In the 1920s, von Neumann introduced the notion of amenability for groups, and he showed that a group is nonamenable whenever it contains the free group . The question of whether this characterizes (non)amenability became known as von Neumann’s problem, and it was finally answered negatively by Olshanskii in 1980: There exist nonamenable groups that do contain no subgroup isomorphic to .
The group von Neumann algebra is the von Neumann subalgebra of bounded operators on generated by the left regular representation of on . There is a notion of amenability for von Neumann algebras, and Connes showed that is amenable if and only if is. Further, if is a subgroup of , then naturally is a sub-von Neumann algebra of . Thus, if contains , then contains the free group factor as a sub-von Neumann algebra. The analog of von Neumann’s problem in this setting remains open:
Question: If is a nonamenable group, does contain ?
The C*-algebraic version of von Neumann’s problem also remains open:
Question: If is a nonamenable group, does the reduced group C*-algebra contain ?
Background: A (discrete) group is said to be amenable if it admits a finitely additive, left invariant probability measure. To quote Brown-Ozawa, there are approximately different characterizations of amenability for groups .
It is known that every nuclear C*-algebra is exact, and that exactness passes to sub-C*-algebras. It follows that every sub-C*-algebra of a nuclear C*-algebra is exact. For separable C*-algebras, the converse holds. In fact, Kirchberg’s -embedding theorem shows that a separable C*-algebra is exact if and only if it embeds into the Cuntz algebra (which is nuclear), Theorem 6.3.11 in .
At the Kirchberg Memorial Conference in Münster, July 2023, Simon Wassermann asked if this result can be generalized to the nonseparable case:
Question: Does every (nonseparable) exact C*-algebra embed into a nuclear C*-algebra?
Does there exist a purely infinite C*-algebra that admits a tracial weight taking a finite, nonzero value?
Here a C*-algebra is purely infinite if every element is properly infinite ( is Cuntz subequivalent to in ). This notion was introduced and studied by Kirchberg-Rørdam in  and . Further, a weight on a C*-algebra is a map that is additive and satisfies for all and . A weight is tracial if for all .
If is a (not necessarily closed) two-sided ideal that is strongly invariant (for , we have if and only if ), then the map given by if and otherwise, is a tracial weight. Note that these tracial weights are trivial in the sense that they only take the values and .
The question is if there exists a nontrivial tracial weight on a purely infinite C*-algebra.
It is well-known that every lower-semicontinuous tracial weight on a purely infinite C*-algebra is trivial. In particular, purely infinite C*-algebras do not admit tracial states. The question is about tracial weights that are not lower-semicontinuous. Sometimes, such tracial weights are called singular traces.
Question: Is every element of trace zero in a factor a commutator?
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 . 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  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
that is, an element is a commutator if and only if its image in the simple C*-algebra (if and is separable, then this is the Calkin algebra ) is not a nonzero scalar multiple of the identity. This was shown by Halpern in . In particular, is neither closed, nor a subspace, but every element in is a sum of two commutators and thus .
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 . In particular, every nilpotent element in is a commutator, and every normal element with vanishing trace and with atomic spectral measure is a commutator.
Question: Are maximal ideals in C*-algebras closed?
Let be a C*-algebra. By an ideal in we mean a two-sided ideal that is not necessarily closed. We say that is a maximal ideal if the only ideals satisfying are and . An ideal is proper if .
If is unital, then every maximal ideal is closed. Indeed, in this case, using that a proper ideal does not contain any invertible element of , and using that the set of invertible elements is open, we see that if is a proper ideal, then so is its closure . Thus, if is maximal, then , and is closed.
Further, it is known that every maximal left ideal in a C*-algebra is closed; see for example Proposition 3.5 in . (This even holds in Banach algebras admitting a bounded approximate unit.) In commutative C*-algebras, every maximal ideal is also a maximal left ideal and therefore closed.
Assume is a non-closed, maximal ideal in a C*-algebra (if it exists). Then is dense and therefore contains the Pederson ideal of . Further, the quotient is a simple -algebra. It is easy to see that is radical, that is, , because has no nonzero maximal (modular) left ideals. One can show that is hereditary (if in , and belongs to , then so does ), strongly invariant (if , then ), and invariant under powers: if and , then . For the arguments see this post in mathoverflow.
In , Blackadar and Handelman made two conjectures:
Conjecture 1: (Below Theorem I.2.4 in ) Let be a unital C*-algebra. Then the set of lower-semicontinuous dimension functions is dense in the set of dimension functions.
Conjecture 2: (Below Theorem II.4.4 in ) Let be a unital C*-algebra. Then the compact, convex set is a Choquet simplex.
Here, a dimension function on a unital C*-algebra is a map that associates to every matrix over a positive number and satisfying properties that generalize the classical properties of the rank of complex matrices:
If and are orthogonal matrices (that is, ), then .
If is Cuntz-dominated by (that is, there exist sequences and such that ), then .
Such a dimension function is said to be lower-semicontinuous if it is lower-semicontinuous with respect to the norm-topology, that is, whenever is a sequence in converging to some , then . We equip with the topology of pointwise convergence. This gives the structure of a compact, convex set. We note that the subset is usually not closed (indeed, the conjecture is that it is dense). There is another (natural) topology on giving it the structure of a compact, convex set that is even a Choquet simplex. Here, a compact, convex set is a Choquet simplex if the set of continuous, affine functions satisfies Riesz interpolation, that is, given satisfying for there exists such that for . Choquet simplices have the property that every element can be represented in a unique way by a boundary measure. Important examples of Choquet simplices are the Bauer simplices: Given a compact, Hausdorff space , the set of positive, Borel probability measures on is a Choquet simplex with boundary , and .
If is a commutative C*-algebra, then naturally corresponds to the set of finitely-additive probability measures on , while naturally corresponds to the set of (-additive) probability measures on . For a general C*-algebra , we therefore consider the dimension functions on as „noncommutative, finitely-additive probability measures“, and similarly are the „noncommutative probability measures“ on . It is easy to see that the probability measures on a compact, Hausdorff space are dense in the set of finitely-additive probability measures (see the proof of Theorem I.2.4 in ), and it is a classical result that the finitely-additive probability measures on a compact, Hausdorff space form a Choquet simplex. The Blackadar-Handelman conjectures predict that theses results generalize to the noncommutative setting.
The first Blackadar-Handelman conjecture has been verified in the following cases: if is commutative (Theorem I.2.4 in ); if is simple, exact, stably finite and has strict comparison of positive elements (Theorem B and 6.4, and Remark 6.5 in ); if has finite radius of comparison (Theorem 3.3 in ).
The second Blackadar-Handelman conjecture has been verified in the following cases: if is commutative; if is simple, exact, stably finite and -stable (Theorem B in ); if has real rank zero and stable rank one (Corollary 4.4 in ); in , the assumption of real rank zero was removed from the result in , thus verifying the second Blackadar-Handelman conjecture for all C*-algebras of stable rank one.
With view to the results in  it is natural to ask if the first Blackadar-Handelmann conjecture can be verified for all C*-algebras of stable rank one.
N.P. Brown, F. Perera, A.S. Toms, The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras, Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal). (2008). https://doi.org/10.1515/crelle.2008.062.
K. De Silva, A note on two Conjectures on Dimension funcitons of C*-algebras, ArXiv. (2016) 1601.03475.
Let be a unital, simple C*-algebra of stable rank one. Does there exist a commutative sub-C*-algebra such that for every lower-semicontinuous function there exists an open subset such that for ? Here, denotes the Choquet simplex of normalized -quasitraces on (if is exact, then this is just the Choquet simplex of tracial states on ), and denotes the probability measure on induced by the restriction of to .
More specifically, one may ask if this is always the case for a Cartan subalgebra of .
Question: Is every separable C*-algebra an inductive limit of semiprojective C*-algebras?
This question was first raised by Blackadar in . If we think of C*-algebras as noncommutative topological spaces, then semiprojective C*-algebras are noncommutative absolute neighborhood retracts (ANRs). It is a classical result from shape theory that every metrizable space is homeomorphic to an inverse limit of metrizable ANRs , that is, . This means that is isomorphic to an inductive limit of the , that is, . The question is if the noncommutative analog of this result also holds.
It has been verified for many classes of C*-algebras that they are inductive limits of semiprojective C*-algebras. For example, Enders showed in  that every UCT-Kirchberg algebra is an inductive limit of semiprojective C*-algebras. In , it was shown that the class of C*-algebras that are inductive limits of semiprojective C*-algebras is closed under shape domination, and in particular under homotopy equivalence. One deduces, for example, that if is a contractible, compact, metrizable space, and if is an inducitve limit of semiprojective C*-algebras, then so is . It also follows that every contractible C*-algebra is an inductive limit of semiprojective C*-algebras – in fact, even of projective C*-algebras, as was shown in .
The commutative C*-algebra is probably the easiest C*-algebra where it is currently unknown if it is an inductive limit of semiprojective C*-algebras. Equivalently, it is unknown if is an inductive limit of semiprojective C*-algebras. By Example 4.6 in , we know that the stabilization is an inductive limit of semiprojective C*-algebras.
Question: Let be a C*-algebra that is complemented in its bidual by a *-homomorphism, that is, there exists a *-homomorphism such that for all . Is a von Neumann algebra?
The converse is true: Let be a von Neumann algebra. Then has a (unique) isometric predual . Let be the natural inclusion of the Banach space in its bidual. We naturally identify the dual of with , and the dual of with . Then the transpose is a *-homomorphism that complements in .
Question: Let be a -factor. Is the unitary group contractible when equipped with the strong operator topology?
Background/Motivation: By Kuiper’s theorem, if is an infinite-dimensional Hilbert space, then the unitary group of the -factor is contractible in the norm topology. This was generalized by Breuer  to (certain) and von Neumann algebras, and eventually Brüning-Willgerodt  showed that the unitary group of every properly infinite von Neumann algebra is contractible in the norm topology. This is no longer true for finite von Neumann algebras: The unitary group of a finite matrix algebra is not contractible. Similarly, the unitary group of a factor is not contractible in the norm topology since its fundamental group does not vanish – in fact, it was shown in  that for every factor .
As noted in the introduction of , the unitary group of every properly infinite von Neumann algebra is also contractible in the strong operator topology. This naturally leads to the above question, which was considered by Popa-Takesaki in . They showed that is contractible in the strong operator topology if is a separable factor such that the associated factor admits a trace scaling one-parameter group of automorphisms. This includes all McDuff factors ( is McDuff if for the hyperfinite factor ) and all factors that satisfy , where is the group von Neumann algebra of the free group on infinitely many generators.