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 A note on local commutators in division rings with involution Bull. Korean Math. Soc.Published online 2019 Mar 04 Mai Hoang Bien University of Science, VNUHCM Abstract : In this paper, we consider a conjecture of I. N. Herstein for local commutators of symmetric elements and unitary elements of division rings. For example, we show that if $D$ is a finite dimensional division ring with involution $\star$ and if $a\in D^*=D\backslash\{0\}$ such that local commutators $axa^{-1}x^{-1}$ at $a$ are radical over the center $Z(D)$ of $D$ for every $x\in D^*$ with $x^\star=x$, then either $a\in Z(D)$ or $\dim_{Z(D)}D= 4$. Keywords : division ring; involution; symmetric element; unitary element; local commutator MSC numbers : 16K20; 16R50 Full-Text :