Fack, Thierry

Annales de l’institut Fourier (AIF)

ANNALES DE L’INSTITUT FOURIERTHIERRY FACKFinite sums of commutators in C∗-algebrasAnnales de l’institut Fourier, tome 32, no 1 (1982), p. 129-137<http://www.numdam.org/item?id=AIF_1982__32_1_129_0>© Annales de l’institut Fourier, 1982, tous droits réservés.L’accès aux archives de la revue « Annales de l’institut Fourier »(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-tion commerciale ou impression systématique est constitutive d’une in-fraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/Ann. Inst. Fourier, Grenoble32,1 (1982), 129-137FINITE SUMS OF COMMUTATORS IN C*-ALGEBRASby Thierry FACKIntroduction.Let A be a C*-algebra and putAQ = \ x G A | x = ^ x^ x^ - x^x^ ; norm convergence( n>\By [4] (theorem 2.6), A() is exactly the null space of all finitetraces on the self-adjoint part of A.For von Neumann algebras, AQ is spanned by finite sums ofthe above type (see for example [6]). This is not always true for C*-algebras, as it is shown by Pedersen and Petersen ([8], lemma 3.5)for a very natural algebra. A reasonable question is then : when canthis happen for C*-algebras ?The aim of this paper is to show that AQ is spanned by finitesums for stable algebras and C^-algebras with "sufficiently manyprojections" like infinite simple C*-algebras or simple A.F-algebras(with unit).We use the usual terminology of C*-algebras as in [7]. Acommutator of the form [ x , x * } ^ x x * — x * x is called a self-adjoint commutator.* * •* **I ' d like to thank G. Skandalis for fruitful discussions and G.K.Pedersen who originally asked this question.130 T.FACK1. Stable C*-algebras.Recall that a C'"-algebra A is stable if A ^ A ® 9C , where3<. is the C^-algebra of compact operators. We haveTHEOREM 1.1. - Let A be a stable V-algebra. Then, every her-mitian element of A is the sum of five self-adjoint commutators.Every simple A . F-algebra A without non zero finite tracebeing stable, it follows that Ao is spanned by finite sums of self-adjoint commutators.The proof of theorem 1.1 is based on the following lemmas.LEMMA 1.2. -Let A be a C*-algebra and x = x * C A . Letp be a projection in M(A). Then, there exists v C A such thatx -pxp + (1 - p ) x ( l - p ) + [ v , v * ] .Proof - Putv = 1/2|(1 - p ) x p \ l / 2 - |(1 - p ) x p \ l / 2 u * +u |(1 - p ) x p \ l / 2where u is the phase of ( l - p ) x p . As p G M ( A ) , we havev E A . By direct calculation, we have px(l - p) + (1 -p)xp == [ v , v * ] .LEMMA 1.3. - Let A be a C*-algebra with unit and x = x* E A.Let (\i, . . . , \) be a sequence of real numbers satisfyingk0< ^ \ ,<1 ( A : = l , . . . , ^ -1)i= iandi \ = o .1=1Then, there exists u E M^ (A), || u \\ < \\ x \\1/2, such that[\,x 01=[^*L0 • ^4Proof. - Write x = x+ - x_ and put.+ =/ ^ i. \^ .1/2A:' 1 = 1ri'(£ \)'"'t'1FINITE SUMS OF COMMUTATORS IN C*-ALGEBRAS 1311/2^ = ( 1 \) ^/2 ( A : = l , . . . ^ - 1).Y , = l /n-1Take M = ^ (^ ® ̂ ^ + ̂  ® ̂ ^), where ( ^ ) i < / , , < «fc=iis the canonical system of matrix units. As x + x _ = 0, we get theresult by direct calculation, oLet e be a rank one projection in Wi .LEMMA 1.4. - Let A be a C*-algebra and x = x* € A. Then,x ® e is the sum of two self-adjoint commutators of A ® 9C .Proof. -where (\,(-)„>! is the12Write1' 2 'x <2sequence14 'S e1414 ' 4 '".X:0\^x•>10'X,;cro 0X,xX^^LO1 1 \- g , . . . , - ^ - , . . . ^ .8 termsThe result follows from lemma 1.3.Proof of theorem 1.1. — Let x be a hermitian element of A ®3<.Take a projection p G M(3C) with p ^ 1 — p ^ 1 .By lemma 1.2, there exists v G A ®3^C such that^=pxp + ( 1 -^(1 -p)+ h,i;*].By lemma 1.4, pxp and (1 — p)x (1 — p) are both sums oftwo self-adjoint commutators. Q2. Infinite simple C*-algebras.The main result of this section is the followingTHEOREM 2.1. - Let A be a C*-algebra with unit. Supposethat there exist two orthogonal projections e and f such thate ^ f ~ 1 in A. Then, each hermitian element of A is the sumof five self-adjoint commutators.132 T.FACKRecall that a simple C*-algebra with unit is said to be infiniteif it contains an element x such that x*x = 1 and xx* ^ 1 . Fromtheorem 2.1, we deduceCOROLLARY 2.2. - Let A be an infinite simple C*-algebrawith unit. Then each hermitian element of A is the sum of five self-adjoint commutators.Apply theorem 2.1 and proposition 2.2 of [1]. The proof oftheorem 2.1 is based on the following lemma :LEMMA 2.3. - Let A , e and f be as in theorem 2.1. Let pbe a rank one projection in Wi. Then, there exists a homomorphism^ : A ® 3<: —> A such that^p(x ® p ) =x for each x € ( l -/)A(1 -/).Proof — Let u , v be partial isometries such thatu*u = v*v = 1 ; uu* = e , vv* == f.Put H^ = 1 - / + vf and w^ = vu^ ~1 v(n > 2).The w^ are isometries with pairwise orthogonal ranges. Let(^y) be a system of matrix units for ffC , with e^ = p . Put then^p(z 0 e,y) = w, zwy" (z e A). aProof of the theorem 2.1. - Let x = x* G A . By lemma 1.2,there exists y € A such that x = exe + (1 - e)x (1 - e) + [ y , y * ] .By lemmas 2.3 and 1.4, both exe and (1 — e)x (1 — e) are sumsof two self-adjoint commutators (note that exe G ( l — / )A(1— /)). aFor non simple infinite C^-algebras with unit, we may combinecorollary 2.2 with the following obvious lemma:LEMMA 2.4. - Let 0 —> ] —> A ——> B —> 0 be anexact sequence of C* algebras. Suppose that each hermitian elementof 1 (resp. of B) is a sum of n (resp. k) self-adjoint commutators.Then, any hermitian element of A is the sum of n + k self-adjointcommutators.Example. - Let A = (A(i,;)),. ̂  be a transition matrixon a finite set 2 . Assume that A has no zero columns or rows.For z , / E 2 , write i<j if the transition from / to i is possibleFINITE SUMS OF COMMUTATORS IN C*-ALGEBRAS 133(cf. [2]). We call i and 7 equivalent if f < / < / . Let F be theset of maximal states : F.= {; E 2 | V / € 2 i < / ==^ f < i}.F is an union of equivalence classes and every element of 2 ismajorized by an element of F.Assume that the restriction A^ of A to each equivalenceclasse 7 of F is not a permutation matrix. Then ©^ is definedin [2], [3] as the C*-algebra generated by any system (S,),e2 °fnon zero partial isometries with pairwise orthogonal ranges satisfyings*s,= ^ AO\/)S,S; ae2)./e=2We claim that each hermitian element of ©^ ls ̂ e sum °^ ten se^"adjoint commutators.Put A' = AS_F and A" = Ap.As ©A" ls a finite direct sum of ©g with B irreducible, eachhermitian element of Q^" is the sum of five self-adjoint commutatorsby corollary 2.2 and theorem 2.14 of [3]. But it is easy to see thatthere exists an exact sequenceo —> o^ ® 3C —> e^ —> e^ —> oand the result follows from lemma 2.4 and theorem 1.1.3. Simple A.F-algebras.In this section, we shall prove the following result :THEOREM 3.1. — Let A be a simple approximately finite dimen-sional C*-algebra with unit. Then, each element of AQ is the sumof seven self-adjoint commutators.The proof is based on the following technical lemmas :LEMMA 3.2. - Let A be a C*-algebra and x = ^ * G A . Letp, q, r be orthogonal projections in A with p + q + r = 1. Then,there exists u € A, || u \\ < 2 v/T|| x ||1/2 , such thatx — pxp — qxq — rxr = [u, u*}.134 T.FACKProof. - Putu = p - r - -^ (pxq - qxp) - , (pxr - rxp) - - (qxr - rxq).We have x — pxp — qxq — rxr = [u, u*] by direct calculation.Moreover, ||.x'||<2 implies | |^| |<4. The lemma follows, nLEMMA 3.3. - Let A be a C*-algebra and x = J C * G A . Letp , q , r be orthogonal projections in A mth p + q + r = 1 andP ^ Q ^ r . Then, there exists u ^ A , \\u\\ ̂  3 IMI172 and y E Asuch thatx = [u, u*] + ypyp = qyq =. 0\\ryr\\<3\\x\\.Proof. - Let v and w be partial isometries such that vv* = p ,i;*t; < ^7 , vw* = q , w*w < r. Putu = ^/(pxp)^.v + v*\/(px^)_ -^^/{qxq + u*xi;)+w+ v^*^/(qxq + i;*xi;)_and ^ = j c — [ ^ , ^ * ] . We have ||^|| < 3 H^H 1 7 2 , PYP = qyq = 0and ||r^r|| < 3 ||.v|| by direct calculation, aLEMMA 3.4. - Let A be a C*-algebra and x = x * E A . Letp, q, r be orthogonal projections in A with p + q + r == 1 andp ^ q ^ r . Then, there exist u,vCA\ \\u\\ < 3 \\x\\112,H v l l <13||JC||1/2 ^c/z that x - [ u , u * ] - [ v , v * ] e r A r and\\x-[u,u*]-[v,v*]\\<3\\x\\.Proof. — By lemma 3.3, we have x = [u, u*} + y withIMI Olljcll172, p y p = q y q = Q and ||ryr|| < 3 ||jc||. Wededuce \\y\\ < 19 ||.x:||, and the result follows from lemma 3.2. aLEMMA 3.5. - Let B be a finite dimensional C*-ulgebra and; cGBo. Then, there exists uCB, \\u\\ < ̂ /T^x^2, such thatx = [u, u*].Proof. — Using the decomposition of B into simple compo-nents, we can assume that B = M^ (C). One may also suppose xis diagonal. The proper values of x are real numbers X ^ , . . . ,\FINITE SUMS OF COMMUTATORS IN C*-ALGEBRAS 135nwith ^ \ = 0. As there exists a permutation r of { 1 , . . . ,n}f=1 ksuch that 0 < ^ \a) < 2 sup | \ \ for k = 1 , . . . , » ,, =: 1 1 < I < MM fcwe can assume that ^ = ^ \e^ and 0< ^ X,<2| |x | |i^i i^i(A: = 1,. . . , n), where (^7)1 <,,/<„ is some system of matrix units.Apply then lemma 1.3. aLEMMA 3.6. — Let A be a simple A.F-algebra with unit.Suppose that A is non isomorphic to M^(C). Then, there existsequences (?„)„>!, OzA>i and (^)n>i of projections such thati )Pi + Qi + r\ = 1i i )P«^^^ ' 'M (">!)iii) rt^ r^ are mutually orthogonal,iv)^-i =?„ +^ (^ >2) .Proof. - It suffices to show that there exists, for each pro-jection p ^ 0, an element q € Ko(A)+ such that 2q <p <3q .Passing to pAp , we may assume that p = 1 . By [5] (lemma A.4.3),Ko(A) is the limit of a system Z^0 -^ Z^ ^-> • • • • havingthe following properties :i) the <^ are strictly positive, i.e. ^ = (ajp with o .̂ > 0,ii) there exist order units u^ E Z''00 such thatu, —> ^ —^ • • • —^ 1.One then may choose q E Ko(A)+ such that 2q < 1 < 3<?. oProof of theorem 3.1. — The case A = M^(C) is trivial, sothat we can assume A ^ M ^ ( C ) . Let x be in Ao. Let (?„)„>!,(^y»)M>i an^ (^^^i 10e sequences of projections as in lemma 3.6.Apply first lemma 3.4 to get x^r^Ar^ \\x^\\ < 3 \\x\\,and M . V G A such that ;c = [u, u*] + [u, v*] + .x:^. As r^ isan order unit in Ko(A)+, any finite trace on r\Ar^ extendsuniquely to a finite trace on A, so that x^ E (^ Ar^)o .Starting from x ^ , we are going to construct sequences (x^\^^,( ^ )n> i» (^)n>i and Wn>i satisfying136 T.FACKa) xn = K > ^1 + ["« > ^1 + [^ . w?] + ^n+1 >^) ^ e ?•„ Ar,, ; v^,w^e (/•„ + ^+1) A(^ +^+i),7) ^G(^A^)o,^ II II ^ 3 1 1 ^ 1 15) ll-H-nll < ————e) ||Mj|<2||x,||1/2 and v^, w^ -^ 0 ( w — ^ o o ) .Suppose ( X t , . . . , x ^ _ i , x ^ ) , ( U i , . . . , M , , _ i ) , ( V i , . . . ,v^_t)and ( W i , . . . , w^_,) constmcted.Put a = —— • As x^ G (/•„ A^),,, we have^ = I [Cp,^]p>iwhere CpGr^Ar^ and the sum being norm convergent. By approxi-mation,we can find a finite dimensional subalgebra B of /•„ Ar,,a n d ^ G B o such that l l j ' l l < 2 H^|| and ||^->'||<a.By lemma 3.5, there exists u^ € /•„ Ar^ ,llr„ll<\/2•||>'||l/2<2||A•Jll/2such that x^ = [u^, u^] + z , where z = x^ - y .Note that z € ((^ + r^,) A(r« + r̂ i)),, .By lemma 3.4, there exist ^ , w,, €(/•„ + r,,̂ ) A(r^ + ̂ ^.i)such that z = [^ , v^} + [w^, w^] + x^^ where x^+i er,,̂  A^+iandll^llOHzll^Oa^||wJI<13||^|l l /2<13a l/2.We havexn = ["» >";?] + K > ̂ 1 + [w« , w^] + ̂ iand hence x„+l e(^+i A^+i)o. Moreover,| |^^| |<31|z||<3a<^1 1^1 1 .w + 1By induction, the existence of four sequences satisfying a), ft), 7), 5)and e) is then proved.PutU= S «„n>lFINITE SUMS OF COMMUTATORS IN C -ALGEBRAS 137^ev = "L V2n^ ^od = S ^+1 .n>\ n>0W^ = 1 : w^ ; W^ = ^ w^i .n>\ n>QThese sums make sense because they involve elements withdisjoint support and norm converging to zero. Moreover, we havex == [u, u^} + [v,v^] + [U, U*] + [V,, , V;J + [V^ , V^]+[W,,,W;J ^[W^,W:J.The proof of theorem 3.1 is complete. DBIBLIOGRAPHY[1] J. CUNTZ, The structure of multiplication and addition in simpleC*-algebra Math. Scand., 40 (1977).[2] J. CUNTZ, A class of C^-algebras and topological Markov chainsII : Reducible Markov chains and the Ext-functor for C*-algebras,Preprint Univ. Heidelberg, n° 57 (March 1980).[3] J. CUNTZ and W. KRIEGER, A class of C*-algebras and topologicalMarkov chains, InventionesMath., 56 (1980), 251-268.[4] J. CUNTZ and G.K. PEDERSEN, Equivalence and traces on C*-algebras,/. Functional Analysis, to appear.[5] E.G. EFFROS, Dimensions and C*-algebras, Preprint UCLA (1980).[6] T. PACK et P. DE LA HARPE, Sommes de commutateurs dans lesalgebres de von Neumann finies continues, Ann. Inst. Fourier,Grenoble, 30,3 (1980), 49-73.[7] G.K. PEDERSEN, C^-algebras and their Automorphism Groups,Academic Press, New-York (1979).[8] G.K. PEDERSEN and N.H. PETERSEN, Ideals in a C*-algebra, Math.Scand., 27(1970), 193-204.Manuscrit re^u Ie 25 mars 1981.Thierry FACK,Laboratoire de Mathematiques FondamentalesUniversite Pierre et Marie Curie4, Place JussieuTour 45-46, 3^ etage75230 Paris Cedex 05.

Finite sums of commutators in $C^*$-algebras

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Finite sums of commutators in $C^*$ -algebras

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Finite sums of commutators in C∗C^*C∗-algebras

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Finite sums of commutators in $C^*$ -algebras