MFF UK

Open problems

The problems given below are open, at least for me. If someone finds a solution, I would be very interested in that. Therefore, the first person who provides me a correct complete solution (preferrably by e-mail - see the homepage) may be awarded by a prize (a beer, a wine, an apple juice or something similar). The prize may be received only in the Czech Republic.

Topological problems on Valdivia compacta


Definition. Let K be a compact Hausdorff space.

  • A subset A⊂ K is said to be a Σ-subset of K if there is a homeomorphic injection h:K→RΓ such that
              A={x∈ K; {γ∈Γ ; h(x)(γ)≠0} is countable}.
  • K is said to be Valdivia if it has a dense Σ-subset.
  • K is said to be Corson if it is a Σ-subset of itself.


Definition. A topological space is said to be scattered if any its nonempty subset has a relatively isolated point.


Problem V1. Is there a scattered Valdivia compact space containing a homeomorphic copy of the ordinal segment [0,ω2]?


Problem V2. Let K and L be nonempty compact spaces such that their cartesian product K×L is Valdivia. Are K and L Valdivia, too?
Remark: Yes, if each one of them has at least one Gδ-point (easy) or if at least one of them has a dense set of Gδ-points (not easy).


Problem V3. Let K be a Valdivia compact space and h:K→K a homeomorphism such that h○h=id. Is there a dense Σ-subset of K which is invariant for h?
Remark. Note that if A is a dense Σ-subset of K, then so is h(A); hence the answer is positive if there is only one dense Σ-subset (for example if K has a dense set of Gδ-points). A concrete space for which the asnwer is not clear is the Tychonoff cube [0,1]Γ for uncountable Γ.

Functional-analytic problems on Valdivia compacta and related Banach spaces


Definition. Let X be a Banach space.

  • A subspace S⊂ X* is said to be a Σ-subspace of X* if there is M⊂ X linearly dense (i.e., whose linear span is dense in X) such that
              S={x*∈ X*; {x∈M ; x*(x)≠0} is countable}.
  • X is said to be 1-Plichko if X* admits a 1-norming Σ-subspace.
  • X is said to be Plichko if X* admits a norming Σ-subspace.
  • X is said to be weakly Lindelöf determined if X* is a Σ-subspace of itself.


Problem P1. Let X be a complex Banach space. Denote by XR its real version, i.e., the same Banach space considered as a real space. Assume that XR is 1-Plichko. Is X 1-Plichko, too?
Remark. Yes, if X=C(K), the space of continuous complex-valued functions on a compact space, equipped with the sup-norm.


Problem P2. Let X=l1(Γ) where Γ is an uncountable set. (This space is considered with the standard norm ||x||=Σγ∈Γ|x(γ)|.) Is any subspace of X 1-Plichko?
Remark. Yes for subspaces of codimension one (at least in the real case); the question is open even for subspaces of codimension two, see the following problem.


Problem P3. Let X=l1([0,π/2]) and Y={x∈ X; Σt∈[0,π/2]x(t)cos(t)=0 and Σt∈[0,π/2]x(t)sin(t)=0} ( ={sin,cos}⊥). Is the dual unit ball of Y, in its weak* topology, a Valdivia compact space?
Remark. There are three possible interesting answers:
a) No.
b) Yes, and there is a dense Σ-subset which is absolutely convex. In this case Y is 1-Plichko.
c) Yes, but no dense Σ-subset is absolutely convex.

Problems on homeomorphisms


Problem H1. Is there an onto homeomorphism h:[0,1]N →[-1,1]N such that ||h(x)||∞=||x||∞ for each x∈[0,1]N?


Problem H2. Let 1<p<q<∞. Are the sequence spaces lp and lq, equipped with their weak topologies, homeomorphic?
Remark. The unit balls are weakly homeomorphic by the Mazur map. This map extends to a homeomorphism of whole spaces equipped with the pointwise convergence topology.


Problem H3. Are there two Banach spaces which are not isomorphic but which are weakly homeomorphic?

Other problems


Problem O1. Let A be a commutative unital Banach algebra, Δ(A) its spectrum (i.e., the multiplicative functionals equipped with the weak* topology), y↦ŷ be the Gelfand transform. Assume that K⊂ Δ(A) is a compact subset such that for each y∈A we have ŷ(K)=σ(y) (where σ(y) denotes the spectrum of y). Is necessarily K=Δ(A)?
Remark. If K is not compact, the answer is negative. Take, for example, A=C([0,ω1]), the space of continuous functions on the ordinal interval [0,ω1], and K=[0,ω1) (more precisely, the respective Dirac measures).