Kategorie-Archive: Open Problems

Scottish Book Problem 166

Von | Oktober 7, 2020

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 »

Scottish Book Problem 155

Von | Oktober 7, 2020

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 »

Purely infinite rings and C*-algebras

Von | September 23, 2020

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 »

Liftable normal elements

Von | September 22, 2020

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 »

Simple, Z-stable, projectionless C*-algebras

Von | September 22, 2020

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 »

Automorphisms of the Calkin algebra

Von | Juni 20, 2020

Does the Calkin algebra admit an automorphism that induces the flip on ? Background/Motivation: Let be a separable, infinite-dimensional Hilbert space. The Calkin algebra is the quotient of the bounded, linear operators on by the closed, two-sided ideal of compact operators. The important problem of whether the Calkin algebra has outer automorphisms was eventually shown… Weiterlesen »