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Fix a compact Hausdorff space $K$ and think about $C(K)$ as a C*-algebra acting on a Hilbert space $H$. Suppose that $C(K)$ is closed in $\mathcal{B}(H)$ in:

  • $\sigma$-strong
  • $\sigma$-strong*

topology. Must $K$ be extremelly disconnected?

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Please use this spelling: "extremally disconnected". Thank you. –  Gerald Edgar May 15 '11 at 23:37

1 Answer 1

up vote 5 down vote accepted

Recall that if two compact spaces $K_1$, $K_2$ are such that $C(K_1)\cong C(K_2)$, then $K_1\cong K_2$. The space $K$ is called the spectrum of the abelian C*-algebra $C(K)$.

Since $C(K)$ is closed in the σ-strong topology, it is a von Neumann algebra (that condition is equivalent to being closed in the σ-strong* topology).

Now, the spectrum of an abelian von Neumann algebra is indeed an extremely disconnected space. So yes: $K$ has to be extremely disconnected. This kind of space is also called hyperstonean space.



By the way, here's one way to visualize the hyperstonean space associated to the von Neumann algebra $L^\infty ([0,1])$:
For every measurable partition of $[0,1]$ into finitely many subsets $$[0,1]=X_1\cup\ldots \cup X_n$$ (where each $X_i$ is well defined up to measure zero sets), we form the space $$ \overline{X_1}\sqcup\ldots \sqcup \overline{X_n} $$ where $\overline{X_i}$ is the closure of $X_i$ (more precisely, it is the intersection of all closures of sets that are equal to $X_i$ up to a measure zero set) and $\sqcup$ denotes disjoint union. The assignment $$ X_1\cup\ldots \cup X_n \mapsto \overline{X_1}\sqcup\ldots \sqcup \overline{X_n} $$ is a functor from the poset of measurable partitions of $[0,1]$ to the category of compact topological spaces. The hyperstonean space associated to $L^\infty ([0,1])$ is the inverse limit of that functor.

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Just to clarify the third paragraph: Every hyperstonean space is extremally disconnected, but not every extremally disconnected space is hyperstonean, because there are additional conditions (e.g., existence of sufficiently many normal measures) involved in the definition of hyperstonean spaces. –  Dmitri Pavlov May 16 '11 at 4:24
    
Another helpful source of intuition about hyperstonean spaces: Clopen (closed and open) subsets of a hyperstonean space are in bijective correspondence to equivalence classes of measurable sets modulo null sets of the corresponding measurable space. –  Dmitri Pavlov May 16 '11 at 4:28
    
Thank you all. Is there a purely topological characterization of the hyperstonian space corresponding to a given compact space? If we are interested in the Gleason cover of a compact space then indeed, there are at least two interesting characterizations. –  Tomek Kania May 16 '11 at 8:56
    
@Tomasz: The construction you are referrring to is known as the hyperstonean cover. A search in Google Scholar on “hyperstonean cover” reveals quite a few papers on this matter and one of them is titled “Topological characterization of the hyperstonean cover”. –  Dmitri Pavlov May 16 '11 at 15:22

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