This is the ongoing story about our understanding of a formula for the Fuglede-Kadison determinant of a matrix-valued semicircle when a constant matrix is added, and consequences of this for our understanding of the Brown measure for general matrix-valued circular versions.
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Doing Mathematics with AI — but How?
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No comments on Doing Mathematics with AI — but How?AI is revolutionizing mathematics, and it will not go away. So somehow we have to arrange ourselves with it and find our way of doing mathematics under these changed conditions. I am of course also fascinated by asking AI (ChatGPT in my case) to solve all my problems. And sometimes there are indeed answers which…
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A Story of the Determinant of Operator-Valued Semicircular Elements
Operator-valued semicircular elements are the powerhouse of free probability and random matrix theory. In particular, the intrinsic freeness principle typically replaces random matrices by operator-valued semicircular elements and shows that, in many respects, the former are close to the latter. Thus it is important to understand as much as possible about operator-valued semicirculars. An operator-valued…
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The Story of the Determinant: Why the Determinant Formula is True
post by Vladislav Kargin In his post Roland asked whether the Mai–Speicher formula for the Fuglede–Kadison determinant of a matrix semicircle survives when a constant matrix is added. With Sa=a⊗1+∑jAj⊗sjS_a=a\otimes1+\sum_jA_j\otimes s_j and η(X)=∑jAjXAj\eta(X)=\sum_jA_jXA_j, the proposed formula is where the infimum is over positive matrices bb with 𝒞b−η⪰0\mathcal C_b-\eta\succeq0. It is true, for every real zz.…
