Examples and Observations

Abstract

1°/ Let X be a locally compact space, μ a positive measure on X with support X and H the Hilbert space L² (X, μ). Let A_i be a sequence of μ-measurable functions on X with values in [1, +∞] such that μ (x ∈ X|A_i(x) = +∞) = 0. Let L^∞((A_i), X, μ) be the set of all functions of the form gA_i with g moving in L^∞ (X, μ) and i moving in IN. The linear space D = {g ∈ H/A_ig ∈ H} for all i ≥ 0 is necessarily dense in H and each f ∈ L^∞ ((A_i), X, μ) is identified to the - closeable - linear operator T_f defined by T_fg = fg for all g ∈ D . Assuming that each A ²_i is dominated a.e. by some A_j, we see that L^∞((Ai), X, μ) is an abelian ⋆-algebra ultraweakly closed relative to D $\hat \otimes$ D, with natural domain D and cofinal sequence T_{A i} ≡ A _i, and the condition A_iD = D is satisfied. Conversely, a general, ultraweakly closed abelian ⋆-algebra with condition I or II is isomorphic to such an object. In paragraph 6 it is seen that derivations on such function spaces are identically zero.