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Volume 186, No. 2, (1998), 285-313
Branka Pavlovic
Defining Metric Spaces via Operators from Unital C*-Algebras
Abstract:
For a unital C*-algebra A and an operator T with
Dom(T) in A, range(T) in a normed space,
and ker T = C1, we consider the metric dT on
S(A), the state space of A, given by
dT(phi,psi)
= sup {|phi(a)-psi(a)| : a in A and ||Ta|| <= 1},
for phi, psi in S(A).
This is a generalization of the definition given by A. Connes for defining
a metric on S(A) via unbounded Fredholm modules over
A.
The main problem of our investigation, posed by M. Rieffel, is the
relationship between thus defined metric topology
TdT,
and the weak-* topology Tw* on S(A).
We give two different complete characterizations of those operators
for which
TdT = Tw*.
First, we establish the relevance to this relationship of the induced
one-to-one operator (T^): Dom(T)/C1 map range(T),
and
B1 = { a in Dom(T): ||Ta|| <= 1 } / C1
which is the inverse image under (T^) of the unit ball of
range(T).
We show that:
(1) dT is bounded if and only if B1 is bounded,
if and only if (T^)-1 is bounded;
(2) TdT = Tw*
if and only if -B1- is
compact, if and only if (T^)-1 is compact.
Furthermore, we consider the de Leeuw derivation DdT
associated
to T, which is defined by (f(y)-f(x))/dT(x,y),
x,y in S(A), and is an operator from C(S(A)) into
Cb(Y),
Y = {(x,y) in S(A) \times S(A): x not = y},
whose domain is the Lipschitz algebra Lip(S(A),dT).
We show that TdT = Tw*
if and only if DdT
is unbounded on every infinite dimensional subspace of its domain.
In particular, we use all these results to characterize those unbounded
Fredholm modules over A whose metric topology coincides with the
weak-* topology on S(A).
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