Open problems raised during the Workshop in Analysis and Probability
Open problems raised during the Workshop in Analysis and Probability
July-August 2008
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Problem lists from previous years: 2007 | Older 1 | Older 2
The problems here were either submitted specifically for the purpose of inclusion in this list, or were taken from talks given during the Workshop in Linear Analysis and Probability.
Problem 1 (Submitted by Yun-Su Kim)
Let and be abstract operator spaces. Is dense in ?
Let and be abstract operator spaces. Is dense in ?
Problem 2 (Submitted by Bentuo Zheng)
Let and be a separable reflexive Banach space. Assume that satisfies an asymptotic lower--tree estimate, and that is a bounded linear operator that satisfies an asymptotic upper--tree estimate. Does factor through a subspace of a reflexive space with an asymptotic FDD?
Let and be a separable reflexive Banach space. Assume that satisfies an asymptotic lower--tree estimate, and that is a bounded linear operator that satisfies an asymptotic upper--tree estimate. Does factor through a subspace of a reflexive space with an asymptotic FDD?
Problem 3 (Submitted by Rachid El Harti)
Let be any non-simple -algebra. Is there a non-trivial pro--algebra structure for ?
Let be any non-simple -algebra. Is there a non-trivial pro--algebra structure for ?
Problem 4 (Submitted by Rachid El Harti)
Let be a non-trivial, non-commutative unital pro--algebra. Is there a non-unital -algebra such that is the multiplier algebra of ?
Let be a non-trivial, non-commutative unital pro--algebra. Is there a non-unital -algebra such that is the multiplier algebra of ?
Problem 5 (Submitted by Rachid El Harti)
Let be a pro--algebra that is the bounded part of the inverse limit of a system of -algebras . If each is RR0, is necessarily also RR0?
Let be a pro--algebra that is the bounded part of the inverse limit of a system of -algebras . If each is RR0, is necessarily also RR0?
Problem 6 (Submitted by Hector Salas)
Let be an infinite dimensional Hilbert space, and let denote the subclass of consisting of the hypercyclic operators on Let be compact such that each component intersects the unit circle Must there exist such that its spectrum ? More generally, for each infinite dimensional separable Banach space characterize those for which there exists with
Let be an infinite dimensional Hilbert space, and let denote the subclass of consisting of the hypercyclic operators on Let be compact such that each component intersects the unit circle Must there exist such that its spectrum ? More generally, for each infinite dimensional separable Banach space characterize those for which there exists with
Problem 7 (Submitted by Hector Salas)
Let be an infinite dimensional separable Banach space such that its dual is also separable. An operator is dual hypercyclic if (such operators exist). Identify the compact subsets of which are the spectra of dual hypercyclic operators.
Let be an infinite dimensional separable Banach space such that its dual is also separable. An operator is dual hypercyclic if (such operators exist). Identify the compact subsets of which are the spectra of dual hypercyclic operators.
Problem 8 (Submitted by Hector Salas)
Let be a topological space such that multiplication is a continuous mapping with and What are the conditions on and continuous for which has a dense orbit for each Although the question so posed is quite vague, a particularly interesting case is the infinite torus
Let be a topological space such that multiplication is a continuous mapping with and What are the conditions on and continuous for which has a dense orbit for each Although the question so posed is quite vague, a particularly interesting case is the infinite torus
Problem 9 (Submitted by Simon Cowell)
For a separable Banach space , has implies that has . Under what extra hypotheses on are they equivalent? In particular, are they equivalent provided that does not contain ?
For a separable Banach space , has implies that has . Under what extra hypotheses on are they equivalent? In particular, are they equivalent provided that does not contain ?
Problem 10 (Submitted by Deping Ye)
Let the Hilbert space to be . What is the exact Hilbert-Schmidt volume of separable states on ? What is the exact Bures volume of Separable states on ?
Let the Hilbert space to be . What is the exact Hilbert-Schmidt volume of separable states on ? What is the exact Bures volume of Separable states on ?
Problem 11 (Submitted by Stephen Dilworth)
- (a)
- For which smooth Banach spaces do the XGA and/or the DGA converge?
- (b)
- Do these algorithms converge in uniformly smooth spaces?
- (c)
- Does the XGA converge in (or even ) for , ?
- (d)
- Does the XGA converge in ( , ) when the dictionary is the Haar system?
Problem 12 (Submitted by Piotr Nowak)
Let be a finitely generated group. Knowing that the fundamental class vanishes in for of growth slower than linear, does there exist an aperiodic tiling of ? (see J.Block, S.Weinberger, JAMS 1992 5 (4) pp. 907-918. for the case const).
Let be a finitely generated group. Knowing that the fundamental class vanishes in for of growth slower than linear, does there exist an aperiodic tiling of ? (see J.Block, S.Weinberger, JAMS 1992 5 (4) pp. 907-918. for the case const).
Problem 13 (Submitted by Piotr Nowak)
Let be a finitely generated group. Is there a growth type sufficiently slow such that for any , if the fundamental class vanishes in for slower than then it vanishes in for const?
Let be a finitely generated group. Is there a growth type sufficiently slow such that for any , if the fundamental class vanishes in for slower than then it vanishes in for const?
Problem 14 (Submitted by Brett Wick)
Give an intrinsic characterization of the set of functions such that is -Carleson.
Give an intrinsic characterization of the set of functions such that is -Carleson.
Problem 15 (Submitted by Antoine Flattot)
Does the Bishop operator have an invariant subspace for every irrational number?
Does the Bishop operator have an invariant subspace for every irrational number?
Problem 16 (Submitted by Miguel Martín)
- Find more sufficient conditions for a set to be SCD. For instance, if has a 1-symmetric basis, is an SCD set?
- Let be a Banach space with unconditional basis. Is SCD?
- Let and be SCD spaces. Are and SCD?
- If and are SCD operators, is SCD?
- If is an SCD operator, is there an space such that factors through ?
Problem 17 (Submitted by Stefan Richter)
Characterize the extremals for the families of -commuting contractions, -contractions and -isometries.
Characterize the extremals for the families of -commuting contractions, -contractions and -isometries.
Problem 18 (Submitted by Lawrence Fialkow)
- (a)
- For the truncated moment problem in the plane
(2 real variables), it is known that if the degree of the
problem is 2, then there is a representing measure
if and only if the moment matrix for the data, , is
positive
semidefinite. If the degree is 6, it is known that there
are
examples where the moment matrix, , is positive
definite,
but there is no measure. If the degree is 4 and the moment
matrix, , is positive definite, is there a
representing measure?
- (b)
- Suppose is a real polynomial with
,
and suppose the restriction of to the closed disk,
, is positive. It can be shown that is of the form
where and are each sums of squares of polynomials of degree at most 1. It is known that for , there are polynomials of degree 6, with positive, such that in any representation of as in (*), the degree of some summand in the sums of squares has degree . Does there exist , such that if and is positive, then admits a representation (*) where each summand has degree at most ?
Problem 19 (Submitted by Søren Eilers)
- (a)
- Understand why the -web is sufficient for classifying Cuntz-Krieger algebras in spite of their ideal lattice being non-linear.
- (b)
- Augment the invariant to arrive at a complete classification of purely iinfinite -algebras with finitely many ideals.
- (c)
- Classify the sofic -gap shifts.
Problem 20 (Submitted by Chris Phillips)
- (a)
- Let . Assume that is a MASA in for all . Doest it follow that is a MASA in the direct limit of the 's?
- (b)
- What is (i.e. the commutation constant of the diagonal matrices in the matrices)?
- (c)
- Let be a unital -algebra, and let be a ``good'' MASA. Does it follow that ? Does it follow that ?
Problem 21 (Submitted by Jesse Peterson)
If , then do we have that if and only if ?
If , then do we have that if and only if ?
Problem 22 (Submitted by David Sherman)
- (a)
- (Dixmier) Let be two representations of a -algebra such that is unitarily equivalent to for all in . Are and equivalent?
- (b)
- Does locally inner imply inner for a separable -algebra ?
- (c)
- Are the concepts of -local innerness different for -algebras?
Problem 23 (Submitted by Ilijas Farah)
- (a)
- Assuming CH or Martin's axiom, there exist an ultrafilter such that . Can we remove the axioms?
- (b)
- Does there exist a nonprincipal ultrafilter on such that ?
- (c)
- What can be said about the structure of in general?
- (d)
- Is equivalent to is flat?
Problem 24 (Submitted by Narutaka Ozawa)
- (a)
- Is there a von Neumann algebra that fully remembers the group action?
- (b)
- Is there a von Neumann algebra that fully remembers the group?
- (c)
- Is for ?
For further submissions or corrections, send an email to
jcdom@math.tamu.edu