I ntrnat. J. Math. & Math. Sci.
Vol. 5 No. 2 (1982) 407-409
407
THE DUAL OF THE MULTIPLIER ALGEBRA OF
PEDERSEN’S IDEAL
KELLY McKENNON
Department of Pure and Applied Mathematics
Washington State University
99164
Pullman, Washington
(Received November 24, 1980)
ABSTRACT.
It is shown that the dual of the multiplier algebra of Pederson’s ideal is
not always spanned by its positive elements.
KEY WORDS AND PHRASES.
Multiplier algebra, Pederson’s ideal.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
I.
Primary 46L05.
INTRODUCTION.
In [I] Lazar and Taylor study the multiplier algebra F(K) of Pedersen’s minimal
dense ideal K of a C*-algebra A.
Equipped with its canonical strict topology, F(K)
is a locally convex space, and Lazar and Taylor have demonstrated that the dual
can be identified with the set of all linear functionals A-F
F, G 6 A’ ([I] 6.1).
positive element of
+ G-A
where A
F(K)’
K+ and
They have also shown that, under this identification each
F(K)’
is of the form A*-F-A for some A
This note answers negatively their question, whether or not
K and F 6 A’ ([I] 6.5).
F(K)’ is the span of its
positive elements.
2.
MAIN RESULTS.
Let H denote the Hilbert space
canonical Hilbert basis.
sending each x
H to
2(Z),
<,>
its inner product, and
For vectors v, w 6 H, let v
<x,w>v.
Denote by (B,
* ,II B
(R)
{bn
n6Z
its
w* denote the linear operator
the C*-algebra of all bounded
408
K. McKENNON
The identity transformation I has a decomposition P
linear transformations of H.
2(I)
where P is the orthogonal projection of H onto
onto
+ Q
and Q the orthogonal projection
2 (Z/l).
Let A be the C*-algebra of all bounded sequences A
Anll B + IIAnPll B
liP
lim
n-oo
We write
If
B such that
(2.1)
0.
for the norm on
flAIl
IIAnlIB
sup
nN
(V A 6 A).
Let K be the set of all A 6A such that
{n
Anll + IIAnPII
# 0}
(2.2)
is finite.
Pedersen’s ideal in A is just K.
PROPOSITION.
PROOF.
liP
I
That K is an ideal is trivial.
For each A
ll, let A
and n
Am(n) Am (Vm
be the element of K defined by
(n)
n),
1,2
(n)
QQ
(Vk
n
+ i,
n
+ 2
).
A,
Then, for each A
li-- lIA-
A(n)ll
IIPQII B
lim sup
n
k>n
+
QQI]B -<
lIAk
lira sup
n
k>n
n
IIQPII B
+
llPPll B
0
by (2. I), which proves that K is dense in A.
A
Since the minimal dense ideal of
that .B
N+,
A for some B
contains all positive elements A
it will suffice in showing
demonstrate that K is spanned by elements of this sort.
A such
K is Pedersen’s ideal to
Since K is evidently spanned
by its positive elements, it will be sufficient to examine an arbitrary positive
element A of
m > n.
A
Let B
B
Then, since
For such A
K.
k
Am
I
+
+,
K
there exists n 6 N such that PA
m
be defined by
(Vk
QAmQ
n), Bm
1,2
(Vm
Q
for all m > n, AB
A.
n
+ I, n + 2
Q.E.D.
Let F be the bounded linear functional on A defined by
F(A)
0
7.
n=l
<
A
n
(bl)’
b
-i
>
2
-n
(VA
Let D 6 K be the partial unitary operator defined by
D
n
Q
(Vn
).
A).
).
A P for all
m
DUAL OF THE MULTIPLIER ALGEBRA OF THE PEDERSEN’S IDEAL
A’
For G
A, the
and A
linear functionals A-G and A.G.A are defined by
G(AB) and A’G’A(B)
A’G(B)
G(ABA)
K+ and
THEOREM.
GIF-A’
A’
of the form A*-G.A such
A’.
G
The linear functional
D-F is
not in S.
+
Then there exists a finite subset F of K
Assume false.
PROOF.
A).
(VB
Let S be the linear span of the set of all elements of
that A
409
and a map
such that
E
D-F
A*.GA.A.
(2.3)
A{F
N such that
Choose n
PA --AP=0
Let B
F;
(VA
m
m
n, n
m
+ I, ...).
A be defined by
B
b
n
l*
-I
b
Bn
and
B
and
k
(Vk
0
k
n)
and note that
QBn
we
QBnQ
0.
have
D-F(B)
For each A
that B A
n n
F(DB)
F, we have
0.
An
<(DB)n(bl), b_l >
-n-QAQ; therefore,
2
-n
2
-n
(b_l
since
# 0.
(R)
bl*)Q_
(2.4)
0, it follows
Thus
BA
A*-GA-A(B)
0
and
GA(A*BA)
0
(A
But (2.3), (2.4), and (2.5) are incompatible.
F).
Q.E.D.
REFERENCE
[1] Lazar, A. J.,
and Taylor, D. C., Multipliers of Pedersen’s Ideal,
Mem. Am. Math. Soc. 169, (1976).
(2.5)