Huruya, Tadasi

English

Huruya Tadasi

University of Szeged

Acta Sci. Math, SO (1986), 183—189 Decompositions of completely bounded maps TADASI HURUYA 1. Introduction. Let cp: A—B be a bounded linear map between C"'-alge-bras, and let q>®id„: A®Mn—B<&M„ be the associated maps («=1,2 , . . . ) , where M„ is the full matrix algebra of order n. The map <p is said to be completely positive if each q>®id„ is positive, and completely bounded if ||<p||cb=sup [|p<g>idJ<oo; n in this case ||<p||Cb is called the completely bounded norm of (p. It is known that a completely positive map q> is completely bounded and ||<p||cb=||<p||. A linear map (p: A—B between C*-algebras is said to have a positive (resp. completely positive) decomposition if q> can be written as a linear combination of positive (resp. com-» pletely positive) linear maps. A C*-algebra A is injective if and only if for any C*-algebra B such that B^A, there exists a projection of B onto A of norm one [ 2 ; Theorem 5 . 3 ] . W I T T S T O C K [ 1 5 ] proved that every completely bounded map of a unital C*-algebra into an injec-tive C*-algebra has a completely positive decomposition (see, also; [8]). In [3] we proved, as a limited converse of Wittstock's theorem, that given a separable C*-algebra B, every bounded linear map of any C*-algebra into B has a positive decom-position if and only if B is finite-dimensional, namely, injective. In this paper, we show that given a separable unital C*-algebra B, every completely bounded map of any unital C*-algebra into B has a completely positive decomposition if and only if B is finite-dimensional, namely, injective. We also prove that if A and B are separable, infinite-dimensional, unital C*-algebras and A contains a. self-adjoint ele-ment such that the set of limit points of its spectrum is infinite, then the span of positive linear maps of A into B is nowhere dense in the Banach space of all bounded linear maps of A into B. Throughout this paper, it is assumed that all C*-algebras are unital. If S is a compact Hausdorff space, we denote by C(S) the C*-algebra of continuous func-tions on S. We mean by <xN the one-point compactification of the set N of positive integers with the point at infinity. .. Received May 20, 1983, 184 Tadasi Huruya We remark that some of the results of this paper were announced in [12]. The author would like to thank Professor J. Tomiyama for his interest and encouragement. 2. Bounded linear maps between commutative C*-algebras. For each n in N let Xn={Xn,i> •••i*»,m. •••>*»,»} and Y„={y„tl, —,y„,mi ...}. Denote by X the one-point compactification of the topological sum of the sequence {X„}"=1 of copies aN; denote by Y the one-point compactification of the sequence {F„}"=1 of copies N with the point j«, at infinity. The space Y is homeomorphic to a N. From now on, we use X, Y, X„, xn-m, x„„, ynm and y„ in the above situation. We consider linear maps of C(X) into C(Y). Proofs of Lemmas 2 and 3 are based on an idea due to KAPLAN [ 4 ] and Tsui [13]. We first recall Tsui's example [13; 1.3.4, Example II]. Lemma 1. Let C(X)-*C(Y) be the self-adjoint linear map defined by $(f)(y„,m) =f(x„,J —/(*„,m+i) and <P(f)(y„)=0. Then $ has no positive decom-position. Lemma 2. For any positive integer k, there exists a self-adjoint linear map ¥ k : C(X)—C(F) with 11̂ 11 = 1 satisfying the following properties. (1) Tk has a positive decomposition. (2) If Wk is decomposed as the difference of two positive linear maps 3 / + , then || f + | | s i t /2 and ||y-||&jfc/2. Proof . We define the map Wk: C(X)-*C(Y) by J = ( l /2)( / fe ,m)- / (xB > m + 1)) if « S f c , ¥k(f)(y*,m) = 0 if V k ( f ) ( y J = 0. It is easy to check that ¥ k ( f ) is continuous on Y and ||!Pt|| = 1. (1) We define k positive linear maps I¡/¡: C(X)—C(Y) by ^ ( / ) O „ , J = 0 if n+m^i, Mf)(yn,m) = (l/2)/(* i>n+m-i) if n+m > i, * « ( / ) 6 0 = d /2 )/(*,,»)• Since Mf)(yi .m)=(V2)f(xt,m)> we have and hence <Pk has a positive decomposition. (2) For a linear map ij/: C(X)-*C(Y), let ^(B,m) denote the linear functional on C(X~) defined by ^,m)(f)=ip(f)(ytt,J. If nsk', then (l/2)6(xn ,m)Si' (+m) and Decompositions of completely bounded maps 185 (l/2)<5 (*„_ m+1) ̂  m), where S(x) denotes the point measure at x. Let en be the characteristic function of the subset Xn of X. Then 1/2 = (l/2)d(x„,m)(en) == = ^+(«n)(Jn,m). 1/2 = (l/2)5(x„im+1)(e„) ^ m)(e„) = J. If j=+, —, we have 1/2 S lim = m Therefore, f J ' ( l ) 0 0 s 1 y'(OCK-) s fc/2, B=1 so that ||!P+||sfc/2 and | |!P-||s*/2. Lemma 3. Ŵ z'/Zi $ as m Lemma 1, if <p is a bounded linear map of C(X) into C(Y) satisfying || <p—<1/2, then (p has no positive decomposition. Proof . Suppose that <p has a positive decomposition. Then the self-adjoint part T of (p has a positive decomposition T = T + — . For a linear map ip: C(X)-*-C(Y), as in Lemma 2, we define the linear functional on C(X) by ^ ( n , m ) ( / )= =rl/(f)(yn m). Since X is countable, x(„>m) can be written as *{„,»> = 2 F(x)d(x), 0 =s /?>(*)€ R, j = +, *€X where ¿(x) denotes the point measure at x. We then have 1 / 2 > \\<p-n S | t—<P| | S ¡ | T ( n , r a ) - ^ ( „ , m ) [ l = = II2 P+(x)S(x)~ 2 p-(x)5(x)-5(xn,m)+S(x„,m+1)\\ s xZX Hence (l/2)5(xn in)ST{+ m). Let e„ be the characteristic function of the subset X„ of X. Then 1/2 = (l/2)5(^>m)(en) S r(+n>m)(en) = T + ( O G O , so that l /2s l im t+(e„)(y„ m)=T+(en)(yJ). Therefore TO n=1 for any positive integer A:. This implies the unboundedness of t + . 186 Tadasi Huruya 3. The main results. We recall that X and Y are the one-point compactifica-tions of the topological sums of sequences of copies aN and N, respectively. In order to extend maps obtained in Section 2 to non-commutative C'-algebras, we construct completely positive maps with range algebras C(X), C(Y). Lemma 4. If a separable C*-algebra A contains a self-adjoint element a such that the set of limit points of the spectrum of a is infinite, then there exist united com-pletely positive maps nA: A—C(X) and v^: C(X)—A such that nAovA is the identity map on C(X). Proof . Let S denote the spectrum of a. Since S is a compact subset of real numbers, choose a point sm and a sequence { J„}™=1 of limit points of S such that 3 | j „—i 1 I + 1 | < | i „ - j I I | for all n. For each n take a sequence {•yn i}|'l1 of distinct points of S such that 3|J„—^.¡l-«^»— i„| for all i and lim.sni=.yn. Put S„= OO M = K,1 , . . . ,sn ,m , . . . , and ^={ioo}U(U S„). If s£S, we choose a state gs on n=l A such that gs(f)=f(s) for a l l / i n C(S) because C(S) is the C*-subalgebra gen-erated by a and 1. We then define the positive linear map n of A into the C*-algebra of all bounded functions on S by n(b)(s)=gs(b) for s in S and b in A. Since A is separable, so is the C*-subalgebra C*(n(A)) generated by n(A). There exists a compact metric space T with metric d such that C(T)=C*(n(A)). Then S is can-onically regarded as a subset of T. For each n let t„ be a limit point of the subset S„ of T and choose a subsequence of {in,i}°li such that lim sn>i=t„. If / e C ( 5 ) , then n(f)ifn) = lim 7 t ( / ) O J = lim /(i„, ;) =/(s„). Hence tny±tm if n ^ m . We again choose a point t„ and a subsequence {^(n)}"=1 of {?„}"=1 such that 3d(U, tHn+1))*=d(t„, thW). For each n take a subsequence {/A(„),i}™1 of {?*„). J ^ such that 3d(tMn), th(n)J)-=d(t„, th(n)). Put T„={tHn)>1, tH„hm, ..., th(n)} and oo -?={*<»}U(IJ T„). By its construction, X is canonically regarded as the space X. Let q>: X=XSQS be defined by <P(tKn),d = tKn),i> Vi'hw) = s»(n), <p(U) = So=: For / in C(S), Tt(f)(thW)=f(shm) and n(f)(t„)=f(sj). The map q> is one-to-one and continuous. Then there exists, by [1; Theorem 3.11], a unital positive linear map vA: C(X)-C(S)QA such that vA(f)ocp=f for a l l / i n C ( X ) . Define the unital positive linear map nA: A — C(X) by nA(b)=n(b)\x, the restriction to X=X of n{b). Then nA(f)=fo<p for all / i n C(S). Hence nAovA, Decompositions of completely bounded maps 187 is the identity map on C(X). Both nA and vA are completely positive [10; Chapter IV, Corollary 3.5]. Lemma 5. If B is a separable, infinite-dimensional C*-algebra, then there exist tmital completely positive maps nB: B—C(Y) and vB: C(Y)—B such that nBovB is the identity map on C(Y). Proof . There exists a self-adjoint element a in B with infinite spectrum S [7]. Denote by C*(a, 1) the C*-subalgebra generated by a and 1. Then C(S)=C*(a, 1). Since S is a compact metrizable space, we choose a point s„ and a sequence {i„}™=1 of distinct points in S with lim s„=s„ and {an}"=1 of C(5) such that a„(i„)=l, 0 S a „ S l and apaq=0 for p^q. Put §={si, ..., sn,..., s„}. If s£§, we take a stategs on B such that g s ( / ) = =f(s) for all / in C(S). We define the unital positive linear map n of B into the C*-algebra of all bounded functions on S by n(b)(s)=gs(b). Since B is separable, so is the C*-subalgebra C*(it(B)) generated by n(B). There then exists a compact metrizable space T such that C(T)=C*(n(B)). Then S is canonically regarded as a subset of T. We choose a point in T and a subsequence of {in}"=1 such that lim sh(n)=sh(m). Put sh(n),..., .?,,(„)}. Then Y is canonically regarded as the space Y because ¥ is homeomorphic to a N. We define the unital positive linear map nB: B—C(Y) by itB(b)=n(b)\Y, the restriction to Y=Y of n(b). We also define the unital positive linear map vB: C(T) — +C(S)QB by vB(b) = 2 №(«Mn)) -h(SA(»))1 aKn> + b(SA(~)) 1 • n = 1 ; Then nBovB is the identity map on C(Y) and both nB and vB are completely positive [10; Chapter IV, Corollary 3.5]. Theorem 6. Let A andB be separable, infinite-dimensional C*-algebras. Assume that A contains a self-adjoint element a such that the set of limit points of the spectrum of a is infinite. Then (1) There exists a completely bounded map $: A—B such that each bounded linear map (p: A—B with \\(p — $\\ <1/2 has no positive decomposition. (2) There exists a self-adjoint linear map A-+B having a completely positive decomposition such that for any positive decomposition ¥= — we have || W+1| > >||^||cb and USMIHIPIL. Proof . We use maps <P, W4, nA, vA, nB and vB constructed in Lemmas 1, 2, 4 and 5. 188 Tadasi Huruya (1) We put $=vBo$onA. By [6; Lemma 1], $ is completely bounded. Then \\7iBo(povA-0l = lnBoq> ovA—nBo$ ovAl S ¡iP-^ll < 1/2. By Lemma 3; nBo<povA has no positive decomposition. If q> has a positive deicom-position, so does nBoq>ovA. This is a contradiction. (2) We put ¥=\boW4otia. By Lemma 2 and [10; Chapter IV, Corollary 3:5], ^ is a self-adjoint linear map of A into B having a completely positive decomposi-tion and ' 1 = ll^ll = H « B o S > o v J ^ | l S » J e b = K o ! P 4 o W J e b s n ^ l U H I ^ I I , where the last equality follows from [6; Lemma 1]. Hence | | ? | | c b =l . If T has a positive decomposition W=W+—W~, then W4 has a positive decomposition f 4 = =TtBo*F+ovA—TiBoip~ovA. By Lemma 2 we have ¡ P + l s ^ o p + o v j = 4/2 ¡i'lcbj and similarly, R e m a r k 7. Let A1 and B1 be C*-algebras. Suppose that there exist unital completely positive maps n^. Al-*C(X), v1:C(X)—A1, C(Y), v2: C(Y)-+Bi such that TijOVi and 7t2ov2 are the identity maps on C(X) and C(Y), respectively. If we replace A and B by A1 and Blt Theorem 6 remains true from the same argument in its proof (cf. [9; Theorem 2.6]). We recall that the set of self-adjoint elements of an injective C*-algebra is con-ditionally complete [11; Theorem 7.1]. Hence a separable C*-algebra A is injective |f and only if A is finite-dimensional. Coro l l a ry 8. Let B be a separable C*-algebra. The following statements are equivalent; (1) B is injective; (2) Every completely bounded map of any C*-algebra into B has a completely positive decomposition; (3) Every linear map (p having a completely positive decomposition of any C*-algebra into B has a completely positive decomposition such that <p=(pi—q>2+i (<p3—(p4) with M^McbO'=l, -,4). Proof . By [15; Satz 4.5] we have (1)=>(2) and (1)=>(3). Combining the above remark about injective, separable C*-algebras with Theorem 6, we see that (2)=>(1) and (3)=>(1). In the category of partially ordered Banach spaces, WICKSTEAD [ 1 4 , Theo-rem 3.15] obtained a result similar to Corollary 8. Decompositions of completely bounded maps 189 Addition. After this paper was written, the author discovered an example of a non-injective, non-separable C*-algebra B such that every completely bounded map of any C*-algebra into B has a completely positive decomposition [16]. References [1] W. G. BADE, The Banach space C(S), Aarhus University Lecture Notes Series, No. 26, 1971. [ 2 ] E . G . EFFROS and E . C . LANCE, Tensor products of operator algebras, Adv. in Math., 2 5 (1977) , 1 — 3 4 . [3] T . HURUYA and J. TOMIYAMA, Completely bounded maps of C*-algebras, J. Operator Theory, to appear. [4] S. KAPLAN, An example in the space of bounded operators from C(X) to C(Y), Proc. Amer. Math. Soc., 38 (1973), 595—597. [5] H. E . LACEY, The isometric theory of classical Banach spaces, Springer-Verlag (Berlin—Heidel-berg—New York, 1974). [6 ] R . I. LOEBL, Contractive linear maps on C*-algebras, Michigan Math. J., 2 2 ( 1 9 7 5 ) , 3 6 1 — 3 6 6 . [7] T. OGASAWARA, Finite dimensionality of certain Banach algebras, J. Sci. Hiroshima Univ., 17 (1954), 359—364. [8] V. I. PAULSEN, Completely bounded maps on C-algebras and invariant operator ranges, Proc. Amer. Math. Soc., 86 (1982), 91—96. [9] R. R. SMITH, Completely bounded maps between C*-algebras, J. London Math. Soc., (2) 27(1983), 157—166. [10] M. TAKESAKI, Theory of operator algebras. I, Springer-Verlag (Berlin—Heidelberg—New York, 1979). [11] J. TOMIYAMA, Tensor products and projections of norm one in von Neumann algebras. Lecture Notes (mimeographed), University of Copenhagen, 1970. [12] J. TOMIYAMA, Recent development of the theory of completely bounded maps between C*-algebras, to appear. [13] S.-K. J. Tsui, Decompositions of linear maps, Trans. Amer. Math. Soc., 230 (1977), 87—112. [14] A. W. WICKSTEAD, Spaces of linear operators between partially ordered Banach spaces, Proc. London Math. Soc., (3) 28 (1974), 141—158. [15] G. WrrrsrocK, Ein operatorwertiger Hahn—Banach Satz, / . Funct. Anal., 40 (1981), 127—150. [16] T. HURUYA, Linear maps between certain non-separable C*-algebras, to appear. FACULTY OF EDUCATION NIIGITA UNIVERSITY NNGITA 950-21, JAPAN 

Decompositions of completely bounded maps

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Decompositions of completely bounded maps

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