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.
This is one of the few problems from the Scottish book that are still open. In slightly modernized form, and correcting the typo (in the book, and should be switched in the last sentence) the problem is:
Let be a topological manifold, and let be a continuous function. Let denote the subgroup of homeomorphisms that satisfy . Let be another manifold that is not homeomorphic to . Does there exist a continuous function such that is not isomorphic to ?
This is one of the few problems from the Scottish book that are still open. In modern terminology, the problem is:
Let and be Banach spaces, and let be a bijective map with the following property: For every there exists such that for the sphere , the restriction is isometric. Does it follow that is isometric?
It is noted in the Scottish Book that the answer is „yes“ whenever is continuous, which is automatic if is finite-dimensional, or if has the property that for any two elements satisfying and there exists such that .