Defining metric spaces via operators from unital C∗-algebras

Branka Pavlović
1998 Pacific Journal of Mathematics  
For a unital C * -algebra A and an operator T with DomT ⊆ A, RangeT in a normed space, and ker T = Cmathrm1, 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 ∈ A & T a ≤ 1}, for φ, ψ ∈ 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 ,
more » ... ic 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 operatorT : DomT /C1 → RangeT , and B 1 = {a ∈ DomT : T a ≤ 1}/C1, which is the inverse image underT of the unit ball of RangeT . We show that: (1) d T is bounded if and only if B 1 is bounded, if and only if T −1 is bounded; (2) T dT = T w * if and only if B 1 is compact, if and only ifT −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 ∈ S(A), and is an operator from C(S(A)) into C b (Y ), Y = {(x, y) ∈ 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).
doi:10.2140/pjm.1998.186.285 fatcat:4t2qb2yfs5atrl3gyqucxddcxm