Is there a commutative subalgebra $A\subset B(H)$ containing the 1 dimensional scalars with the following property:



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> The algebra $A$ has trivial intersection with the set of commutator elements $xy-yx$
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A subalgebra of $B(H)$ which does not contain a commutator element
From the results of


*Brown, Arlen; Pearcy, Carl*, [**Structure of commutators of operators**](https://doi.org/10.2307/1970564), Ann. Math. (2) 82, 112-127 (1965). [ZBL0131.12302](https://zbmath.org/?q=an:0131.12302).


an element of $B(H)$ is a commutator $xy-yx$ if and only if it is not of the form $\lambda I + C$ for some non-zero scalar $\lambda$ and compact $C$. So your algebra would have to contain an element of the form $\lambda I + C$ but not $C$ (unless $C=0$), and so cannot contain the scalars unless it consisted solely of the scalars, which are indeed not commutators by the classical results of [Wintner](https://mathscinet.ams.org/mathscinet-getitem?mr=20724) and [Wielandt](https://mathscinet.ams.org/mathscinet-getitem?mr=30701).


