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5

(This answer expands the ideas in the comments by Mikael de la Salle and myself). There is a family of matrix norms that can be defined as follows: given $p \geq 1,k \leq n$, $$ \|A\|_{p,k} = \left(\sum_{i=1}^k \sigma_i(A)^p\right)^{1/p}, $$ where we denote by $\sigma_i(A)$ the $i$th singular value of $A$ (taken in nonincreasing order, $\sigma_1(A)\geq ...


2

Let $A$ be a square matrix of dimension $n\times n$ and consider the following norm for $1< p<\infty$: $$\|A\|_{p} = \max_{x \neq 0} \frac{\|Ax\|_p}{\|x\|_p}.$$ Let $\psi_p(x):=\big(|x_1|^{p-1}\operatorname{sign}(x_1),\ldots,|x_n|^{p-1}\operatorname{sign}(x_n)\big)$ and write $p'$ the Hölder conjugate of $p$. Then for every $x$ we have $$\max_{x\neq ...


4

The following argument seems easier, but there might be a still more fundamental one. Notice that $ \phi: A^{\sim} \to \mathbb{C} $ above is also a $ C^{*} $-algebraic homomorphism. As $ C^{*} $-algebraic homomorphisms are automatically contractive (which is a consequence of a not-too-difficult spectrum argument), we have $$ \forall (a,\lambda) \in A \times ...



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