Characterizing group $C^ast$-algebras through their unitary groups: the Abelian case
We study to what extent group $C^\ast$-algebras are characterized by their unitary groups. A complete characterization of which Abelian group $C^\ast$-algebras have isomorphic unitary groups is obtained. We compare these results with other unitary-re…
Authors: Jorge Galindo, Ana-Maria Rodenas
CHARA CTERIZING GR OUP C ∗ -ALGEBRAS THR OUGH THEIR UNIT AR Y GR OUPS: THE ABELIAN CASE JOR GE GALINDO AND ANA MARÍA RÓDENAS Abstra t. W e study to what exten t group C ∗ -algebras are haraterized b y their unitary groups. A omplete haraterization of whi h Ab elian group C ∗ -algebras ha v e isomorphi unitary groups is obtained. W e ompare these results with other unitary-related in v arian ts of C ∗ (Γ) , su h as the K -theoreti K 1 ( C ∗ (Γ)) and nd that C ∗ -algebras of nonisomorphi torsion-free Ab elian groups ma y ha v e isomorphi K 1 -groups, in sharp on trast with the w ell-kno wn fat that C ∗ (Γ) (ev en Γ ) is haraterized b y the top ologial group struture of its unitary group when Γ is torsion-free and Ab elian. 1. Intr odution The index theorem states that ev ery on tin uous f : T → T is homotopi to the funtion t 7→ t n for some n ∈ Z (its winding n um b er). As a onsequene the quotien t of the unitary group of C ∗ ( Z ) b y its onneted omp onen t is isomorphi to Z . This iden tiation an b e extended in a funtorial fashion to nitely generated Ab elian groups and their indutiv e limits. Sine ev ery torsion-free Ab elian group is an indutiv e limit of nitely generated groups, the follo wing theorem, that w e tak e as the departing p oin t of our pap er, follo ws. Theorem 1.1 (see Theorem 8.57 of [ 10 ℄) . If Γ is a torsion-fr e e A b elian gr oup the quotient U / U 0 of the unitary gr oup U = U ( C ∗ (Γ)) by its onne te d omp onent U 0 is isomorphi to Γ . Hen e, two torsion-fr e e A b elian gr oups Γ 1 and Γ 2 with top olo gi al ly isomorphi unitary gr oups U ( C ∗ (Γ 1 )) and U ( C ∗ (Γ 2 )) must alr e ady b e isomorphi. Theorem 1.1 suggests the usage of U ( C ∗ (Γ)) as an in v arian t for C ∗ (Γ) . T o determine its strength it is neessary to kno w to what exten t the top ologial group struture of U ( C ∗ (Γ)) determines C ∗ (Γ) . As a rst step in this diretion, w e dev ote Setion 4 to haraterize when t w o Ab elian groups Γ 1 and Γ 2 ha v e isomorphi unitary groups. The groups U ( C ∗ (Γ 1 )) and U ( C ∗ (Γ 2 )) are sho wn to b e top ologially isomorphi if and only if | Γ 1 /t (Γ 1 ) | = | Γ 2 /t (Γ 2 ) | =: α and ⊕ α Γ 1 /t (Γ 1 ) is group- isomorphi to ⊕ α Γ 2 /t (Γ 2 ) , where t (Γ i ) stands for the torsion subgroup of Γ i . Another unitary-related in v arian t of C ∗ (Γ) of great imp ortane is the K 1 -group, K 1 ( C ∗ (Γ)) . Sine K 1 ( C ∗ ( Z m )) = Z 2 m − 1 , t w o torsion-free nitely generated Ab elian groups are isomorphi whenev er their K 1 -groups are. The w a y this fat is pro v ed do es not ho w ev er allo w a funtorial extension to indutiv e limits and, indeed, w e Date : No v em b er 2, 2018. 2000 Mathematis Subje t Classi ation. 19L99, 22D15, 22D25, 43A40, 46L05, 46L80. Key wor ds and phr ases. group C ∗ -algebra, unitary group, top ologial group, K 1 -group, exte- rior pro dut. Resear h partly supp orted b y the Spanish Ministry of Siene (inluding FEDER funds), gran t MTM2004-07665-C02-01 and Generalitat V aleniana GV/2007/012. 1 2 JOR GE GALINDO AND ANA MARÍA RÓDENAS onstrut in Setion 3 t w o nonisomorphi torsion-free Ab elian groups Γ 1 and Γ 2 with isomorphi K 1 -groups, thereb y sho wing that Theorem 1.1 is not v alid for K 1 - groups instead of unitary groups. W e nd therefore that U ( C ∗ (Γ)) is a stronger in- v arian t than K 1 ( C ∗ (Γ)) , for torsion-free Ab elian groups. F or general (ev en Ab elian) groups this is no longer true, K 1 ( C ∗ (Γ)) distinguishes b et w een groups with dier- en t nitely generated torsion-free quotien ts, while U ( C ∗ (Γ)) need not, see Setion 5 . 2. Ba k gr ound This pap er is onerned with group C ∗ -algebras. The C ∗ -algebra C ∗ (Γ) of a group Γ is dened as the en v eloping C ∗ -algebra of the on v olution algebra L 1 (Γ) and, as su h, eno des the represen tation theory of Γ , see [4, P aragraph 13℄. W e analyze in this pap er to what exten t a group Γ , or rather the C ∗ -algebra struture of C ∗ (Γ) , is determined b y the top ologial group struture of U ( C ∗ (Γ) ) . The unitary groups U ( C ∗ (Γ)) are ob viously related to another in v arian t of C ∗ (Γ) of greater imp ortane, the K 1 -group of K -theory . K -theory for C ∗ -algebras is based on t w o funtors, namely , K 0 and K 1 , whi h asso iate to ev ery C ∗ -algebra A , t w o Ab elian groups K 0 ( A ) and K 1 ( A ) . The group K 1 ( A ) is in partiular dened b y iden tifying unitary elemen ts of matrix algebr as o v er A . It is allo wing matries o v er A (instead of elemen ts of A ) that mak es K 1 -groups Ab elian. If Γ is a disrete group, there is a natural em b edding of Γ in U ( C ∗ (Γ)) , this ma y b e omp osed with the anonial map ω : U ( C ∗ (Γ)) → K 1 ( C ∗ (Γ)) . K 1 ( C ∗ (Γ)) b eing Ab elian, the resulting homomorphism fators through the Ab elianization of Γ , yielding a homomorphism κ Γ : Γ / Γ ′ → K 1 ( C ∗ (Γ)) . κ Γ w as sho wn to b e rationally injetiv e in [7 ℄, see also [1 ℄. No w and for the rest of the pap er w e restrit our atten tion to disrete Ab elian groups. When Γ is a disrete Ab elian group, C ∗ (Γ) is a omm utativ e C ∗ -algebra with sp etrum homeomorphi to the ompat group b Γ , the group of haraters of Γ . W e ma y th us iden tify C ∗ (Γ) with the algebra of on tin uous funtions C ( b Γ , C ) and the Gelfand transform oinides with the F ourier transform. The unitary group U ( C ∗ (Γ)) an therefore b e iden tied with the top ologial group of T -v alued fun- tions C ( b Γ , T ) . Hene relating U ( C ∗ (Γ)) to Γ amoun ts in this ase to relating Γ to C ( b Γ , T ) . Also, for omm utativ e A (as is the ase with C ∗ (Γ) , with Γ Ab elian), the de- terminan t map ∆ : K 1 ( A ) → U ( A ) / U ( A ) 0 is a right inverse of the anonial map ω : U ( A ) / U ( A ) 0 → K 1 ( A ) (see [14 , Setion 8.3℄) and the link b et w een K 1 ( A ) and U ( A ) is stronger. The ommonly used notation K ∗ ( A ) = K 1 ( A ) ⊕ K 0 ( A ) will also b e adopted in this pap er. A kno wledgemen t: W e w ould lik e to thank Pierre de la Harp e for some sug- gestions and referenes that help ed us to impro v e the exp osition of this pap er. 3. A torsion-free Abelian gr oup Γ not determined by K 1 ( C ∗ (Γ)) As stated in the in tro dution, there is a group isomorphism In : C ( T , T ) /C ( T , T ) 0 → Z assigning to ev ery f ∈ C ( T , T ) its winding n um b er. In other w ords, ev ery ele- men t of C ( T , T ) is homotopi to exatly one harater of T . This p oin t of view CHARA CTERIZING GR OUP C ∗ -ALGEBRAS THR OUGH THEIR UNIT AR Y GR OUPS 3 an b e arried o v er to T n and then, taking pro jetiv e limits, to ev ery ompat on- neted Ab elian group, ultimately leading to Theorem 1.1 , after iden tifying C ( b Γ , T ) with U ( C ∗ (Γ)) . Despite the strong relation b et w een U ( C ∗ (Γ)) and K 1 ( C ∗ (Γ)) w e onstrut in this setion t w o nonisomorphi torsion-free Ab elian groups Γ 1 and Γ 2 with K 1 ( U ( C ∗ (Γ 1 ))) isomorphi to K 1 ( U ( C ∗ (Γ 2 ))) . 3.1. The struture of K 1 ( C ∗ (Γ)) for torsion-free Ab elian Γ . A oun table torsion-free Ab elian group Γ an alw a ys b e obtained as the indutiv e limit of torsion- free nitely generated Ab elian groups. Simply en umerate Γ = { γ n : n < ω } , dene Γ n = h γ j : 1 ≤ j ≤ n i and let φ n : Γ n → Γ n +1 dene the inlusion mapping, then Γ = lim − → (Γ n , φ n ) . Ea h homomorphism φ n then indues a morphism of C ∗ -algebras φ ∗ n : C ∗ (Γ n ) → C ∗ (Γ n +1 ) , and C ∗ (Γ) = lim − → ( C ∗ (Γ n ) , φ ∗ n ) The funtor K 1 omm utes with indutiv e limits, see for instane [14 ℄. If K 1 ( φ n ) : K 1 ( C ∗ (Γ n )) → K 1 ( C ∗ (Γ n +1 )) denotes the homomorphism indued b y the morphism φ ∗ n , w e ha v e K 1 ( C ∗ (Γ)) = lim − → ( K 1 ( C ∗ (Γ n )) , K 1 ( φ n )) . The groups Γ n in the ab o v e disussion are all isomorphi to Z k ( n ) , for suitable k ( n ) , and it is w ell-kno wn that K ∗ ( C ∗ ( Z k )) is isomorphi to the exterior pro dut ∧ Z k . Sine this realization of K 1 ( C ∗ (Γ)) through exterior pro duts will b e essen tial in determining our examples, w e next reall some basi fats ab out them. The k -th exterior, or w edge, pro dut ∧ k ( Z n ) of a nitely generated group Z n with free generators e 1 , . . . , e n is isomorphi to the free Ab elian group generated b y e i 1 ∧ · · · ∧ e i k : { i 1 , . . . , i k } ⊂ { 1 , . . . , n } . A group homomorphism φ : Z n → Z m indues a group homomorphism ∧ k ( φ ) : ∧ k ( Z n ) → ∧ k ( Z m ) in the ob vious w a y ∧ k ( φ )( e i 1 ∧ · · · ∧ e i k ) = φ ( e i 1 ) ∧ · · · ∧ φ ( e i k ) . If Γ = lim − → (Γ i , h i ) is a diret limit, ∧ k (Γ) an b e obtained as ∧ k (Γ) = lim − → ( ∧ k (Γ i ) , ∧ k ( h i )) . Other elemen tary prop- erties of exterior pro duts are b est understo o d taking in to aoun t that ∧ Γ is isomorphi to the quotien t of N Γ b y the t w o-sided ideal N spanned b y tensors of the form g ⊗ g . The referene [2 ℄ is a lassial one onerning exterior pro duts. The follo wing result is w ell-kno wn ([3, 6℄), w e supply a pro of for the reader's on v eniene. Lemma 3.1 ([6 ℄, P aragraph 2.1) . L et Γ b e a torsion-fr e e disr ete A b elian gr oup. Then K 1 ( C ∗ (Γ)) ∼ = ∧ o dd Γ := ∞ M j =0 ∧ 2 j +1 Γ . Pr o of. Reall in rst plae that there is a unique ring isomorphism R : ∧ Z n → K ∗ ( C ∗ ( Z n )) resp eting the anonial em b eddings of Z n in b oth K ∗ ( C ∗ ( Z n )) and ∧ Z n . Sine K ∗ ( C ∗ ( Z n )) = K 0 ( C ∗ ( Z n )) ⊕ K 1 ( C ∗ ( Z n )) and the ring struture K ∗ ( C ∗ ( Z n )) is Z 2 -graded (whi h means that x ∈ K i ( C ∗ ( Z n )) , y ∈ K j ( C ∗ ( Z n )) implies that xy ∈ K i + j ( C ∗ ( Z n )) with i, j ∈ Z 2 ), w e ha v e that the isomorphism R maps ∧ o dd Z n on to K 1 ( C ∗ ( Z n )) . No w put Γ = lim − → (Γ n , φ n ) with Γ n ∼ = Z j n . The uniqueness of the ab o v e men tioned ring-isomorphism, together with the fat that w edge pro duts omm ute with diret limits implies that K 1 ( C ∗ (Γ)) is isomorphi to ∧ o dd Γ . 4 JOR GE GALINDO AND ANA MARÍA RÓDENAS Sine the groups Γ n are alw a ys isomorphi to Z k ( n ) a omparison b et w een Γ and K 1 ( C ∗ (Γ)) turns in to a omparison of t w o indutiv e limits, lim − → ( Z k ( n ) , φ n ) and lim − → ( Z 2 k ( n ) − 1 , K 1 ( φ n )) . When Γ has nite rank m it ma y b e assumed without loss of generalit y that k ( n ) = m for all n . If in addition m ≤ 2 , it is easy to see (f. Lemma 3.5 ) that K 1 ( φ n ) = φ n . W e ha v e th us: Corollary 3.2. If Γ is a torsion-fr e e A b elian gr oup of r ank ≤ 2 , then K 1 ( C ∗ (Γ)) is isomorphi to Γ . Corollary 3.2 sho ws that t w o nonisomorphi torsion-free Ab elian groups Γ i with K 1 ( C ∗ (Γ 1 )) isomorphi to K 1 ( C ∗ (Γ 2 )) m ust ha v e rank larger than 2. F or our oun terexample w e will deal with t w o groups of rank 4. If Γ is su h a group, then K 1 ( C ∗ (Γ)) is isomorphi to ∧ 1 (Γ) ⊕ ∧ 3 (Γ) . Our seletion of the examples is determined b y the follo wing theorem of F u hs and Lo onstra. Theorem 3.3 (P artiular ase of Theorem 90.3 of [8℄) . Ther e ar e two nonisomor- phi gr oups Γ 1 and Γ 2 , b oth of r ank 2, suh that Γ 1 ⊕ Γ 1 ∼ = Γ 2 ⊕ Γ 2 . W e then ha v e: Theorem 3.4. L et Γ 1 , Γ 2 b e the gr oups of The or em 3.3 and dene the 4-r ank gr oups, ∆ i = Z ⊕ Z ⊕ Γ i . Then K 1 ( C ∗ (∆ 1 )) and K 1 ( C ∗ (∆ 2 )) ar e isomorphi, while ∆ 1 and ∆ 2 ar e not. W e shall split the pro of of Theorem 3.4 in sev eral Lemmas. W e b egin b y observ- ing ho w Lemma 3.1 mak es the groups K 1 ( C ∗ (∆ i )) easily realizable. Lemma 3.5. If Γ is a torsion-fr e e A b elian gr oup of r ank 2 and ∆ = Z ⊕ Z ⊕ Γ , then K 1 ( C ∗ (∆)) ∼ = Z ⊕ Z ⊕ Γ ⊕ Γ ⊕ ∧ 2 Γ ⊕ ∧ 2 Γ . Pr o of. As ∆ has rank 4, (1) ∧ o dd ∆ = ∧ 1 ∆ ⊕ ∧ 3 ∆ ∼ = ∆ ⊕ ∧ 3 ∆ . Put Γ = lim − → (Γ n , φ n ) , with Γ n ∼ = Z 2 . Then, dening id ⊕ id ⊕ φ n : Z ⊕ Z ⊕ Γ n → Z ⊕ Z ⊕ Γ n +1 in the ob vious w a y , w e ha v e that ∆ = lim − → ( Z ⊕ Z ⊕ Γ n , id ⊕ id ⊕ φ n ) and ∧ 3 ∆ = lim − → ( ∧ 3 ( Z ⊕ Z ⊕ Γ n ) , ∧ 3 (id ⊕ id ⊕ φ n )) . If e n j , j = 1 , 2 are the generators of Z ⊕ Z and f n j , j = 1 , 2 are the generators of Γ n , ∧ 3 ( Z ⊕ Z ⊕ Γ n ) = h e n 1 ∧ e n 2 ∧ f n 1 , e n 1 ∧ e n 2 ∧ f n 2 , e n 1 ∧ f n 1 ∧ f n 2 , e n 2 ∧ f n 1 ∧ f n 2 i . The images of ea h of these generators under the homomorphism ∧ 3 (id ⊕ id ⊕ φ n ) are: ∧ 3 (id ⊕ id ⊕ φ n ) e n 1 ∧ e n 2 ∧ f n j = e n +1 1 ∧ e n +1 2 ∧ φ n ( f n j ) , j = 1 , 2 ∧ 3 (id ⊕ id ⊕ φ n ) e n j ∧ f n 1 ∧ f n 2 = e n +1 j ∧ ∧ 2 ( φ n )( f n 1 ∧ f n 2 ) , j = 1 , 2 . In the limit, the thread formed b y the rst t w o generators will yield a op y of Γ while the one formed b y ea h of the other t w o will yield a op y of ∧ 2 Γ . This and (1 ) giv e the Lemma. CHARA CTERIZING GR OUP C ∗ -ALGEBRAS THR OUGH THEIR UNIT AR Y GR OUPS 5 W e no w tak e are of ∧ 2 (Γ) . This is a rank one group. Ab elian groups of rank one are ompletely lassied b y their so-alled t yp e. The t yp e of an Ab elian group A is dened in terms of p -heigh ts. Giv en a prime p , the largest in teger k su h that p k | a is alled the p -height h p ( a ) of a . The sequene of p -heigh ts χ ( a ) = ( h p 1 ( a ) , . . . , h p n ( a ) , . . . ) , where p 1 , . . . , p n , . . . is an en umeration of the primes, is then alled the har ateristi or the height-se quen e of a . T w o harateristis ( k 1 , . . . , k n , . . . ) and ( l 1 . . . , l n , . . . ) are onsidered equiv alen t if k n = l n for all but a nite n um b er of nite indies. An equiv alene lass of harateristis is alled a typ e . If χ ( a ) b elongs to a t yp e t , then w e sa y that a is of typ e t . In a torsion-free group of rank one all elemen ts are of the same t yp e (su h groups are alled homo gene ous ). F or more details ab out p -heigh ts and t yp es, see [8 ℄. The only fat w e need here is that t w o groups of rank 1 are isomorphi if and only if they ha v e non trivial elemen ts with the same t yp e, Theorem 85.1 of [ 8 ℄. W e no w study the t yp e of groups Γ ∧ Γ with Γ of rank 2. Lemma 3.6. L et Γ b e a torsion-fr e e gr oup of r ank 2 and let x 1 , x 2 ∈ Γ . The element x 1 ∧ x 2 ∈ Γ ∧ Γ is divisible by the inte ger m if and only if ther e is some element k 1 x 1 + k 2 x 2 ∈ Γ divisible by m with either k 1 or k 2 oprime with m . Pr o of. W e an without loss of generalit y assume that the subgroup generated b y x 1 , x 2 is isomorphi to Z 2 and that Γ is an additiv e subgroup of the v etor spae spanned o v er Q b y x 1 , x 2 . No w x ∧ y will b e divisible b y m if and only if there are elemen ts u 1 , u 2 in Γ su h that u i = α i 1 x 1 + α i 2 x 2 with det( α ij ) = 1 /m (note that u 1 ∧ u 2 = det( α ij ) x 1 ∧ x 2 ). T o get that determinan t w e learly need some denominator m and w e an assume (b y on v enien tly mo difying the α ij 's) that α 11 = k 1 /m and α 12 = k 2 /m with either k 1 or k 2 oprime with m . The elemen t of Γ w e w ere seeking is then k 1 x 1 + k 2 x 2 . Lemma 3.7. L et Γ 1 and Γ 2 b e two r ank 2, torsion-fr e e A b elian gr oups. If Γ 1 ⊕ Γ 1 ∼ = Γ 2 ⊕ Γ 2 , then ∧ 2 (Γ 1 ) ∼ = ∧ 2 (Γ 2 ) . Pr o of. Let { v 1 , w 1 } and { v 2 , w 2 } b e maximal indep enden t sets in Γ 1 and Γ 2 , re- sp etiv ely and denote b y φ : Γ 1 ⊕ Γ 1 → Γ 2 ⊕ Γ 2 the h yp othesized isomorphism. By on v enien tly re-dening the elemen ts v i and w i it ma y b e assumed that φ ( v 1 , 0) = ( α 11 v 2 + α 12 w 2 , β 11 v 2 + β 12 w 2 ) φ ( w 1 , 0) = ( α 21 v 2 + α 22 w 2 , β 21 v 2 + β 22 w 2 ) , with α ij , β i,j ∈ Z , i, j ∈ { 1 , 2 } . W e will no w nd a nite set of primes F su h that v 2 ∧ w 2 is divisible b y p k whenev er v 1 ∧ w 1 is divisible b y p k , for ev ery prime p / ∈ F . Sine the whole pro ess an b e rep eated for φ − 1 , this will sho w that v 1 ∧ w 1 and v 2 ∧ w 2 ha v e the same t yp e. Sine φ is an isomorphism, the matrix M = α 11 α 21 α 12 α 22 β 11 β 21 β 12 β 22 has rank t w o. A t least one of the follo wing submatries m ust then ha v e rank 2 as w ell: M 1 = α 11 α 21 α 12 α 22 , M 2 = β 11 β 21 β 12 β 22 or M 3 = α 11 α 21 β 11 β 21 . 6 JOR GE GALINDO AND ANA MARÍA RÓDENAS Let p b e an y prime not dividing det( M 1 ) , det( M 2 ) or det( M 3 ) and supp ose p k divides v 1 ∧ w 1 . By Lemma 3.6 there is an elemen t A = k 1 v 1 + k 2 w 1 ∈ Γ 1 divisible b y p k with either k 1 or k 2 oprime with p . Then φ ( A, 0) = k 1 φ ( v 1 , 0) + k 2 φ ( w 1 , 0) = (2) ( k 1 α 11 + k 2 α 21 ) v 2 + ( k 1 α 12 + k 2 α 22 ) w 2 , ( k 1 β 11 + k 2 β 21 ) v 2 + ( k 1 β 12 + k 2 β 22 ) w 2 ∈ Γ 2 × Γ 2 Supp ose for instane that M 1 has rank 2. The only solution mo dulo p to the system α 11 x + α 21 y = 0 α 12 x + α 22 y = 0 is then the trivial one. The in tegers k 1 and k 2 annot therefore b e a solution to the system (they are not b oth oprime with p ). It follo ws that one of the in tegers k 1 α 11 + k 2 α 21 or α 12 k 1 + α 22 k 2 is not a m ultiple of p . If M 2 or M 3 ha v e rank t w o w e argue exatly in the same w a y . A t the end w e nd that at least one of the k 1 α 1 i + k 2 α 2 i or k 1 β 1 i + k 2 β 2 i is not a m ultiple of p . W e kno w b y (2) that b oth ( k 1 α 11 + k 2 α 21 ) v 2 + ( k 1 α 12 + k 2 α 22 ) w 2 and ( k 1 β 11 + k 2 β 21 ) v 2 + ( k 1 β 12 + k 2 β 22 ) w 2 are divisible b y p k and w e onlude with Lemma 3.6 that v 2 ∧ w 2 is divisible b y p k . Pro of of Theorem 3.4 T o see that K 1 ( C ∗ (∆ 1 )) ∼ = K 1 ( C ∗ (∆ 2 )) , simply put together Lemma 3.7 and Lemma 3.5 . Sine Γ 1 and Γ 2 are not isomorphi and nitely generated Ab elian groups ha v e the anellation prop ert y , ∆ 1 and ∆ 2 annot b e isomorphi, either. R emark 3.8 . The argumen t of Lemma 3.5 sho ws that K 0 ( C ∗ (∆)) is (again) isomor- phi to Z ⊕ Z ⊕ Γ ⊕ Γ ⊕ ∧ 2 (Γ) ⊕ ∧ 2 (Γ) (this time K 0 ( C ∗ (∆)) ∼ = ∧ 0 ∆ ⊕ ∧ 2 ∆ ⊕ ∧ 4 ∆ with ∧ 2 ∆ ∼ = Z ⊕ Γ ⊕ Γ ⊕ ∧ 2 Γ and ∧ 4 ∆ ∼ = ∧ 2 Γ ). The group C ∗ -algebras C ∗ (∆ 1 ) and C ∗ (∆ 2 ) of Theorem 3.4 ha v e therefore the same K -theory . 4. Rela ting U ( C ∗ (Γ)) and Γ This Setion is dev oted to evidene what is the relation b et w een t w o C ∗ -algebras C ∗ (Γ 1 ) and C ∗ (Γ 2 ) with top ologially isomorphi unitary groups. A result lik e Theorem 1.1 annot b e exp eted for general Ab elian groups, as for instane all oun tably innite torsion groups ha v e isometri C ∗ -algebras. The righ t question to ask is ob viously whether group C ∗ -algebras are determined b y their unitary groups. Ev en if this question also has a negativ e answ er, t w o group C ∗ -algebras C ∗ (Γ 1 ) and C ∗ (Γ 2 ) are strongly related when U ( C ∗ (Γ 1 )) and U ( C ∗ (Γ 2 )) are top ologially isomorphi as the on ten ts of this Setion sho w. Our main to ols here will b e of top ologial nature and w e shall regard U ( C ∗ (Γ)) as C ( b Γ , T ) . W e b egin with a w ell-kno wn observ ation. Denote b y C 0 ( X, T ) the subgroup of C ( X, T ) onsisting of all n ullhomotopi maps, that is, C 0 ( X, T ) is the on- neted omp onen t of the iden tit y of C ( X, T ) . Let also π 1 ( X ) denote the quotien t C ( X, T ) /C 0 ( X, T ) , also kno wn as the rst ohomotop y group of X and often de- noted as [ X , T ] . It is w ell kno wn that C 0 ( X, T ) oinides with the group of fun- tions that fator through R , that is, C 0 ( X, T ) is the range of the exp onen tial map exp : C ( X , R ) → C ( X, T ) . CHARA CTERIZING GR OUP C ∗ -ALGEBRAS THR OUGH THEIR UNIT AR Y GR OUPS 7 Lemma 4.1 (Setion 3 of [ 13 ℄, see page 405 of [ 9℄ for this form) . If X is a om- p at Hausdor sp a e, the strutur e of C ( X, T ) is desrib e d by the fol lowing exat se quen e 0 → C ( X , Z ) → C ( X, R ) → C 0 ( X, T ) → C ( X, T ) → π 1 ( X ) . In addition C 0 ( X, T ) is op en and splits, i.e., C ( X, T ) ∼ = C 0 ( X, T ) ⊕ π 1 ( X ) . Our seond observ ation is that, as far as group C ∗ -algebras are onerned, all disrete Ab elian groups ha v e a splitting torsion subgroup. Theorem 4.2 (Corollary 10.38 [10 ℄) . The onne te d omp onent G 0 of a omp at gr oup G , splits top olo gi al ly, i.e, G is home omorphi to G 0 × G/G 0 . The harater group of a oun table disrete group Γ is a ompat metrizable group b Γ and the set of haraters that v anish on its torsion group, t Γ , oinides with the onneted omp onen t of b Γ , in sym b ols t Γ ⊥ = b Γ 0 . F urther, the dualit y b et w een disrete Ab elian and ompat Ab elian groups iden ties b t Γ with the quotien t b Γ / b Γ 0 . It follo ws therefore from Theorem 4.2 that (3) b Γ ∼ b t Γ × ( t Γ) ⊥ and, hene, that C ∗ (Γ) is isometri to C ∗ ( t Γ ⊕ Γ /t Γ) . W e no w turn our atten tion to groups with splitting onneted omp onen t. 4.1. The struture of unitary groups of ertain omm utativ e C ∗ -algebras. W e b egin b y noting that the additiv e struture of a omm utativ e C ∗ -algebra on- tains v ery little information on the algebra. This fat will b e useful in lassifying unitary groups. Theorem 4.3 (Milutin, see for instane Theorem I I I.D.18 of [15 ℄) . If K 1 and K 2 ar e un ountable, omp at metri sp a es, then the Banah sp a es C ( K 1 , C ) and C ( K 2 , C ) ar e top olo gi al ly isomorphi. Lemma 4.4. L et K and D b e omp at top olo gi al sp a es, K onne te d and D total ly dis onne te d. The fol lowing top olo gi al isomorphism then holds: (4) C ( K × D, T ) ∼ = C ( K × D, R ) × C ( D , T ) × ⊕ w ( D ) π 1 ( K ) , wher e w ( D ) denotes the top olo gi al weight of D 1 Pr o of. W e rst observ e that C ( K × D , T ) is top ologially isomorphi to C ( D, C ( K, T )) . F rom Lemma 4.1 w e dedue that (5) C ( K × D, T ) ∼ = C ( D, C 0 ( K, T )) × C ( D , π 1 ( K )) . There is a top ologial isomorphism from the Bana h spae C ( K, R ) on to the Bana h spae C • ( K, R ) of funtions sending 0 to 0. It is no w easy to he k that the mapping ( f , t ) 7→ t · exp( f ) iden ties C • ( K, R ) × T with C 0 ( K, T ) and hene C 0 ( K, T ) ∼ = C ( K, R ) × T . Along with (5) w e obtain C ( K × D , T ) ∼ = C ( D, C ( K, R ) × T ) × C ( D , π 1 ( K )) = C ( D × K , R ) × C ( D , T ) × C ( D , π 1 ( K )) . 1 By the top ologial w eigh t of a top ologial spae X w e mean, as usual, the least ardinal n um b er of a basis of op en sets of X . 8 JOR GE GALINDO AND ANA MARÍA RÓDENAS No w π 1 ( K ) is a disrete group and ea h elemen t of C ( D, π 1 ( K )) determines an op en and losed subset of D . An analysis iden tial to that of [ 5 ℄ for C ( X, Z ) then yields C ( D, π 1 ( K )) ∼ = ⊕ w ( D ) π 1 ( K ) , and the pro of follo ws. The follo wing lemma an b e found as Exerise E.8.14 in [10 ℄. Lemma 4.5. If D is a total ly dis onne te d omp at sp a e, C ( D, T ) = C 0 ( D , T ) and C ( D, T ) is onne te d. Theorem 4.6. L et K i , i = 1 , 2 , b e two omp at onne te d metrizable sp a es and let D i , i = 1 , 2 , b e two omp at total ly dis onne te d metrizable sp a es. Dening X = K 1 × D 1 and Y = K 2 × D 2 ,the fol lowing assertions ar e e quivalent. (1) C ( X, T ) ∼ = C ( Y , T ) . (2) (a) L w ( D 1 ) π 1 ( K 1 ) ∼ = L w ( D 2 ) π 1 ( K 2 ) , wher e w ( D 1 ) and w ( D 2 ) ar e the top olo gi al weights of D 1 and D 2 , r esp e tively, and (b) C ( D 1 , T ) ∼ = C ( D 2 , T ) . Pr o of. It is ob vious from Theorem 4.3 (observ e that K i × D i is unoun table as so on as K i is non trivial) and Lemma 4.4 that (2) implies (1). W e no w use the deomp osition of Lemma 4.4 to dedue (2) from (1). Assertion (a) follo ws from fatoring out onneted omp onen ts in (4) (note that C ( K i × D i , R ) × C ( D i , T ) is onneted, use Lemma 4.5 for C ( D i , T ) ). The onneted omp onen ts C ( K 1 × D 1 , R ) × C ( D 1 , T ) and C ( K 2 × D 2 , R ) × C ( D 2 , T ) will b e top ologially isomorphi as w ell. Let H : C ( K 1 × D 1 , R ) × C ( D 1 , T ) → C ( K 2 × D 2 , R ) × C ( D 2 , T ) denote this isomorphism. Consider no w the homomorphism b H : C ( K 2 × D 2 , R ) b × C ( D 2 , T ) b → C ( K 1 × D 1 , R ) b × C ( D 1 , T ) b that results from dualizing H . When D is a totally disonneted ompat group, the only on tin uous haraters of C ( D, T ) are linear om binations with o eien ts in Z of ev aluations of elemen ts of D , i.e., the group C ( D, T ) b is isomorphi to the free Ab elian group A ( D ) on D [13 ℄ (see [9℄ for more on the dualit y b et w een C ( X, T ) and A ( X ) based on the exat sequene in Lemma 4.1 )). There is on the other hand a w ell-kno wn isomorphism b et w een C ( K 1 × D 1 , R ) b and the additiv e group of the v etor spae of all on tin uous linear funtionals on C ( K 1 × D 1 , R ) . The group C ( K 1 × D 1 , R ) b is therefore a divisible. Sine free Ab elian groups, su h as A ( D i ) , do not on tain an y divisible subgroup, b H ( C ( K 1 × D 1 , R ) b m ust equal C ( K 2 × D 1 , R ) b . W e dedue th us, taking quotien ts, that C ( D 1 , T ) and C ( D 2 , T ) are top ologially isomorphi. 4.2. The group ase. W e no w sp eialize the results in the previous paragraphs for the ase of a ompat Ab elian group. When T is a torsion disrete Ab elian group, b T is a ompat totally disonneted group and hene homeomorphi to the Can tor set. The group C ∗ -algebras of all oun tably innite torsion Ab elian groups will therefore b e isometri. These fats are summarized in the follo wing lemma. Lemma 4.7. L et T 1 and T 2 b e ountable torsion disr ete A b elian gr oups. Then the fol lowing assertions ar e e quivalent: (1) The gr oup C ∗ -algebr as C ∗ ( T 1 ) and C ∗ ( T 2 ) ar e isomorphi as C ∗ -algebr as. CHARA CTERIZING GR OUP C ∗ -ALGEBRAS THR OUGH THEIR UNIT AR Y GR OUPS 9 (2) The unitary gr oups of C ∗ ( T 1 ) and C ∗ ( T 2 ) ar e top olo gi al ly isomorphi. (3) The omp at gr oups c T 1 and c T 2 ar e home omorphi. (4) The gr oups T 1 and T 2 have the same ar dinal. Hene, the main result asserts: Theorem 4.8. L et Γ 1 and Γ 2 b e ountable disr ete A b elian gr oups. The fol lowing ar e e quivalent: (1) The unitary gr oups of C ∗ (Γ 1 ) and C ∗ (Γ 2 ) ar e top olo gi al ly isomorphi. (2) | t Γ 1 | = | t Γ 2 | = α and M α Γ 1 t Γ 1 ∼ = M α Γ 2 t Γ 2 . Pr o of. Using the homeomorphi iden tiation in (3), page 7 , and Lemma 4.4 w e ha v e: (6) U ( C ∗ (Γ i )) ∼ = C ( c t Γ i × ( t Γ i ) ⊥ , T ∼ = C ( c t Γ i × ( t Γ i ) ⊥ , R ) × C ( c t Γ i , T ) × M w ( c t Γ i ) π 1 (( t Γ i ) ⊥ ) , where ( t Γ i ) ⊥ are ompat onneted and c t Γ i are ompat totally disonneted Ab elian groups. Supp ose rst that U ( C ∗ (Γ 1 )) and U ( C ∗ (Γ 2 )) are top ologially isomorphi. By Theorem 4.6 , C ( c t Γ 1 , T ) is top ologially isomorphi to C ( c t Γ 2 , T ) . It follo ws from Lemma 4.7 that c t Γ 1 and c t Γ 2 are homeomorphi. Let α = w ( c t Γ 1 ) . By statemen t (a) of Theorem 4.6 , M α π 1 (( t Γ 1 ) ⊥ ) ∼ = M α π 1 (( t Γ 2 ) ⊥ ) , No w π 1 ( t Γ ⊥ i ) is isomorphi b y Theorem 1.1 to the torsion-free group Γ i /t (Γ i ) . The ab o v e isomorphism th us b eomes (7) M α Γ 1 t Γ 1 ∼ = M α Γ 2 t Γ 2 and w e are done. Supp ose on v ersely that assertion (2) holds. W e ha v e then from Lemma 4.7 that C ( c t Γ 1 , T ) and C ( c t Γ 1 , T ) are top ologially isomorphi. On the other hand, the isomorphism L α Γ 1 t Γ 1 ∼ = L α Γ 2 t Γ 2 implies, b y w a y of The- orem 1.1 , that ⊕ α π 1 (( t Γ 1 ) ⊥ ) is isomorphi to ⊕ α π 1 (( t Γ 2 ) ⊥ ) . It follo ws then from Theorem 4.6 that C ( c Γ 1 , T ) and C ( c Γ 2 , T ) , that is U ( C ∗ (Γ 1 )) and U ( C ∗ (Γ 2 )) , are top ologially isomorphi. 5. Conluding remarks Theorem 1.1 sho ws ho w strongly the top ologial group struture of U ( A ) ma y happ en to determine a C ∗ -algebra A . Theorem 4.8 then preises the amoun t of information on A that is eno ded in U ( A ) , for the ase of a omm utativ e group C ∗ -algebra. This rev eals some limitations on the strength of U ( A ) as an in v arian t of A that will b e made onrete in this Setion. 10 JOR GE GALINDO AND ANA MARÍA RÓDENAS F rom Theorem 1.1 and Lemma 4.7 w e ha v e that C ∗ (Γ) is ompletely determined b y its unitary group when Γ is either torsion-free or a torsion group. This is not the ase if Γ is a mixed group. Example 5.1. Two nonisometri A b elian gr oup C ∗ -algebr as with top olo gi al ly iso- morphi unitary gr oups. Pr o of. Let Γ 1 and Γ 2 b e the groups in Theorem 3.3 . Dene ∆ i = Γ i ⊕ Z 2 . Iden ti- fying as usual C (∆ i , T ) with U ( C ∗ (∆ i )) and applying Lemma 4.4 , w e ha v e that U ( C ∗ (∆ i )) ∼ = C (∆ i , R ) × T 2 × (Γ i ⊕ Γ i ) . The eletion of Γ i and Milutin's theorem sho w that U ( C ∗ (∆ 1 )) is top ologially isomorphi to U ( C ∗ (∆ 2 )) . The algebras C ∗ (∆ 1 ) and C ∗ (∆ 2 ) are not isometri, sine their sp etra, c Γ 1 × Z 2 and c Γ 2 × Z 2 , are not homeomorphi (their onneted omp onen ts are not homeo- morphi). This example also sho ws that simple "dupliations" of torsion-free groups are not determined b y the unitary groups of their C ∗ -algebras: Example 5.2. Two nonisomorphi torsion-fr e e A b elian gr oups Γ 1 and Γ 2 suh that U ( C ∗ (Γ 1 ⊕ Z 2 )) and U ( C ∗ (Γ 2 ⊕ Z 2 )) ar e top olo gi al ly isomorphi. Finally , Example 5.3. Two A b elian gr oups Γ 1 and Γ 2 of dier ent torsion-fr e e r ank with U ( C ∗ (Γ 1 )) top olo gi al ly isomorphi to U ( C ∗ (Γ 2 )) . Pr o of. Let Γ 1 = Z ⊕ ( ⊕ ω Z 2 ) and Γ 2 = ( Z ⊕ Z ) ⊕ ( ⊕ ω Z 2 ) . The argumen t no w is as in Example 5.1 . In the ab o v e example one an ob viously replae Γ 2 b y ( ⊕ ω Z ) ⊕ ( ⊕ ω Z 2 ) and ha v e an example of t w o Ab elian groups with U ( C ∗ (Γ 1 )) top ologially isomorphi to U ( C ∗ (Γ 2 )) while the torsion-free rank of one of them is nite and the torsion-free rank of the other is innite. 5.1. In v arian ts. The unitary group U ( C ∗ (Γ)) is an in v arian t of the group C ∗ (Γ) , and as su h an b e ompared with other w ell kno wn unitary-related in v arian ts, lik e for instane K 1 ( C ∗ (Γ)) . W e an also men tion here related w ork of Hofmann and Morris on free ompat Ab elian groups [11 ℄. This is part of a more general pro jet of atta hing a ompat top ologial group F C ( X ) to ev ery ompat Hausdor spae X . The free ompat Ab elian group on X is onstruted as the harater group of the disr ete group C ( X, T ) d . F or an Ab elian group Γ , this pro ess pro dues an in v arian t of C ∗ (Γ) , namely the group U ( C ∗ (Γ)) d equipp ed with the disrete top ology . The harater group of U ( C ∗ (Γ)) d is preisely the free ompat Ab elian group on b Γ . Being the same ob jet but with no top ology , this in v arian t is w eak er than U ( C ∗ (Γ)) . It is easy to see that it is indeed stritly w eak er, simply tak e Γ 1 = Q and Γ 2 = ⊕ ω Q . In general there is a op y of the free Ab elian group generated b y X , densely em b edded in F C ( X ) , F C ( X ) is, atually (a realization of ) the Bohr ompatiation of the free Ab elian top ologial group on X (see [9℄ for detailed referenes on free Ab elian top ologial groups and their dualit y prop erties). Sine t w o top ologial spaes with top ologial isomorphi free Ab elian top ologial groups m ust ha v e the same o v ering dimension [ 12 ℄, Example 5.2 is somewhat unexp eted. CHARA CTERIZING GR OUP C ∗ -ALGEBRAS THR OUGH THEIR UNIT AR Y GR OUPS 11 The omparison with K 1 ( U ( C ∗ (Γ))) is ri her. As w e sa w in Setion 3, the group algebras C ∗ (Γ 1 ) and C ∗ (Γ 2 ) of t w o nonisomorphi torsion-free Ab elian groups Γ 1 and Γ 2 an ha v e isomorphi K 1 -groups, while their unitary groups m ust b e top o- logially isomorphi b y Theorem 1.1 . The opp osite diretion do es not w ork either. W e nd next t w o disrete groups whose group C ∗ -algebras ha v e isomorphi unitary groups while their K 1 -groups fail to b e so. W e rst see that from Theorem 4.2 and with a simple appliation of the Künneth theorem, the K 1 -group of a group C ∗ -algebra dep ends exlusiv ely on its torsion-free omp onen t. Lemma 5.4. L et Γ b e an A b elian disr ete gr oup. Then K 1 ( C ∗ (Γ)) ∼ = K 1 ( C ∗ (Γ /t Γ)) Pr o of. F rom Theorem 4.2 , b Γ is homeomorphi to b Γ / b Γ 0 × b Γ 0 , where b Γ / b Γ 0 ∼ = b t Γ and b Γ 0 ∼ = t Γ ⊥ ∼ = [ Γ /t Γ . Therefore, (8) C ∗ (Γ) ∼ = C ∗ ( t Γ) ⊗ C ∗ (Γ /t Γ) . Applying the Künneth form ula to (8), w e obtain, K 1 ( C ∗ (Γ)) ∼ = K 1 ( C ∗ ( t Γ) ⊗ C ∗ (Γ /t Γ)) ∼ = K 0 ( C ∗ ( t Γ)) ⊗ K 1 ( C ∗ (Γ /t Γ)) ⊕ K 1 ( C ∗ ( t Γ)) ⊗ K 0 ( C ∗ (Γ /t Γ)) ∼ = Z ⊗ K 1 ( C ∗ (Γ /t Γ)) ∼ = K 1 ( C ∗ (Γ /t Γ)) , sine K 0 ( C ( D )) = Z and K 1 ( C ( D )) = 0 for a innite totally disonneted ompat group D . Example 5.5. Two A b elian gr oups Γ 1 and Γ 2 whose gr oup C ∗ -algebr as have top o- lo gi al ly isomorphi unitary gr oups, wher e as their K 1 -gr oups ar e nonisomorphi. Pr o of. T ak e Γ 1 and Γ 2 from Example 5.3. Applying Lemma 5.4 and Lemma 3.1 , w e ha v e that K 1 ( C ∗ (Γ 1 )) ∼ = K 1 ( C ∗ ( Z )) ∼ = Z and K 1 ( C ∗ (Γ 2 )) ∼ = K 1 ( C ∗ ( Z ⊕ Z )) ∼ = Z ⊕ Z . The top ologial groups U ( C ∗ (Γ 1 )) and U ( C ∗ (Γ 2 )) are top ologially isomorphi as w as pro v ed in Example 5.3 . As a onsequene, w e see that none of the in v arian ts U ( C ∗ (Γ)) and K 1 ( C ∗ (Γ)) , of a group algebra C ∗ (Γ) is stronger than the other. The groups in Theorem 3.4 also sho w that t w o nonisometri (Ab elian) C ∗ -algebras an ha v e top ologially isomorphi unitary groups and isomorphi K 1 -groups. T ak e Φ i = ∆ i × Z 2 with ∆ i dened as in Theorem 3.4 . The same argumen t of Example 5.1 sho ws that U ( C ∗ (∆ i )) ∼ = C (Φ i , R ) × T 2 × ∆ i × ∆ i and, hene, that U ( C ∗ (Φ 1 )) ∼ = U ( C ∗ (Φ 2 )) . T o see that K 1 ( C ∗ (Φ 1 )) ∼ = K 1 ( C ∗ (Φ 2 )) simply note that, b y Lemma 5.4 , K 1 ( C ∗ (Φ i )) ∼ = K 1 ( C ∗ (∆ i )) and that K 1 ( C ∗ (∆ 1 )) ∼ = K 1 ( C ∗ (∆ 2 )) b y Theorem 3.4 . Referenes [1℄ Hela Bettaieb and Alain V alette. Sur le group e K 1 des C ∗ -algèbres réduites de group es disrets. C. R. A ad. Si. Paris Sér. I Math. , 322(10):925928, 1996. [2℄ N. Bourbaki. Éléments de mathématique. A lgèbr e. Chapitr es 1 à 3. Hermann, P aris, 1970. [3℄ Berndt Brenk en. K -groups of solenoidal algebras. I. Pr o . A mer. Math. So . , 123(5):1457 1464, 1995. [4℄ Jaques Dixmier. L es C ∗ -algèbr es et leurs r epr ésentations . Gauthier-Villars & Cie, Éditeur- Imprimeur, P aris, 1964. 12 JOR GE GALINDO AND ANA MARÍA RÓDENAS [5℄ Katsuy a Eda, Shizuo Kamo, and Haruto Oh ta. Ab elian groups of on tin uous funtions and their duals. T op olo gy Appl. , 53(2):131151, 1993. [6℄ G. A. Elliott. On the K -theory of the C ∗ -algebra generated b y a pro jetiv e represen tation of a torsion-free disrete ab elian group. In Op er ator algebr as and gr oup r epr esentations, V ol. I (Neptun, 1980) , v olume 17 of Mono gr. Stud. Math. , pages 157184. Pitman, Boston, MA, 1984. [7℄ George A. Elliott and T oshik azu Natsume. A Bott p erio diit y map for rossed pro duts of C ∗ -algebras b y disrete groups. K -The ory , 1(4):423435, 1987. [8℄ László F u hs. Innite ab elian gr oups. Vol. II . A ademi Press, New Y ork, 1973. Pure and Applied Mathematis. V ol. 36-I I. [9℄ Jorge Galindo and Salv ador Hernández. P on try agin-v an Kamp en reexivit y for free ab elian top ologial groups. 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W o jtaszzyk. Banah sp a es for analysts , v olume 25 of Cambridge Studies in A dvan e d Mathematis . Cam bridge Univ ersit y Press, Cam bridge, 1991. Dep ar t amento de Ma temá tias, Universid ad Ja ume I, Campus Riu Se, 12071, Castel- lón, Sp ain, E-mail addr ess : jgalindomat.uji.es E-mail addr ess : arodenasmat.uji.es
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