For various reasons I became interested in bounds on the norm of commutators of operators. For instance, if $B(H)$ is the algebra of bounded operators on a Hilbert space, one may ask for a lower bound on $||[A,B]||$ in terms of properties of $A$ and $B$. (It is clear that one can not have simple norm estimates since for $A=1$ one has $[A,B]=0$.) One may consider questions like: Is there a subspace $\Omega$ of $B(H)$ such that $||[A,B]||\geq ||A||$ for all $A\in\Omega$ (keeping $B\in B(H)$ fixed)? More generally, I'm also interested in similar questions for the $L^p$-norm $||A||\_p=(\operatorname{tr}|A|^p)^{1/p}$. In the literature, one finds a flora of results concerning upper bounds on norms of commutators (however, for the $L^p$-norm, results are sparse). Is anyone familiar with these types of questions, concerning lower bounds? Can someone point out relevant sources?



Lower bounds for norms of commutators
This doesn't address the full generality of your question but it might at least suggest places to hunt in the literature.


Thinking of $B$ as fixed, you're asking for a lower bound on the norm of $\operatorname{ad}\_B(A)$ as $A$ varies in some subspace. If we take $A$ to range over all of $B(H)$ then MathSciNet directs me to



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> J. G. Stampfli. *The norm of a derivation.*
> [Pacific J. Math. Volume 33, Number 3 (1970), 737–747](http://projecteuclid.org/euclid.pjm/1102976839)
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in which it's proved that $\Vert \operatorname{ad}\_B \Vert$ is **equal to** twice the distance of $B$ from the subspace ${\mathbb C} I$.


One can consider similar problems for norms of inner derivations on other C\*-algebras (the von Neumann case seems apparently behaves just like $B(H)$, see L. Zsido, *The norm of a derivation in a $W^\ast$-algebra*, Proc. Amer. Math. Soc. 38 (1973), 147–150). In the unital case, it seems like a good place to find out what's known is



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> [MR2274022 (2008f:46071)](http://www.ams.org/mathscinet-getitem?mr=2274022)
>  R. J. Archbold, D. W. B. Somerset. *Measuring noncommutativity in $C^\ast$-algebras.* J. Funct. Anal. 242 (2007), no. 1, 247–271.
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