Abstract

For a unital C^*-algebra A and an operator T with DomT ⊆ A, RangeT in a normed space, and kerT = C1, we consider the metric d_T on S(A), the state space of A, given by d_T(φ,ψ) = sup{|φ(a) −ψ(a)| : a in A & ∥Ta∥≤ 1}, for φ,ψ 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 T_dT, and the weak-* topology T_w^* on S(A). We give two different complete characterizations of those operators for which T_dT = T_w^*. First, we establish the relevance to this relationship of the induced one-to-one operator T : DomT ∕ C1 → RangeT, and B₁ = {a in DomT : ∥Ta∥≤ 1} ∕ C1, which is the inverse image under T of the unit ball of RangeT. We show that: (1) d_T is bounded if and only if B₁ is bounded, if and only if T⁻¹ is bounded; (2) T_dT = T_w^* if and only if B₁ is compact, if and only if T⁻¹ is compact. Furthermore, we consider the de Leeuw derivation D_dT associated to T, which is defined by (f(y) − f(x)) ∕ d_T(x,y), x,y in S(A), and is an operator from C(S(A)) into C_b(Y ), Y = {(x,y) in S(A) ×S(A) : x≠y}, whose domain is the Lipschitz algebra Lip(S(A),d_T). We show that T_dT = T_w^* if and only if D_dT 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).

Authors