# 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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### A bounded extension operator

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary $\partial\Omega$. Consider the harmonic extension operator $E :L^2(\partial \Omega) \rightarrow H^{1/2}(\Omega)$ which assigns ...
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### Tensor contraction (vector-valued trace) on $\bigotimes_{i=1}^k\mathcal L(E_{i-1},E_i)$

If $E_i$ is a $\mathbb R$-vector space, then the vector-valued trace $\operatorname{tr}_{E_1}:(E_2\otimes E_1^\ast)\otimes(E_1\otimes E_0)\to E_1\otimes E_0$ (or tensor contraction) is the ...
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### trace expressions for matrix quadratic forms [migrated]

Let $A$ be a real symmetric $n \times n$ matrix. Which quadratic forms in $A$ can be written in trace form? Such an expression would naturally generalise some invariant random matrix ensembles. ...
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### Computational complexity of computing the trace of a matrix product under some structure

I have two problems related to computing some trace, and some (possibly suboptimal) answers. My question is about a potential more efficient algorithm for each one. (More interested in an answer to ...
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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' ...