Given elements and in a ring, the element is called an (additive) commutator. Given a C*-algebra , we use to denote the additive subgroup (equivalently, the linear subspace) generated by the set of additive commutators in . Note that is not necessarily a closed subspace. Given additive subgroups and , it is customary to use… Weiterlesen »
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… Weiterlesen »
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… Weiterlesen »
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 [1] and [2]. Further, a weight on a C*-algebra is… Weiterlesen »
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 ,… Weiterlesen »
Surprisingly, the following question is open: 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,… Weiterlesen »
In [1], Blackadar and Handelman made two conjectures: Conjecture 1: (Below Theorem I.2.4 in [1]) 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 [1]) Let be a unital C*-algebra. Then the compact, convex set is a Choquet… Weiterlesen »
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 ),… Weiterlesen »
Question: Is every separable C*-algebra an inductive limit of semiprojective C*-algebras? This question was first raised by Blackadar in [1]. 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… Weiterlesen »
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… Weiterlesen »
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