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 »
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 [1] to (certain) and von Neumann algebras, and eventually Brüning-Willgerodt… Weiterlesen »
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… Weiterlesen »
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… Weiterlesen »
Given groups and , let us say that the pair has property (*) if any two injective homomorphisms are conjugate if (and only if) they are pointwise conjugate. Problem 1: Describe the class of groups such that has (*) for every group . Problem 2: Describe the class of groups such that has (*) for… Weiterlesen »
Question 1: If is a *-subalgebra of bounded linear operators on a separable Hilbert space such that is purely infinite as a ring, is the norm-closure purely infinite as a C*-algebra? This is Problem 8.4 in [1]. As noted in Problem 8.6 in [1], this is even unclear if is a unital, simple, purely infinite… Weiterlesen »
Given a C*-algebra and a closed, two-sided ideal , is the image of the normal elements in under the quotient map a closed subset of ? Equivalently, if is a sequence of normal elements in that converge to , and if each admits a normal lift in , does admit a normal lift? This question… Weiterlesen »
Do simple, -stable, stably projectionless C*-algebras have stable rank one? Definitions: A C*-algebra is said to be -stable if it tensorially absorbs the Jiang-Su algebra , that is, . Further, a simple C*-algebra is projectionless if it contains no nonzero projections, and it is stably projectionless if is projectionless. (In the nonsimple case, one should… Weiterlesen »
(Based on the talk „Thoughts on the classification problem for amenable C*-algebras“ of George Elliott, 30. June 2020, at the Zagreb Workshop on Operator Theory.) Let us consider the class of unital, separable, simple, nuclear C*-algebras. The Toms-Winter conjecture predicts that for such an algebra , the following conditions are equivalent: has finite nuclear dimension.… Weiterlesen »
Given a separable, infinite-dimensional Hilbert space , what is the real rank of the minimal tensor product ? Background/Motivation: The real rank is a noncommutative dimension theory that was introduced by Brown and Pedersen in [1]. It associates to each C*-algebra a number (its real rank) . The lowest and most interesting value is zero.… Weiterlesen »
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