# Questions tagged [traces]

For questions involving the trace of a square matrix, i.e. the sum of the elements on the main diagonal.

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### Necessary and sufficient conditions on kernels of trace-class operators

Question: Let $K \in L^2(R^n\times R^n)$. Are "explicit" necessary and sufficient conditions known such that $K$ is the kernel of some trace-class operator $A \in TC(L^2(R^n))$? We know that ...
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### Loss function for matrix with fixed trace

I have a matrix $P \in \mathbb{R}^{n \times m}$ where $n \gg m$ and each row of $P$ has norm $1$. It is not hard to see that the $P^T P$ has a fixed trace $n$. I am try to find a loss function to push ...
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### Frobenius algebras and traces of modules

$\newcommand{\Hom}{\mathscr{Hom}}$ Let $C$ be a cocomplete closed symmetric monoidal category, and the tensor product preserves colimits in each variable; Let $A$ be a commutative algebra in $C$, ...
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### Derivative of trace uniformly bounded

I have that $\boldsymbol{\Omega{(\boldsymbol{\theta})}}$ is symmetric, positive definite, and uniformly bounded where $\boldsymbol{\theta} \in \boldsymbol{\Theta}$ with $\boldsymbol{\Theta}$ a ...
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### Proof of Araki-Lieb-Thirring inequality

I would like to know a proof of Araki-Lieb-Thirring inequality. Let $H$ be a separable Hilbert space. Let $A$ and $B$ be positive and self-adjoint (not necessary bounded) operators on $H$. Let $BAB$ ...
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### Is the secondary Euler characteristic a categorical trace?

Context: The ordinary Euler characteristic of a complex (satisfying appropriate finiteness conditions so that all cohomology groups are finite-dimensional over some field ''k'', say, and only finitely ...
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### Why trace is more natural than (preferred to) determinant for smooth map $f:M\to N$?

Cross-post from MSE. For a continuous map $f:(M,g)\to (N,h)$, between Riemannian manifolds $(M,g)$ and $(N,h)$ we can pullback $h$ by $f$. Most experts take the trace from this new tensor and work ...
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### Closed paths, closed trails and traces

Let $A$ be the adjacency matrix of a (non-oriented) graph $\Gamma$. Then $\textrm{Tr} A^k$ equals both the sum $\sum_i \lambda_i^k$ of $k$th powers of eigenvalues of $A$, on the one hand, and the ...
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### Why is the relative trace of Sobolev norms finite?

I am reading the 2009 Paper on Effective Equidistribution by Einsiedler, Margulis and Venkatesh (EMV). I do not understand Section 5.3 on the proof of (3.10). They want to prove that the relative ...
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### Can we extend c.p. normal maps on a finite von Neumann algebra $M$ to $L_0(M)_+$?

Suppose that $M$ is a von Neumann algebra with a finite, normal, faithful trace $\tau$. Let $T\colon M\to M$ be a completely positive, normal map. Can $T$ be extended to a `positively linear map' ...
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Let $\mathbb{F}_{q^\ell}/\mathbb{F}_{q}$ be the extension of finite filed $\mathbb{F}_{q}$, where $\ell$ be a odd prime and $(\neq q)$. Take $\zeta\in\mathbb{F}_{q^\ell}$. Does there exist different $... 0answers 144 views ### Traces in finite extensions of integrally closed domains$\def\fp{\mathfrak{p}}\def\fq{\mathfrak{q}}$I'm looking for a reference for the following commutative algebra fact. Let$A$be an integrally closed integral domain, with field of fractions$K$. Let$...
Let $F$ be a finite extension of $\mathbb{Q}_p$ (p is prime) and $K/F$ be a unramified extension of prime degree $\ell (\neq p)$. Denote $\mu_K$ be the group of roots of unity in $K.$ Does there exist ...