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 Apr 24 comment A curious eigenvalue inequality With the notation I used in my previous comment the conjectured inequality $tr((U^*AU+B)^2) \le tr((A+B)^2)$ is then equivalent to $tr((R B^2 R)^{1/2} B) \le tr((B R^2 B)^{1/2} B)$ . Apr 24 comment A curious eigenvalue inequality If we set $R=B^{-1} A U$ then $R$ is Hermitian and we have to show : $\rho((R B^2 R)^{1/2} + B)\le \rho((B R^2 B)^{1/2} + B)$ . Apr 23 comment A curious eigenvalue inequality Could there exist a completely positive map that maps $A+B$ to $U^* A U + B$ and the identity to itself ? If yes this would be the solution. Mar 23 comment A log inequality for positive definite trace-one matrices Can you also prove the last inequality of the question ? Mar 2 comment How to prove this determinant is positive-II? What one needs is $A_i^T J + J A_i \ge 0$ but I haven't checked whether this is a generalization. Feb 24 answered How to prove this determinant is positive-II? Feb 5 comment How to prove this determinant is positive-II? The quadratic form is $q(x,y) = x^T J y$ where $J = diag(I_n,-I_n)$ and the semi group is given bei the real S with $q(Sx, Sx) \ge q(x,x)$ for all $x$ . Now $\frac{d}{dt} q(e^{t A_i} x,e^{t A_i} x) = 2 x^T diag (E_i,F_i) x \ge 0$ at $t = 0$, therefore the $e^{A_i}$ are in the semi group. Feb 4 comment How to prove this determinant is positive-II? The $e^{A_i}$ increase the quadratic form that the split orthogonal group leaves invariant . So maybe this is the generalization of Part I : The split orthogonal group is replaced by the semi group that increases the quadratic form. Feb 3 comment How to prove this determinant is positive-II? to the warm up : $det(I_n +\prod_i e^{-F_i}) \ge 0$ because $\Vert\prod_i e^{-F_i}\Vert \le 1$ and $F_i$ real . Then also $det(I_n +\prod_i e^{E_i}) = det(\prod_i e^{E_i}) det(I_n +(\prod_i e^{E_i})^{-1}) \ge 0$ . Jan 24 comment Resolvent of the operator Calculate the eigenfunctions : T is the Hamiltonian of an electron in a constant magnetic field perpendicular to the plane. The eigenfunctions are the elements of the Landau levels. Jan 23 comment States in C*-algebras and their origin in physics? Within ZFC there are states that are not represented by a density matrix. Jan 13 comment Does quantum mechanics ever really quantize classical mechanics? And what is the "sample space" in quantum mechanics ? For a sample space you need a Hilbert space basis, but what basis should one use ? According to decoherence theory there are "robust" pointer states. But you have to split the Hilbert space into system and environment and integrate out the environment degrees of freedom if you want to see this. Jan 11 comment Does quantum mechanics ever really quantize classical mechanics? The problem with this model is that superpositions of states last forever, since quantum mechanics is a linear theory, but in the classical world we don't see superpositions of macroscopic states. The cure for this problem could be decoherence theory (already mentioned in the question). Nov 25 comment Number theory and physics Nov 7 awarded Nice Question Nov 7 comment Unital $C^{*}$ algebras which all elements have path connected spectrum Example of a connected Cā algebra : See B.E. Blackadar. A simple unital projectionless Cā-algebra. J. Operator Theory, 5:63ā71, 1981. By the comments of Hannes Thiel and Sam Evington this algebra is an example of a connected C* algebra. But is it also path connected (remark by Ali Taghavi) ? Sep 12 awarded Nice Answer Aug 19 awarded Yearling Jul 8 comment Quantum Fields and Infinite Tensor Products 'tiny separable subset' = GNS construction ? Jun 11 accepted Do non-normal states exist in the Solovay model?