HOW TO SOLVE AN OPERATOR EQUATION

MAI2TIN MATHILU This article summarizes a series of lectures clelivered at the Matherna.tics Departrnent of the University of Leipzig, Cerinany in April 1991, which were te overview teclrniques for solving operator ecluations en C*-algebras connected with methocis developed in a Spanish-Gerrnan research project en "Structure ancí Applications of C*-Algebras oí' Quotients" (SACA) . One of the researchers in this project was Professor Pere Menal until his unexpected death this April . To his mernory this paper shall be dedicated .


Introduction
Solving equations belongs to tlre fundamental tasks of inathematics .Many problems in the sciences load te equations involving numbers, mappings ; ttnd other guantities.In faca, it frequendy occurs that eventually a duestion can be phrased as an "equation", although, at first, it appeared tú be of a rather different nature.To find a solution of an equation generally implies both the existence as well as the unidueness problem.There is no universal procedure for solving ; but the devices invented seem , to be as manifold as the possible questions, asid only allow a rather rough classification such as nurnerical, approximative, algebraic methods etc. Flowever, it is always an irnportant stop to determine the corrrmon features in solving a certain class of exarrrples for the aire of developing a machinery which enables to handle a specified collection of equations at one tiene .
In the present paper; we will be concerned with equations within a non-commutative infinite dimensional setting .To be more specific, they will be of the form where, for each `parameter' a, T«,~1, , ,ten is a linear operator on a C*algebra A (with certain additional properties) and we are looking for elements xj E A solving the equation (1) (or better, this system of equations) .We will firstly collect some examples of questions which can be phrased in an equation such as (1), then describe a general tool to tackle them, and finally indicate solutions which yield answers to the questions listed .As a common feature, the questions in Section 2 lead to equations in a C*-algebra; that is, we are looking for certain elements in a C*-algebra solving the equation, while the conditions typically are formulated in terms of operators defined on the C*-algebra.Needless to say that there are many more instantes which can be settled by the proposed methods .

. Examples
We have selected our examples from the following four classes of operators on C*-algebas: derivations, completely positive operators, centralizing mappings, and generators of dynamical semigroups .

. Derivations .
Let A be a C*-algebra and S a derivation on A, Le. a linear mapping from A into itself satisfying Leibniz' rule S(xy) = x6(y) + b(x)y for all x, y E A .Each derivation S is automatically bounded whence it is meaningful and worthwhile to know under which circumstances 8 is a compact operator, with respect to the norm or a weaker topology.Here, we ask when S is weakly compact, that is, when does S map the unit ball of A into a subset whose closure is compact with respect to the weak topology on A. (This is more closely related to the point of view taken in this paper than the norm compact case, which, however, can be treated similarly .)Specialize to the case A = B(H), the algebra of all bounded linear operators on some Hilbert space H . Since B(H) is the second dual of K(H), the closed ideal of all compact operators on H, and S is continuous with respect to the Q(B(H),K(H)*)-topology, S coincides with (Sj )**, the second adjoint of the restriction 81 of S to K(H) .In the general case we have to replace K(H) by the ideal K(A) of all compact elements in A; and, using appropriate representations, we obtain the following, cf.[ Can & = 0 be solved in K(A)?
Going one stop further we can ask a similar question for the product 6162 of two derivations 51, 62 on A (which ; in general, is no longer a derivation) : when is 5162 (weakly) compact?This question should be related to t11e Dunford-Pettis property of a commutative C*-algebra which implies that T1T2 is a compact operator whenever T1, T 2 are weakly compact on A. By similar arguments as above, it can be formulated in tercos of operator equations as follows .
Questions of this kind are studied in [25] and [27] .

.. Completely positive operators .
Recall that a linear mapping T on a C*-algebra A is said to be com- pletely bounded if the nornls JIT,,11 of the canonical extensions T,, of T to the matrix algebras Al,,(A) over A are all bounded by some real number, and T is completely positivo if all T,, are positivo operators on M (A) .The prototypes of completely bounded operators are t11e elemeratary operators given concretely as mappings of the form 5 :x~-+ xbj with x E A, a1, . . ., a,l , b1, . . .; b,L E 1Vl(A), where A11(A) denotes t11e multiplier algebra, of A. This is justified by the representation theorem for completely bounded operators and the fact that certain completely bounded operators can be approximated by elementary operators, cf.[12] This question has emerged to be not only an example, but of fundamental significante for our approach, cf.[28] .

Centralizing mappings .
Let R be a ring.An additive mapping F : R -> R is centralizing if, for every .xE R, we have [x, F(x)] = xF(x) -F(x)x E Z(R), the center of R. In many cases ; the existente of certain centralizing mappings yie1ds commutativity criteria for R.For example, if R is a prime ring; then R, is commutative if there is a non-zero centralizing derivation on R [38, Theorem 2], see also [30], or if there is a non-identical centralizing automorphism on R [31,Theorem] .In the context of operator algebas, there are analogues of there results as follows .
Proposition 3.There is no non-zero cent7-alizing derivation on a C*-algebra .This seems to be a folklore extension of Singer's classical result that there are no non-zero derivations on commutative C*-algebas.In fact, if b is a centralizing derivation on a C*-algebra A, it easily follows that bA C_ Z(A) .Hence, the restriction 61 of S to Z(A) vanishes so that 62 = 0.The identity Here and in the sequel, we will denote by L, the left multiplication x f--> ax and by R.a the right multiplication x H xa.
We will now reformulate both the assumption as well as the conclusion in terms of operator equations .This will enable us to obtain an extension of Bresar's result to arbitrary C*-algebras in Section 4.
Observe at first that every centralizing additive mapping F on a C*algebra A is in fact commuting, Le. (2) 6F(y) -6yF = 0 for all y E A.
Conversely, if c E CB(A) satisfies (3), then ~= F-L, defines an additive mapping from A into CB(A) .As a result we arrive at the following question.
(1 .4)Suppose that F satisfies (2) for all y E A .Is there an element c E CB(A) for a `suitable' C*-algebra B containing A satisfying (3) for all y E A?
Note that (3) precisely is a system of operator equations of the forro (1) parametrized by all elements in A.

Generators of dynamical semigroups.
Let A be a unital C*-algebra .A bounded hermitian-preserving linear operator L : A -> A with L(1) = 0 is called completely dissipative if, for allnEN, These operators are the generators of norm-continuous one-parameter semigroups (Tt)tER+ of unital completely positive operators T,, en A; which describe the irreversible dynamics of open quantum systems, or, equivalently, serve as transition operators of non-commutative Markov processes .In many concrete situations, they are built from two prototypes : the completely positive operators and the hermitian-preserving generalized inner derivations Sk,k.= R.k + Lk. .The converse question, when a given completely dissipative operator L can be decomposed into L,(x*x) ?x *L,,(x) + L,,(x * ) x (x E M,, (A» .
A more general question would be which a in A" solve the equation with 0 completely positive from A into some possibly larger C*-algebra B and k E B was first studied by Corini, Kossakowski and Sudarshan [18] and Lindblad [21] ; and related to cohomological properties of A in [22] and [11] .If A C_ B(H), then a decomposition (4) of L always exists with OA C A" and k E A" .In general ; this decomposition will not be unique .The uniqueness problem can be reformulated in terms of an operator equation as follows .Suppose that L = 01 + 4,,k, = 02 + 8kz,kz

. Devices
All the above equations (1 .1)through (1.5) can be subsumed under the general form (1) .To motivate our tools for solving them ;.let us furthermore consider a special case of (1 .3) .Let A = B(H) and b E A be giverr .
In our particular situation, the answer is quickly reaclied.If L1Ia,b = 0, then axb~= 0 for all x E A asid 1 E H .If b = 0, obviously all a E A are solutions .If b 7~0; pick ~E II with b~:7~0 and note that b~is cyclic for A, i.e.Ab~= H, and thus a = 0. Clearly, this method only works in the presence of a Hilbert space on which A acts'transitively enotrgh', e.g. if A is irreducible .The algebraic method presented now works without underlying space.
It is convenient to rephrase (1 .3)using the following concept .For every C*-algebra A we let ú(A) be the algebra of all elernentary operators on A .We define a surjective algebra llorrrornorphism (7) 0 is injective if and onlg if A is prime.
Since primitive C*-algebra5 are prime, it is tempting te use representation theory in order to approach thc general case frorn the special one.However, as it emerged, there may be problems in putting the 'local' information together to obtain a, 'global' picture .It, seems adva,rrta,geous to view the prime C*-algebas as the building blocks, which results in regarding a C*-algebra as a serniprirne algebra rather tiran a se-misim,ple one .In fact, similar tecliniques arrd results as those described below are available in the setting of serniprirne rings .
The ideal structure of a prime algebra is distinguislied by the fact that ever y non-zero ideal is essential, Le. intersects cae]-) other nonzero ideal non-trivially.This allows to "nieve a.round frorn one place to another''' within the C*-algebra without loss of information .For an arbitrary C*-algebra.A we therefore denote by 1, and 1, the collections of all essential and all closed essential ideals of A, respectively .Note that there are directed dowrrwards by inclusion, Le. 1 7 12 E 1, irnplies that 11 nI2 E1, .
For every serniprirne ring R,, the rnulttiplier ring AI(R) is defined by its universal property that Id is an essential ideal in M(R) and there is a unique extension p of the inclusion p : R -4 M(R) which makes the following diagram commutative ; whenever R is an ideal in another ring S, R --'-> M(R) in other words, M(R) is the (abstract) idealizer of R. Usually, AII(R) is constructed via double centralizers of R.Moreover, p is injective if and only if R is essential in S .Now, if I, J E 1, and J C_ I, then J will be an essential ideal in M(I) whence, from the above, there is a unique injective *-homomorphism p» : M(I) -> M(J) making the following We may describe pjj as "restricting the double centralizers" .By means of this, we obtain a directed system {M(I) ; prj, J C I} of C*-algebas and inclusions, and its algebraic direct limit alglim M(I) along 1., will be denoted by Qb(A) and called the bounded symmetric algebra of quo- tients of A. This is a pre-C*-algebra with completion Qb(A)-= lim M(I) denoted henceforth by MI,,(A) and called the local multiplier algebra of A.
For each I E 1e let P(I) denote the Pedersen ideal of I [37, 5 .6] .Using the fact that P(I) is *-invariant, belongs to 1, and that P(I)P(J) = P(I) f1 P(J) for all I, .I E 1, we define Q, (A) = alglim M(P(I)) along 1e and observe that this definition leads to the symmetric algebra of quotients of A as defined (slightly differently) in ring theory.lt follows that Qb(A) embeds as a *-subalgebra into Q.,(A) and is in fact the bounded part of Q,(A) [2, Theorem 1 .3] .A stronger relation between Qb(A) and Q, (A) proved in [3, Theorem 2] is that Q, (A) is the central localization of Qb(A) .
Remarks .The construction of M1o,(A) was first performed by Pedersen [361 and Elliott [16] under the name of essential multipliers .They used it to study operator equations of the form a = AJ.,ti., u, E Aho,(A) unitary, that is, to obtain innerness of derivations 8 and *-autornorphisras a in A4h o,(A) .In particular, Pedersen proved that (8) always has a solution if A is separable [36, Proposition 2] .At about the sarne tirne, Kharchenko introduced the syrmnetric ring of quotients for semiprime rings and used it in particular in Galois theory [19] ; [20] .This theme was further pursued by Passrnan [34], [35], Montgoinery [33], and others .It is to be seen in a long tradition going back to tlre 30's in investigating general rings of quotients, cf.[40] .The basic idea -to enlarge a given 'domain' by additional 'nunnbers' (='fractions', 'quotients') in order to be able to solve more equationsalso serves as the motiva.tionfor our approa,ch to operator equations .
We will now compile come of the basic properties of Aho,(A) .Proposition 6 .Let, A be a C*-algebra uwith local rnultiplier algebra Ai1,,:(A) .
(i) A is commutative if and only if A1h,,(A) i.s comrnutative .
(ii) A is prime if and only if M1,(A) has trivial, center.
(iii) For each I E Z, and each unitization B of A we have (iv) Let, tl be the primitive spectrum of A. If A is discrete, then H,,,,(A) = A4-(A) .(v) If A is an AW*-alggebra,, then Aho,(A) = A .rom (7) and (ii) in the above proposition we see that the kernel of 0 is closely related to the center Z = Z(A1h,,(A)) of 1VIlo,(A) .It is therefore important to analyse its structure .The following was proved in [5, Theorem 1 and Corollary 1] and can be viewed as a local version of the well-known Dauns-Hofinann theorem identifying the center Z(111(A)) of NI (A) with the algebra, C(3Á) of all continuous complex-valued functions on the Stone-Cech compactification f3fl of fl .Proposition 7.For every C*-algebra A, the center Z of A1h, (A) is an AW*-algebra and can be i,dentified with C(li E n1 0Í), where the inverse limit (in the category of compact spaces) is taken over all dense open subsets Í of Á .
The key to this result is by observing that Z = Cb, where Cb = alglim Z(M(I)), I E Z,, is the center of Qb(A) and called the bounded extended centroid of A. This one takes the role of the extended centroid C = alglim Z(M(P(I))), I E Ze , being of fundamental importante in This result can be considerably strengthened using appropriate metric structures .Let f.~(A) be endowed with the cb-norm, Le. 11SUb = is an isometry .
sup IIS7LII for all S E ú(A) .Let cAI(A) ®z cM(A) op be endowed with 7L the central Haagerup tensor norm II-IIZh defined by where the infimum is taken over all representations of u in cM(A) ®z c~A4(A)op .Then we have Theorem 10.
This last result was recently obtained in [10, Theorem 2.4], see also [39], for von Neumann algebras acting on separable Hilbert spaces using a number of rlon-trivial results on von Neumann subfactors as well as direct integral theory.
In this final section we will outline answers to the questions raised in Section 2 exploiting the tools described in the previous section .As an immediate consequence of Theoreln 9 we obtain the following answer to (1 .3) .Theorem 12 .Let a = (al . . . ., a,L), b = (b,, . ..b,t ) E M(A)7L be such that {b,, . . ., b, L } is Z-independent.If E"_1 Mai,bj = 0, then a=0.
Now the strategy to describe completely positivo elementary operators is as follows, cf.[7] .If S = E"'1 Maj,bj is completely positive, we may without loss of generality assume that both {a1 ; . . ., a,,} and {b1, . . ., b,L} are Z-independent .Then  From the complete positivity of S we then conclude that Aj > 0 for all 1 < j < n and hence, letting cj = A ~/Zbj : obtain the following answer to the question raised in 2 .2.For simplicity, we stick to the prime case in answering the questions of Sections 2.1 and 2.4 .If A is a prime C*-algebra, then, by Theorem 12, Ra + Lb = 0 for some a, b E 111 (A) if and only if a = -b E Z(M(A)) = C1.Suppose that S is a weakly compact derivation on A .If S = 0; it clearly can be implerrlented by a compact element .If S 7~0, then 6A C_ K(A) (Proposition 1) implies that K(A) :y~{0} and thus A can be faithfully represented as an irreducible algebra, on some Hilbert space H such that K(A) becomes K(H) .By the argumenta used in 2 .1,,ve see that S = &, for some a E B(H) and 6Q.= ba = 0 on the Calkin algebra C(H) = 13(H)/K(H) .Since C(H) is prime, Z(C(H)) = Cl wherefore d = Al, equivalently, a + Al E K(A) .Consequently, we have the following .Proposition 14.Let b be a derivation on a prime C*-algebra A .Then S is weakly compact if and only if S = Sa for some a E K(A).
In fact, this result takes over verbatiln to the case of a general C*algebra, which was first proved by Akemann and Wright [1, Theorem 3.3] using representation theory.As a result, a derivation S is weakly compact if and only if the answer is "yes" in (1.1) .
In a similar vein, 5162 is weakly compact if and only if 51 or 62 is weakly compact, provided A is prime.Hence, 6162 is weakly compact if and only if at least one of the .xiin.(1 .2) can be taken from K(A).The formulation of the answer in the general case is somewhat more complicated, and we refer the reader for this (as well as for the norm compact case) to [25].Note that 6x1 5~12 = Mxix2,1 -Al -I,X2 -Ma2,Xi - MI,x2xi   and therefore (1 .2) is closely related to (1 .3)and a description of (weakly) compact elementary operators which was obtained in [24,Part II] .
Specializing the above observation to the case b = a* we obtain that 6a,," = 0 if and only if a = -a* E i Rl whenever A is prime.Using a slight elaboration of this we obtain the answer to (1 .5) .Theorem 15.Let A be a unital C*-algebra and P a proper closed prime ideal of A. Let L : A -> A be linear.Under the hypothesis ,c%A C_ P, each two decompositions of L of the form L = ~b + 6k,k, with ~b : A A completely positive and k E A only differ by an addition by 5,'-, c E .P. Corollary 16.Let A be a unital infinite dimensional prime C*- algebra and L : A -> A. Then there is at most one decomposition L = 0 + 6k k-with k E A a,nd : A -> A completely positive and compact.
These results are proved in [17] .Corollary 16 was first observed by Davies [13, Theorem 2] in the case A = B(H) .
We finally turn our attention to the structure of centralizing mappings of C*-algebras and the questions raised in Section 2 .3 .Unlike in the other examples, there seems to be no direct connection with equations involving elementary operators such as (1 .3) .The following lernma indeed is the key observation which enables us to solve equation (3) .Lemma 17. lf F is an arbitrary mapping on a ring R such that 6F(,) -5,.F maps R into some ideal J of R, then, for all x, y, u, v E R, we have (12) -A4h1(y)(-),ó .(1)-lVlóu(y),6F( )("»-R C J.
This result was obtained in [8, Lemma 2.2] for commuting additive mappings and J = {0} .Although we are dealing here with C*-algebras only, we give the proof in full generality as an illustration of the techniques and with a hope for future applications .
As a consequence, every mapping satisfying (2) has the property that ( 17 ) 115,(b)(x),ó .(v)-1V16 y (T),~F (u~(~>> = 0 for all x, y, u, v E A. An elaboration of the solution to (1 .3), the details of which are given in [6], then yields a family {ey 1 x, y E A} of elements in C and a family {ez 1 x, y E A} of projections in Cb such that (18) ey 6r(,) (X) -cyy 6. y (x) = 0 for all x, y E A.
It is then the self-injectivity of C which allows to find c E C with cey _ cy, which finally has the property that 6"(y)(x)eby (x) = 0 for all x, y E A, that is, which solves (3) .An additional argument is then needed to show that e can be found in C( that is, we obtain a solution to (1 .4) in 'A .We sunnnarize this in the following statement .
Theorem 18. [6, Theorem 3.2] Let.F : A --> A be a centralizing additive mapping on a C*-algebra A .Then there are e E Z and ara additive mapping ~: A -> Z such that F = L,, + (.Note that, by Proposition 6 (v), this is an extension of Bresar's result (Proposition 5) .Under a natural condition, both c and ~can be chosen uniquely.

. Conclusion
We hope that the-results described abóve may give some evidente that the local multiplier álgebra can serve as a `universé', in which operator equations on C*-algebras, at least those of the forro (1),``can~be solved' by a unified method.

23 ,
Theorem 2 .7] .Proposition 1.A derivation S on a C*-algebra A is 2ueakly compact if and only if b**A** C K(A) .Again, K(A) is b-invariant and thus b induces a derivation b en the generalized Calkin algebra A/K(A) .Corollary 2 .If b is weakly compact, then b = 0 .Suppose b were inner, Le .b = b., where 6,(x) = xaax, and the element a belonged to K(A) .Then, b is weakly compact by [41, Theorem 3 .1] .On the other hand, b** is always inner by Sakai's theorem .Therefore, the original question of weak compactness of b leads to the following operator equation .(1 .1) 2 b(x)y5(x) = 62(Zyx) -xb2 (yx) -b2(xy)x + xb2(y)x (x, y E A) therefore yields Aló(x),a(x) = 0 for all x E A, whence b = 0.The case of automorphisms requires some more work and was first studied by Miers.or equivalently, or equivalently, Proposition 4. [32, Theorem 5] Let ce be a centralizir~g *automorphism on a von Neumann algebra A .There is a central projection e E A such that a(e) = e, aJA, = idA, and A(1-e) is commutative .Whether this result remains true for arbitrary (not necessarily *preserving) automorphisms was answered only recently by Bresar, who also obtained a general structure theorem for centralizing mappings en von Neumann algebras as follows .Proposition 5. [8, Theorem 2.1] Let F be a centralizing additive mapping on a von Neumann algebra A .Then there exist an element c E Z(A) and an additive mapping ( : A ---> Z(A) such that F = L, + ( . [x, F(x)] = 0 for all x E A [9, Proposition 3.1] .Replacing x by x + y therefore gives [x, F(y)] + [y, F(x)] = 0 (x, y E A) (A) ® A11(A)"1> --, ú(A), B(a (9 b) = 111,,b where A11(A) ® AJ(A)" denotes the algebraic tensor product of M(A) with its opposite algebra .The problern now is to determine thc kernel of 0. The following was preved in [24, Part 1; Corollary 4.4] .
ring theory.In analogy to the central closure AC we define the bounded central closure °A by °A = ACb = AZ.The nicest C*-algebras in this framework are those which are boundedly centrally closed, that is 'A = A .They can be characterized as follows .Proposition 8 .A is boundedly centrally closed if and only if 11 is extremally disconnected .The fact that every von Neumann algebra is boundedly centrally closed (which follows in particular from Proposition 7 (v)) allows to incorporate the results on von Neumann algebras in our approach, and the fact that M¿o,(A) is boundedly centrally closed [5, Theorem 2] yields an important stability property .It can be shown that every C*-subalgebra B of M,o,(A) containing both A and Cb has center Z(B) equal to Z [7], and hence may be regarded as a Z-bimodule in a natural way .Applying this to 'M(A), the bounded central closure of M(A), we obtain from (6) an induced homomorphism Bz : w(A) ®z cM(A)on -£2(°A), Bz(a ®z b) = Ma,b, where the tensor product is taken in the category of bimodules .Using the fact that A is boundedly centrally closed if and only if M(A) is, we can now formulate the fundamental result yielding solutions to the operator equations listed in Section 2 .Theorem 9. [7] For every C*-algebra A, we have that ker0 = {u E Al(A) ® ltl(A)" b uz = 0}, where uz is the canonical irnage ofu in °M(A) ®z cM(A)ar.Therefore, if A is boundedly centrally closed, then Bz is injective .
e ., 'A11(A) Oh IM(A)°P inherits the operator space structure of °M(A) Oh ~AI(A) .
12 implies the existente of a self-adjoint matrix A = (A k i) E -A/In(Z) such that (10) S = Aki A/IbZ bj /c,j=1 Since Z is an AW*-algebra by Proposition 7, A can be diagonalized by [14, Corollary 3.3], Le. there is a ilnitary matrix U E M,t (Z) such that U*AU is diagonal with diagonal entries ~~, . . ., A,, .Hence, by putting b = b U* E 'A11(A)' we can write S as S Aj AMI-, ,

Theorem 13 .
[7] An elementary operator S o-n a C*-algebra A is completely positive if and only if there are cl, . . ., en E 'M(A) such that ~n S -Lrj=1 nlc i c i .For prime C*-algebras, this was obtained in [24, Part I, Theorem 4.101 .