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First, let me state some basic facts and definitions for my question. I believe these are well-known among experts working on von Neumann algebras, but let me state them anyway since my question is purely topological and measure-theoretic in nature.

Basic definitions and facts

  1. A compact Hausdorff space $X$ is called stonean (some authors also call it extremely disconnected) if the closure of every open set in $X$ is still open. A stonean space $X$ is called hyperstonean if for any positive $f \in C(X)$, there is a positive normal measure $\mu$ with $\mu(f) \ne 0$. Here a positive measure $\mu$ on $X$ is called normal if $\mu(f) = \sup_{i \in I}\mu(f_i)$ whenever $f = \sup_{i \in I}f_i$ for an increasing bounded net $(f_i)_{i \in I}$ in $C_{\mathbb{R}}(X)$, and $\sup_{i \in I}(f_i)$ is taken in the lattice $C_{\mathbb{R}}(X)$, not point-wise limit of functions, the existence of such a supremum relies on $X$ being stonean. It is a classical result in the theory of operator algebras (See e.g., Section III.1, Theorem 1.18 in Theory of operator algebras by Takesaki) that hyperstonean spaces are exactly the spetrums of abelian von Neumann algebras.

  2. Let $X$ be a stonean space, $Z$ a nowhere dense closed set in $X$, a mapping $f: X\setminus Z \to \mathbb{C}$ is called normal if it is continuous and $\lim_{q \to p,\; q \in X \setminus Z}|f(q)| = \infty$ holds for any $p \in Z$. A real-valued normal function $f$ on $X$ is called self-adjoint. In section 5.6 of Kadison & Ringrose, it is proved (unfortunately the results I am referring to, namely Theorem 5.6.19 there, are in the missing pages in the Google book links I give above. You will have to find a copy of the real book or take my word for it.) that when $X$ is the spectrum of an abelian von Neumann algebra $\mathscr{A}$, i.e., when $X$ is hyperstonean, normal functions on $X$ form an involutive algebra $\mathcal{N}(X)$ whose self-adjoint elements are exactly the self-adjoint functions on $X$, and there is a $\ast$-isomorphism mapping this algebra onto the algebra of unbounded operators affiliated with $\mathscr{A}$, and extending the continuous functional calculus.

Comments before my question

A non-expert might ask here how do we define addition and multiplication in $\mathcal{N}(X)$, since functions there might be only partially defined, which may cause a pointwise definition to violate the limit condition in the definition of normal functions given above (involution is just taking conjugation pointwise, which is easy).

In the book of Kadison & Ringrose mentioned above, one passes first from normal functions to their corresponding unbounded operators, then uses the theory of unbounded operators affiliated with an abelian von Neumann algebra to define their sums and multiplications (and the adjoint operation), then passes back to normal functions and thus obtains the needed definitions.

The above approach, while being very elegant in the eyes of an operator algebraist, does naturally raise the question of whether the problem, which can clearly be formed using only topological and measure-theoretic terms, can be solved also using only topology and measure theory. More precisely, my question is formulated as below.

Question

Suppose $X$ is a hyperstone space, $f$ and $g$ are two self-adjoint functions on $X$, with $f$ defined on $X \setminus Z_f$ and $g$ on $X \setminus Z_g$. Let $f + g$ and $f \cdot g$ be the functions defined on $X \setminus (Z_f \cup Z_g)$ by pointwise addition and multiplication respectively.

  1. Prove that both $f + g$ and $f \cdot g$ can be extended to a unique self-adjoint function on $X$ without using operator theory.
  2. Can we prove this using only topological and measure-theoretic means?
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I take it that the normal, self-adjoint functions are assumed to be continuous on their domains. In that case the functions $f+g$ and $f\cdot g$ are continuous on the dense open set $O=X\setminus(Z_f\cup Z_g)$. A well-known characterization of extremally disconnected spaces that open subsets are $C^*$-embedded, i.e., bounded continuous real-valued functions have continuous extensions to the whole space (see, e.eg., Engelking's General Topology, Problem 6.2.G). In this case: if $h:O\to\mathbb{R}$ is continuous then take the unique continuous extension $\tilde h:X\to\mathbb{R}$ of $\arctan h$ (unique because $O$ is dense). Then $H=\tan \tilde h$ defines a normal self-adjoint function with $Z_H=\{x:|\tilde h(x)|=\frac\pi2\}$. Apply this to $f+g$ and $f\cdot g$ to define the sum and product.

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  • $\begingroup$ Yes, self-adjoint and normal functions are continuous in their domains, as I've already stated in their definitions. And indeed, this answers my question. Now I feel stupid to ask this (I know very well Proposition III.1.7 and Corollary III.1.8 in the book of Takesaki, which, together with the Tietze extension theorem, immediately yield the characterisation you mentioned). $\endgroup$ – Rick Sternbach Sep 5 '18 at 11:23

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