Large cardinals and continuity of coordinate functionals of filter bases in Banach spaces

Assuming the existence of certain large cardinal numbers, we prove that for every projective filter $\mathscr F$ over the set of natural numbers, $\mathscr{F}$-bases in Banach spaces have continuous coordinate functionals. In particular, this applies to the filter of statistical convergence, thereby we solve a problem by V. Kadets (at least under the presence of certain large cardinals). In this setting, we recover also a result of Kochanek who proved continuity of coordinate functionals for countably generated filters (Studia Math., 2012).Comment: 10 p

The Point in Weak Semiprojectivity and AANR Compacta

We initiate the study of pointed approximative absolute neighborhood retracts. Our motivation is to generate examples of C*-algebras that behave in unexpected ways with respect to weak semiprojectivity. We consider both weak semiprojectivity (WSP) and weak semiprojectivity with respect to the class of unital C*-algebras (WSP1). For a non-unital C*-algebra, these are different properties. One example shows a C*-algebra can fail to be WSP while its unitization is WSP. Another example shows WSP1 is not closed under direct sums.Comment: Corrected the statement of Theorem 4.16(b

On Polish Groups of Finite Type

Sorin Popa initiated the study of Polish groups which are embeddable into the unitary group of a separable finite von Neumann algebra. Such groups are called of finite type. We give necessary and sufficient conditions for Polish groups to be of finite type, and construct exmaples of such groups from semifinite von Neumann algebras. We also discuss permanence properties of finite type groups under various algebraic operations. Finally we close the paper with some questions concerning Polish groups of finite type.Comment: 20 page

Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees

We study the computational difficulty of the problem of finding fixed points of nonexpansive mappings in uniformly convex Banach spaces. We show that the fixed point sets of computable nonexpansive self-maps of a nonempty, computably weakly closed, convex and bounded subset of a computable real Hilbert space are precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A uniform version of this result allows us to determine the Weihrauch degree of the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is equivalent to a closed choice principle, which receives as input a closed, convex and bounded set via negative information in the weak topology and outputs a point in the set, represented in the strong topology. While in finite dimensional uniformly convex Banach spaces, computable nonexpansive mappings always have computable fixed points, on the unit ball in infinite-dimensional separable Hilbert space the Browder-Goehde-Kirk theorem becomes Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive mappings may not have any computable fixed points in infinite dimension. We also study the computational difficulty of the problem of finding rates of convergence for a large class of fixed point iterations, which generalise both Halpern- and Mann-iterations, and prove that the problem of finding rates of convergence already on the unit interval is equivalent to the limit operator.Comment: 44 page