arXiv:2108.06406v1 [math.OA] 13 Aug 2021 ABELIAN VON NEUMANN ALGEBRAS, MEASURE ALGEBRAS AND L ∞ -SPACES DAVID P. BLECHER, STANIS LAW GOLDSTEIN, AND LOUIS E. LABUSCHAGNE Abstract. We give a fresh account of the astonishing interplay be- tween abelian von Neumann algebras, L ∞ -spaces and measure algebras, including an exposition of Maharam’s theorem from the von Neumann algebra perspective. 1. Introduction It is a beautiful fact, seemingly attributable to Segal [24], that the class of localizable measure spaces “corresponds” to the abelian von Neumann algebras, with the correspondence given by taking such a measure space to its L ∞ algebra (realized as multiplication operators acting on its L 2 space). Contributions from other authors (see for example [22, 27]) show that the classes of decomposable and Radon measure spaces fulfill essentially the same role, but with localizable spaces being the most general class among these three. Segal demonstrated in this now quite dated paper that localiz- able spaces “constitute the most significant general class of measure spaces properly containing the finite ones, and are of maximum generality consis- tent with usefulness in any large sector of abstract analysis”. He did this by identifying a number of important measure-theoretic and functional analytic properties that he showed to each be equivalent to localizability. Segal also generalized to localizable spaces Dorothy Maharam’s stunning structural classification of finite measure spaces [15]. He showed that such spaces can be decomposed into finite pieces to which Maharam’s result ap- plies (see the lines above [24, Theorem 3.3]). Instead of measure spaces, Maharam’s theorem naturally belongs in the setting of measure algebras, objects which have been thoroughly investigated by Fremlin (see [9, Vol- ume III] and [10]) and which may be derived from measure spaces. See [9, 332B] or [10, Theorem 3.9] for the striking measure algebra approach to the (generalized) Maharam’s theorem. One sees that localizable measure alge- bras are essentially just simple products of measure algebras of the spaces Ω κ = {0, 1} κ for various cardinals κ, with its usual measure (perhaps scaled by some factor). The von Neumann algebraic formulation of Maharam’s theorem is usually phrased as the statement that every abelian von Neu- mann algebra is isomorphic to a direct sum of L ∞ spaces of such measure spaces Ω κ . Although the latter is true (see Corollary 9.21 in conjunction with 9.24), it does not convey the full import of the result, as we shall see. It is a well known saying that ‘von Neumann algebras are noncommu- tative measure theory’. Indeed much of von Neumann algebra theory may DB is supported by a Simons Foundation Collaboration Grant (527078). 1