Question: Is every element of trace zero in a
factor a commutator?
Update (August 2024): A positive answer to the questions was recently given In the following article:
S. Wen, J. Fang, Z. Yao. A stronger version of Dixmier’s averaging theorem and some applications. J. Funct. Anal. 287 (2024).
Background on commutators: It is a classical question to determine the elements in an algebra (over a field
) that are commutators, that is, of the form
for some
. A related (and often easier) question is to determine the commutator subspace
, which is defined as the linear subspace of
generated by the set
of commutators. (Warning: In the literature,
is often used to denote
.)
A linear functional is said to be tracial if
for all
; equivalently,
vanishes on
. It follows that an element in
belongs to
if and only if it vanishes under every tracial functional.
Thus, a commutator has to vanish under every tracial functional. Another obstruction occurs in the normed setting, since the unit of a normed algebra is not a commutator (although it may be a sum of two commutators). It follows that nonzero scalar multiples of the unit are not commutators either. A simple proof was given by Wielandt in [1]. Since commutators are mapped to commutators in quotients, we obtain according obstructions quotients by closed, two-sided ideals.
To summarize, if is a unital Banach algebra, and
, then:
for every tracial functional
.
- for every closed ideal
(equivalently: every maximal ideal
), we do not have
for some
.
We will see below that these are the only obstructions for an element to be a commutator in the case of properly infinite factors, as well as type factors.
Let us specialize to the case that is a unital C*-algebra. In this case, one can consider the space
of tracial states on
. Since every tracial state is continuous, it vanishes on the closure of
. Further, it follows from Theorem 5 in [2] that
In many cases, is a closed subspace. If it is not closed, then the C*-algebra admits non-continuous tracial functionals.
Commutators in von Neumann factors. The set of commutators have been completely characterized in properly infinite factors, and in factors of type . After presenting the results for these cases, we discuss some partial results for the case of a
factor.
Type : The matrix algebra
is simple and has a unique tracial state
. A matrix
is a commutator if and only if
. In particular, the set
of commutators agrees with the commutator subspace. More generally, this holds for every matrix algebra
over a field, which was shown by Shoda in 1936 for the case of characteristic zero, and by Albert-Muckenhoupt in 1957 for arbitrary characteristic.
Types ,
and
: Let
be a properly infinite factor. Then
has no tracial states, and
has a unique maximal ideal
. (If
for some infinite-dimensional Hilbert space, then
is the closure of the ideal of operators whose closed range have dimension strictly less than that of
. In particular, if
is separable, then
is the ideal
of compact operators on
.) Then





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Type : Let
be a
factor. Then
is simple and has a unique tracial state
. It is expected that
. Partial results in this direction have been obtained by Dykema-Skripka in [4]. In particular, every nilpotent element in
is a commutator, and every normal element with vanishing trace and with atomic spectral measure is a commutator. [Update: As noted above, in a recent article by S. Wen, J. Fang, and Z. Yao it is shown that every trace-zero element in a
factor is a commutator.]
- [1]H. Wielandt, �ber die Unbeschr�nktheit der Operatoren der Quantenmechanik, Math. Ann. (1949) 21–21. https://doi.org/10.1007/bf01329611.
- [2]N. Ozawa, Dixmier approximation and symmetric amenability for C*-algebras, J. Math. Sci. Univ. Tokyo. 20 (2013) 349–374.
- [3]H. Halpern, Commutators in properly infinite von Neumann algebras, Trans. Amer. Math. Soc. (1969) 55–73. https://doi.org/10.1090/s0002-9947-1969-0251546-8.
- [4]K. Dykema, A. Skripka, On single commutators in II
–factors, Proc. Amer. Math. Soc. (2012) 931–940. https://doi.org/10.1090/s0002-9939-2011-10953-5.