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How to prove the density relation $\overline{\text{Span}\{\varphi_{x,y}:x ,y\in H\}}^{\Vert\cdot\Vert} = N_*$ for any von Newmann algebra $N$?

$\def\tr{\operatorname{Tr}}$ It is known that all normal functionals are of the form $T\longmapsto \tr(AT)$ for $A$ trace-class. "Trace-class" means that $\tr(|A|)<\infty$. The ...
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Idempotents with arbitrarily large norms in Banach algebras

There are also natural commutative examples. Take $\ell_p(\mathbb N$) with pointwise multiplication. Consider a sequence $q$ that has $n$ ones and all other entries are zero. Clearly, $qq=q$ and $\|q\|...
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Maximal Ideal of non-unital Banach algebra

Let $M$ be a closed maximal ideal of $\ell^1.$ We will show that $M\subset M_n=\{x\in \ell^1\,:\,x_n=0\}$ for some $n.$ Assume that for every $n$ there is $y^{(n)}\in M$ such that $y^{(n)}_n\neq 0.$ ...

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