Defining metric spaces via operators from unital C^∗-algebras

Branka Pavlović

Vol. 186 (1998), No. 2, 285–313

DOI: 10.2140/pjm.1998.186.285

Abstract

For a unital C^∗-algebra A and an operator T with DomT ⊆ A, RangeT in a normed space, and kerT = 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 & ∥Ta∥≤ 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, 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∕ℂ 1 → RangeT, and B₁ = {a ∈ DomT : ∥Ta∥≤ 1}∕ℂ 1, 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 ∈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).

Milestones

Received: 5 February 1997

Revised: 30 January 1998

Published: 1 December 1998