Let x be an element in a C\*-algebra A, is it true that if x approximately commute with every element in A, then x is near the centre of A? More precisely, I want to know whether the following is true： Let x be an element in a C\*-algebra A with norm 1. Then for any $\epsilon>0$, there exist a $\delta>0$ such that the following holds: $\forall y\in A,||y||=1,||xy-yx||<\delta\Rightarrow dist（x, Centre(A)）<\epsilon$. This is true if A is the matrix algebra, but I was wondering whether it can be generalized to any C\*-algebra



Perturbation in C*-Algebra
The present question seems very close to the following one, which has been well-studied in the literature:



> 
> Let $A$ be a C\*-algebra. Does there exist a constant $K>0$ such that, for each $x\in A$, the distance from $x$ to $Z(A)$ is bounded above by
>  $$ K \sup\{ \Vert xy-yx \Vert \;:\; y\in A, \Vert y\Vert \leq 1\} ?$$
> 
> 
> 


A useful discussion of what is known can be found in Section 1 of 


[MR2274022]
R. J. Archbold, D. W. B. Somerset, Measuring noncommutativity in C\*-algebras.
JFA 242 (2007) no. 1, 247--271


According to this paper, the answer turns out to be "yes" if $A$ is a von Neumann algebra, or one of several natural classes of C\*-algebra; but the answer is "no" in general, by an example in


[MR0215111]
R. V. Kadison, E. C. Lance, J. R. Ringrose, Derivations and automorphisms of operator algebras. II.
JFA 1 1967 204–221. 


I haven't been able to read the Kadison-Lance-Ringrose paper, but a similar example seems to be discussed in Example 3.3 of


[MR0482236] 
R. J. Archbold, On the norm of an inner derivation of a C\*-algebra.
Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 2, 273–291.


