ON THE RANGE OF COMPLETELY BOUNDED MAPS

It is shown that if every bounded linear map from a C*-algebra α to a von Neumann algebra β is completely bounded, then either α is finite-dimensional or β⫅𝒞⊗Mn, where 𝒞 is a commutative von Neumann algebra and Mn is the algebra of n×n complex matrices.

dimensional or ; @ M n, where is a commutative von Neumann algebra and M is the algebra of n n complex matrlces.n Let 67, be C*-algebras, and let : 67 be a bounded linear map.
For every positive integer n, we define the map n to be n = (R) idn' the entry-wise map from 67 (R) M to (R) M where M denotes the C*-algebra n n n of n g n complex matrices.We say that is comple.telybounde if IInll < m [i].It is not a priori evident that there are bounded maps which sup n fall to be completely bounded.It follows from the results of this paper that there are almost always such maps.

R.I. LOBEL
Let us denote by BI[2 ] the set of all bounded maps from d to , and by B[7, ] the set of all completely bounded maps from 7 to .We shall describe some of the structure of B[7 ] below.Further, in a previous paper 3], we made the following Conjectures: (I) If BI[, 1 B[, ] for all d, then B (R) M n for some conmutatlve C*-algebra and integer n (2 We shall give an affirmative answer to both these conjectures under the hypothesis that is a yon Neumann algebra .We should remark that the converses to (1) and (2) hold; i.e., if 7 is flnite-dlmenslonal or @ n' then Bl[, ] B[, ] (see below).
Although our proof depends heavily on the hypothesis that the range is avon Neumann algebra, we feel that this is merely a shortcoming of our proof, and not a true reflection of the facts.
We begin with what is, to the best of our knowledge, the only example in the literature of a bounded mapat is not completely bounded.
THEOREM I: Let X be an infinite compact Hausdorff space.Then there is a bounded map : C(X) n M2n such that is not completely bounded.
Further, if C(X) c d where d is a C*-algebra, then has an extension PROOF: The proof of the first assertion can be found in [4, Lemma2.1 and Theorem 2.2], and the second assertion follows from the construction used to produce .We will sketch the highlights of the construction, both for the convenience of the reader and for later reference.
Let C(X).For every integer n, there exist elements AI,...,A n E M n (n) (n) be positive linear functionals on with disjoint closed We now remark hat by Krein's Theorem [5, p. 227], the pn) have norm-preserving positive extensions to 67 , and the extension assertion rests on our demonstrating that the IlPn)ll are the keys to computing sp It is true that for a positive linear functional n [I, Prop. 1.2.10].Thus, sp llk(n)ll i=l llpi(n)ll-n I/4.But in fact, Let --n " "n) where all the functionals p-i (n) are chosen to have dis- joint closed supports, which is possible since X is infinite.Then II,II--,Np I < >II SUPn n" =r2.But n, I I > SnUP ][q2n II- sup n 1/4 +, and thus fails to be completely bounded.
n COROLLARY 2 If / is an infinite dimensional Hilbert space, and X is an infinite compact Hausdorff space, there is a bounded map : C(X) , the algebra of all bounded operators on /, such that is not completely bounded.
If 0:7 is a linear map of C*-algebras, we define the adjoint of , , by *(A) =(A*)*.Then * is a linear map from 7 to , and II*[I ''II11" We say that is self-adJoint if -.*.Every map can be written uniquely as I + i2' where I' 2 are self-adJoint.

PROOF:
It is elementary that for all k, I]kH I]1] and that the sum or scalar multiples of completely bounded maps are completely bounded.
For the second assertion, let us re-examine the proof of Theorem i.Let N ,(N) (n) R.I. LOBEL so (N) .However sp II*(N)II (N) I/4 k bounded.
so each (n) is completely We remark that if one defines lli llJ __l[ kll, he B.[, is closed k in III'III.We suspect that Bm[., 8] is always dense in BI[ ], at least in the weak (i.e., pointwise) topology.
We will denote by Mm the algebra ), where is a separable infinite-dimenslonal Hilbert space.By Corollary 2, Mm ) M n.
n 2 We will now do an analysis of von Neumann algebras, based on their type, that will identify the characteristics we need.We follow the type classlfi- cation of [6, pp.24-25].
LEMMA 4: Let be a von Neumann algebra of type !, II or III.
Then (an isomorphic copy of) M.

PROOF:
By [6, Cor.14], is spatially isomorphic to (R) M but LEMMA 5: Let be a yon Neumann algebra of type II I. Then (an isomorphic copy of))n M2n" PROOF: Let {Pn]n=l be family of non-zero, orthogonal projections in Let e = %, where % iS a homogeneous yon Neumann algebra of degree In I n < w.If sup n w, then D (an isomorphic copy of) n M n.
such that ni 2 i, and thus If sp n " there is a subsequence ni M Then ai e M i.
We are now ready to prove the main result of this paper.
THEOREM 7. Let be a yon Neumann algebra, and suppose that for some infinite compact Hausdorff space X, BI[ C(X), ] Bin[ C(X), ].Then there is a commutative C*-algebra and integer n such that u (R) M n PROOF: We can write as a (unique) direct sum I 2 ) 3 4 )5 where i is of type !m, 2 of type lira' 3 of type III, 4 of type III, and 5 of type I n [6, p. 25].By using Theorem i, Corollary 2, Lemma 4, and Len.na 5, we see that the hypothesis forces i 2 3 4 0. Thus 5 is of type I so , where , is homogeneous of degree n [6, p. 42].By applying Theorem I, Lenxna 6, and the hypothesis, we see that sup n N < ".But then (C(X) @ Mn ) ) (C(X) @ Mn) C (R) , for an appropriate commutative C*-algebra .
We can now give our answer to the first conjecture.
For the sake of completeness, we state a converse to Theorem 7. The proof may be found in [3, Lenna 7].
E n) is equivalent to E n) in the usual sense of equivalence for projections.Let [V )] be partial isometries (inPn,Pn [p. 46,Remark])