If I imagine that (the self-adjoint part of) a C*-algebra $A$ represents the algebra of observables of some quantum system, then [certain][1] [perspectives][2] on algebraic quantum theory would ask me to imagine that each (maximal) commutative C*-subalgebra $C \subseteq A$ provides a (maximal) "classical snapshot" of this quantum system.  Gelfand duality yields that $C \cong C(X)$ for the compact Hausdorff space $X = \mathrm{Spec}(C)$, so I would picture $X$ as a classical state space that (maximally) "approximates" the would-be quantum state space corresponding to $A$.

I would like to know how different these spaces $X$ can be as the maximal commutative $*$-subalgebra $C \subseteq A$ varies.  Specifically, can it happen that these have different cardinalities?

I'm interested in the particular case $A = B(H)$ for a separable Hilbert space $H$ and two of its well-known masas: the continuous one $C \cong L^\infty[0,1] \subseteq A$ and discrete one $D \cong \ell^\infty(\mathbb{N}) \subseteq A$.  Thus I ask:

> **Q:** Is there a bijection between the Gelfand spectra $\mathrm{Spec}(C)$ and $\mathrm{Spec}(D)$ for the continuous and discrete masas $C,D \subseteq B(H)$?

It's possible to describe these spectra in more explicit terms using Boolean algebra.  Note that each of these masas is an (A)W*-algebra.  By a combination of Gelfand and Stone dualities (see [section 2 of this paper][3] for a bit more detail), the spectrum of a commutative AW*-algebra $K$ is the [Stone space][4] of the complete Boolean algebra $\mathrm{Proj}(K)$ of projections in $K$, whose points are the ultrafilters of $\mathrm{Proj}(K)$.

The continuous masa $C \cong L^\infty[0,1]$ has $\mathrm{Proj}(C)$ isomorphic to the Boolean algebra of measurable subsets of $[0,1]$ modulo the null sets.  I have just learned through the magic of Wikipedia that this is called the [random algebra][5]; I will denote it by $B$.

The discrete masa $D \cong \ell^\infty(\mathbb{N})$ has $\mathrm{Proj}(D)$ isomorphic to the power set Boolean algebra $2^\mathbb{N}$.  (Note that an ultrafilter on the Boolean algebra $2^\mathbb{N}$ is alternatively referred to as an ultrafilter on the set $\mathbb{N}$.)

Thus my question is equivalent to:

> **Q':** Is there a bijection between the sets of ultrafilters on the random algebra $B$ and the power set algebra $2^\mathbb{N}$?

I am aware that $\mathrm{Spec}(D)$ has spectrum homeomorphic to the Stone-Cech compactification $\beta\mathbb{N}$ of the discrete space $\mathbb{N}$, and that this space has various properties that depend on set-theoretic assumptions.  Now that I know what the random algebra is called, I see that it bears a relationship to forcing.  Thus I can imagine that the answer to my question could be independent of ZFC.  Nevertheless, as I am not asking exactly what the cardinality of this spectrum is, but whether it is in bijection with some other (possibly complicated) spectrum, I have an ounce of hope that this can indeed be decided in ZFC.

(By the way, the [classification][6] of the possible masas of $B(H)$ implies that, if the answer to my question is affirmative, then the spectra of all masas of $B(H)$ are isomorphic.)


  [1]: http://arxiv.org/abs/1412.2177
  [2]: http://dx.doi.org/10.1007/s00220-009-0865-6
  [3]: http://arxiv.org/abs/1212.5778
  [4]: http://en.wikipedia.org/wiki/Stone%27s_representation_theorem_for_Boolean_algebras
  [5]: http://en.wikipedia.org/wiki/Random_algebra
  [6]: http://en.wikipedia.org/wiki/Abelian_von_Neumann_algebra#Classification