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Jun
11
accepted Do non-normal states exist in the Solovay model?
May
28
awarded  Enthusiast
May
26
comment Do non-normal states exist in the Solovay model?
This should be a nonnormal state on a nonseparable Hilbert space in the Solovay model : Let $H = l^2([0,1])$ and $\lbrace e_x : x \in [0,1] \rbrace$ an orthonormal basis of H . Then $f$ defined by $f(A) = \int_0^1 <e_x, A e_x> dx$ for $A \in B(H)$ is a nonnormal state .
May
26
comment Do non-normal states exist in the Solovay model?
@Ashutosh : to lo.logic + math.physics : See the comments to Arnold Neumaiers answer in physicsoverflow.org/31006/why-isnt-the-path-integral-rigorous
May
25
comment Do non-normal states exist in the Solovay model?
@Ashutosh : Yes, at least for separable Hilbert spaces.
May
25
comment Do non-normal states exist in the Solovay model?
@Ashutosh : According to Gleasons theorem for separable Hilbert spaces of dimension other than 2 states correspond to finitely additive measures on pairwise orthogonal projections but normal states correspond to sigma-finite measures, see ncatlab.org/nlab/show/Gleason%27s+theorem .
May
25
comment Do non-normal states exist in the Solovay model?
@Asaf Karagila : All states have norm 1, but non-normal states are discontinous in some other topologies (e.g. ultraweak), see ncatlab.org/nlab/show/state+on+an+operator+algebra .
May
25
asked Do non-normal states exist in the Solovay model?
May
9
comment How to prove this determinant is positive?
@TerryTao : $T$ could also be of the form $\begin{pmatrix} A & A B \\ 0 & A^{*-1}\end{pmatrix}$, where $B^{*} = - B$ .
May
3
comment How to prove this determinant is positive?
@GH from MO : ok, my mistake was that the Baker–Campbell–Hausdorff formula is not always convergent.
May
3
revised How to prove this determinant is positive?
added 67 characters in body
May
3
comment How to prove this determinant is positive?
@Christian Remling choose $M = \left[\begin{matrix}0 & \pi & 0 & 0\\- \pi & 0 & 0 & 0\\0 & 0 & 0 & \pi\\0 & 0 & - \pi & 0\end{matrix}\right]$
May
3
comment How to prove this determinant is positive?
@user23765 : This formular doesn't evaluate log(-1), it evaluates the log auf the product of 2 exponentials, which is simply $X + Y$ in the commuting case but more complicated in the non-commuting case. But the result is still real, if X and Y are real.
May
3
comment How to prove this determinant is positive?
@user23765 : See en.wikipedia.org/wiki/… : From the Baker–Campbell–Hausdorff formula it follows, that a real M exists.
May
3
revised How to prove this determinant is positive?
added 1 character in body
May
3
answered How to prove this determinant is positive?
May
3
comment How to prove this determinant is positive?
I think T is not connected to the identity (the log T I calculated is complex). This means that it is no counterexample.
May
1
comment Does this recursion preserve monotonicity? (was: A nice problem that I am unable to solve)
$f_n(x)$ seems to be concave on $[0,1]$, or not ?
Feb
25
awarded  Necromancer
Jan
6
comment Examples of potentials for which Schrödinger equation lacks discrete points in continuous spectrum
Maybe this makes sense in the rigged Hilbert space approach ?