209
(1978) 209-215
VO,.
ON THE RANGE OF COMPLETELY BOUNDED MAPS
RICHARD h LOEBL
Department of Mathematics
Wayne State University
Detroit, Michigan 48202
(Received March 28, 1977)
It is shown that if every bounded linear map from a
ABSTRACT.
to a
yon
Neumann algebra
dimensional or
and
Mn
;
@ Mn,
is the algebra of
Let 67,
be
is completely bounded, then either
where
n
C*-algebras,
For every positive integer
n,
the entry-wise map from 67 (R) M
n
of
ng n
complex matrices.
C*-algebra
67
67 is finite-
is a commutative von Neumann algebra
complex matrlces.
n
and let
:
67
be a bounded linear map.
we define the map
to
(R)
We say that
IInll
Mn
n
where
is
Mn
to be
n=
(R)
denotes the
comple.tely
bounde
idn’
C*-algebra
if
< 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
210
BI[2
Let us denote by
the set of all completely bounded maps from 7
]
B[7,
and by
the set of all bounded maps from d
]
B[7 ]
shall describe some of the structure of
C*-algebra
conmutatlve
S
c
]
for all
and integer
d,
We
Further, in a
B (R) Mn
then
for some
n
then either 2
B[2, ],
]
BliP7,
If
(2)
B[,
1
BI[,
If
.
,
3], we made the following Conjectures:
previous paper
(I)
below.
to
to
,.(R) Mn
.
is finite-dimenslonal or
We shall give an affirmative answer to both these conjectures under the
is a yon Neumann algebra
hypothesis that
We should remark that the
converses to (1) and (2) hold; i.e., if 7 is flnite-dlmenslonal or
@
n’
B[,
]
Bl[,
then
]
(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
THEOREM I:
Further, if
C(X)
:
c
C(X)
d
n M2n
where
is not completely bounded.
such that
d is a
Then there
C*-algebra,
then
has an extension
The proof of the first assertion can be found in [4, Lemma 2.1
PROOF:
.
and Theorem 2.2], and
to produce
is not completely bounded.
X be an infinite compact Hausdorff space.
Let
is a bounded map
mapat
the
second assertion follows from the construction used
We will sketch the highlights of the construction, both for
the convenience of the reader and for later reference.
Let
such that:
C(X).
(I)
Ai
For every integer
Ai*;
(2)
AiAj
n,
+
AjAi
there exist elements
25ijI 2n;
and (3)
AI,...,An E M 2 n
Tr(Ai)
0.
211
COHPLETELY BOUNDED MAPS
(n)
(n)
Pi
Let
be positive linear functionals on
n
(n) (f)
(n)
p
Then
(f)A
%0
i__l n)
supports and let
n)ll
that if
n
3/4
for all
,
67
extensions to
IlPn)ll
that the
2 n" 14
II(n)II
pn)
by Krein’s Theorem [5, p. 227], the
2 4 [[p
I I
i.
i,
with disjoint closed
n) I 2
SO
We now remark hat
have norm-preserving positive
and the extension assertion rests on our demonstrating
are the keys to computing
sp
It is true that for a positive linear functional
[I, Prop. 1.2.10].
"
I < >II
--n
Let
Thus,
"n)
n
i=l llpi(n)ll-nI/4.
sp llk(n)ll
joint closed supports, which is possible since
II,II-- ,Np
1/4
sup n
SUPn n"
+, and thus
n
COROLLARY 2
p-i(n)
where all the functionals
If /
=r2.
X
But in fact,
are chosen to have dis-
is infinite.
I > SnUP ][q2n II-
n,
But
Then
fails to be completely bounded.
is an infinite dimensional Hilbert space, and
is an infinite compact Hausdorff space, there is a bounded map
the algebra of all bounded operators on
/, such that
:
X
,
C(X)
is not completely
bounded.
,
,
II*[I
If
is a linear map
0:7
by *(A) =(A*)*. Then
’’II11"
We say that
but
Bm
*
Bm[7,
]
is a
we define the adjoint of
is a linear map from
self-adJoint if
I + i2’
be written uniquely as
PROPOSITION 3:
is
C*-algebras,
of
where
-.*.
I’ 2
self-adJoint
are
7 to
,
Every map
and
can
self-adJoint.
linear subspace of
BI[,
need not be norm-closed.
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.
N
,(N)
(n)
Let
],
212
.
(N)
so
R.I. LOBEL
sp II*(N)II
k
However
(N)
I/4
(n)
so each
is completely
bounded.
We remark that if one defines
in
III’III.
We suspect that
lli llJ
Bm[., 8]
k
__l[ kll,
B.[,
he
is always dense in
BI[
is closed
],
at least in
the weak (i. e., pointwise) topology.
We also should remark that if
ll0o@ll 1[8ooll llll
8
is a
and more generally,
Bl[d
]
unchanged under isomorphism; e.g., if
Z
Thus the classes
k.
integers
B.[, ].
Mm
We will denote by
the algebra
*-isomorphism of C*-algebras, then
llkll
II(O@)kll ll(@O)kll
-
%
-
Bm[, ]
and
and
),
for all
are essentially
i’
where
Bm[6Z
then
]
-
is a separable
) M n.
n 2
We will now do an analysis of von Neumann algebras, based on their type,
infinite-dimenslonal Hilbert space.
By Corollary 2,
that will identify the characteristics we need.
cation of
Let
be a von Neumann algebra of type
(an isomorphic copy of)
LEMMA 5:
Let
be a
[6, p. 45].
J,
and
{Pn]n=l
Let
E
Foreach
n)
for projections.
is spatially isomorphic to
yon
Neumann algebra of type II I.
-lj
(R)
M
but
Then
be family of non-zero, orthogonal projections in
2n
n)
n) for
where
n, write Pn
i--1
[V)]
E
En)E
)E
is equivalent to
Let
or III.
of))n M2n"
(an isomorphic copy
PROOF:
!, II
M.
By [6, Cor. 14],
PROOF:
i
We follow the type classlfi-
[6, pp. 24-25].
LEMMA 4:
Then
Mm
E
n)
in the usual sense of equivalence
be partial isometries (in
-lJ -iJ
E
Pn,Pn
[p. 46, Remark])
so that we can take
V
)*
COMPLETELY BOUNDED MAPS
213
Theft
n P-Pn-- n M2n"
LEMMA 6:
In
of degree
n
of)
Let
e
=
%,
n < w.
I
%
where
If
iS a homogeneous yon Neumann algebra
n
sup
w,
D
then
(an isomorphic copy
M n.
2
PROOF:
Each
is (isomorphically> of the form
is a commutative von Neumann algebra
sp n
If
Thus
2
"
[6, p. 98], and thus
there is a subsequence
M
ai e M
Then
% (R) Mn
ni
such that
1 @
2
ni
%
where
i,
Mn-- Mn.
and thus
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,
C*-algebra
a commutative
)5
where
type
III,
and integer
We can write
PROOF:
i
and
n
]
Bin[ C(X), ].
such that
of type
!m,
2
In
[6, p. 25].
of type
lira’
3
5
degree
,
n
we see that
C (R)
is of type
[6, p. 42].
sup n
N
I
so
Then there is
(R) M
n
I 2 )3 4
of type
4
III,
of
By using Theorem i, Corollary 2,
, ,
. Mn
Lemma 4, and Len.na 5, we see that the hypothesis forces
Thus
u
as a (unique) direct sum
is of type
5
BI[ C(X),
where
i 2 3 4
0.
is homogeneous of
By applying Theorem I, Lenxna 6, and the hypothesis,
<
".
But then
for an appropriate commutative
(C(X)
@
)
)
(C(X)
@
Mn)
C*-algebra
We can now give our answer to the first conjecture.
COROLLARY 8:
7,
BI[7, ]
If
Bm[7, ],
is a yon Neumann.algebra, and for all
then
C*-algebras
@ Mn.
For the sake of completeness, we state a converse to Theorem 7.
may be found in [3, Lenna 7].
The proof
R.I. LOBEL
214
PROPOSITION 9:
then for all
Mn,
C*-algebras 6,
B[, ].
]
BI[,
(R)
If
In order to present our answer to the second conjecture, we need the
following result.
PROPOSITION i0:
for some infinite compact Hausdorff space
C(X)
a variation upon
[2].
THEOREM ii:
If
BI[
C*-algebra.
]
E], and
is a von Neumann algebra, and for some
Bm[, ],
then either
=
(R)
is
C*-algebra
C (R) M
n.
is finite-dimensional or
The proof of Theorem 7 shows that either
PROOF:
Then
X.
This fact is established in the proof of [3, Thin.
PROOF:
6,
be an infinlte-dlmenslonal
Let
Mn
or
M n.
is infinite-dimensional, we see
Suppose the latter; then if
2
from Proposition i0 that
C(X) for some infinite compact Hausdorff space
n
X.
But then by Theorem i, there is a map
bounded but not completely bounded.
,
:
,
which is
M n
2
This contradicts the hypothesis, and
n
completes the proof.
For the sake of completeness, we state a result, which along with
Proposition 9, yields a converse to Theorem ii.
The proof may be found in
3, Lemma 5].
,
PROPOSITION 12:
B i[,
1
is finite-dimensional, then for all
If
C*- algebras
BtdT, ].
We remark that, by Theorem ii, there are almost always bounded maps
between
C*-algebras
that fail to be completely bounded.
We should mention that there is another interesting consequence of the
methods of this paper.
-
THEOREM 13.
Let
be an infinite-dimensional
is a bounded self-adjoint map
+
where
/
M n
C*- algebra
Then there
such that
cannot be written
2
are bounded positive linear maps from
).
q0.
n
COMPLETELY BOUNDED MAPS
PROOF:
the map
:
215
The actual assertion of [4, Lemna 2.1 and Theorem 2.2] is that
C(X)
M n
n
described in the proof of Theorem 1 does not admit
2
such a decomposition.
.
The extension statement in Theorem i and Proposition i0
allow us to pass to an arbitrary infinite-dimensional
Arguments simila to those of Lemnas 4, 5, 6 and Theorem 7 in this paper
were used in the dissertation of Sze-Kai Jack Tsui, University of Pennsylvania,
1975.
He also obtained results relating to Theorem 13 of this paper.
BIBLIOGRAPHY
Subalgebras of
C*-algebras,
Acta Math. 123 (1969), 141-224.
i.
Arveson, W. B.
2.
Hirschfeld, R. A. and Johnson, B. E. Spectral characterization of finitedimensional algebras, Inda. Math. 34 (1972), 19-23.
3.
Loebl, R. I. Contractive linear maps on
22 (1975), 361-366.
4.
C*-algebras, Michigan
Math. J.
A Hahn decomposition for linear maps, Pacific J. Math.,
65 (1976), 119-133.
5.
Rickart, C. E.
1960.
General Theory of Banach Algebras, Van Nostrand, New York,
6.
Topping, D. M.
Lectures on
yon
Neumann Algebras, Van Nostrand Reinhold,
London, 1971.
KEY WORDS AND PHRASES. Comply bounded maps, C*-algebras, von Neumann
lgbas, extension of maps.
AMS(MOS) SUBJECT CLASSIFICATION (1970) CODES.
46L05, 46L10, 46J10.