Gleason's Theorem

[Sam Sanders]

Dear All,

Fred Richman and Douglans Bridges have written a paper on the constructive proof of
Gleason's theorem. I would be interested in this result as follows:

<begin>

Gleason's theorem states that "Every frame function F(x) can be written as F(x)=<Tx,x > where T is a trace operator".

Here, F(x) is a frame function IFF $(\exists W)(\forall (x_i) \in OB ) ( \sum_i f(x_i) = W )$, where OB stands for "orthonormal basis".

Is there an "approximate" version of Gleaon's theorem as follows?

For all e>0,

IF $(\exists W)(\forall (x_i) \in OB ) ( | \sum_i f(x_i) - W | < e)$

THEN there is C>0 such that for all x, we have | F(x) - <Tx, x> | < Ce.

where T is the trace operator from Gleason's theorem.

In particular, how big is C?

<end>

The idea behind the approximate version of Gleason's thm is that
Constructive Analysis produces results which are "continuous in the parameters".

There is a "special case" of the approximate version of Gleason's thm
in the Richman-Bridges paper on Gleason's thm (Lemma 19 on page 303).

With kindest regards,

Sam Sanders

[Bas Spitters]

Ken je deze?

http://philsci-archive.pitt.edu/1263/1/indeterminacy2.pdf

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[Kreinovich, Vladik]

Published in Studies in History and Philosophy of
Modern Physics 35 (2004) 177-194

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[yut...@iu.edu]

I am reading [1]. As far as I read, this paper only care about the original definition of frame function, i.e.

but not the approximation frame function

as Sam Sanders asked. Am I correct?

If I am correct, does anyone have studied the approximation frame function before? Especially, can the approximation frame function be even continuous when ε is small enough?

Reference

[1] Ehud Hrushovski and Itamar Pitowsky. “Generalizations of Kochen and Specker’s theorem and the effectiveness of Gleason’s theorem”. In: Studies in History and Philosophy of Science Part B: Studies
in History and Philosophy of Modern Physics 35.2 (2004), pp. 177–194. issn: 1355-2198. doi: http://dx.doi.org/10.1016/j.shpsb.2003.10.002. url: http://www.sciencedirect.com/science/article/pii/S1355219804000024.