always be achieved; t eve its aalg i the cmmutative case. I the cmmutative case whe A C C (X; ) ad we ca divide X it three parts X ; X 2 ; ad X 3 such that A jx is a cmplex algebra, A jx2 csists f cmplex cjugates f the fuctis frm A jx, ad A jx3 is equal t C R (X 3 ) : It is easy t select X 3 = fx 2 X : (x) = xg ; we may als easily select e pit frm each equivalece class x x 2 i (x ) =

7 7 x 2 t get X ad we ca frm X 2 frm the remaiig pits. Hwever we may t be able t make the sets X ; X 2 clsed s A jx ; A jx2 may t be uifrm algebras - see fr example A = f 2 C C S : f e it = f e i(t+) fr 0 t where S is a uit circle. This is the reas we represet a real cmmutative uifrm algebra as a subalgebra f C C (X; ) rather the, what may seem mre appealig, a direct sum f A jx ; A jx2 ; ad A jx3 : We prved that the algebra H has the fllwig prperty: Ay subalgebra A f C H (X) = C R (X) H that strgly separates the pits f X ad such that ff (x) : x 2 Xg = H fr all x 2 X is equal t C H (X). We als kw (Ste-Weierstrass Therem) that the algebra R has the same prperty. It wuld be iterestig t d ther algebras with the same prperty. The authr wuld like t thak S. H. Kulkari fr his hspitality ad fr several stimulatig discussi durig the authr s visit i the Idia Istitute f Techlgy. Refereces [] B. Aupetit ad J. Zemaek, O the real spectral radius i real Baach algebras, Bull. Acad. Pl. Sci. Ser. Sci. Math. Astrm. Phys., 26 (978) [2] F. F. Bsall, J. Duca, Cmplete Nrmed Algebras, Spriger, Berli, 973 [3] R. Egelkig, Geeral Tplgy, Mathematical Mgraphs, Vl. 60, PWN Plish Scieti c Publishers, Warsaw, pp. [4] S. Heirich, Ultrapwers i Baach space thery, J. Reie Agew. Math., 33 (980), [5] R. A. Hirschfeld ad W. Zelazk, O spectral rm Baach algebras. Bull. Acad. Pl. Sci. Sér. Sci. Math. Astrm. Phys. 6 (968), [6] J. Hlladay, A te the Ste-Weierstrass therem fr quateris. Prc. Amer. Math. Sc. 8 (957), [7] K. Jarsz, Ultraprduct ad small bud perturbatis, Paci c J. Math., 48 (99) [8] K. Jarsz, Ncmmutative Uifrm Algebras, Studia Math. 62 (2004), [9] S. H. Kulkari ad B. V. Limaye, Real Fucti Algebras, Mgraphs ad Textbks i Pure ad Applied Math., 68, Marcel Dekker, Ic., New Yrk, 992. [0] S. V. Lüdkvsky ad F. va Oystaeye, Di eretiable fuctis f quateri variables. Bull. Sci. Math. 27 (2003),. 9, [] T. Palmer, Baach Algebras ad the Geeral Thery f -Algebras. vl. I. Algebras ad Baach algebras, Ecyclpedia f Mathematics ad its Applicatis, 49, Cambridge Uiv. Press, Cambridge, 994. [2] A. Sudbery, Quateriic aalysis. Math. Prc. Cambridge Phils. Sc. 85 (979),. 2, Suther Illiis Uiversity Edwardsville address: kjarsz@siue.edu URL: