On the Geometry of Müntz Spaces

LetΛ={λk}k=1∞satisfy0<λ1<λ2<⋯,∑k=1∞‍1/λk<∞andinfk(λk+1-λk)>0. We investigate the Müntz spacesMpΛ=span¯{tλk:k=1,2,…}⊂Lp(0,1)for1≤p≤∞. We show that, for eachp, there is a Müntz spaceFpwhich contains isomorphic copies of all Müntz spaces as complemented subspaces.Fpis uniquely determined up to isomorphisms by this maximality property. We discuss explicit descriptions ofFp. In particularFpis isomorphic to a Müntz spaceMp(Λ^)whereΛ^consists of positive integers. Finally we show that the Banach spaces(∑n‍⊕Fn)pfor1≤p<∞and(∑n‍⊕Fn)0forp=∞are always isomorphic to suitable Müntz spacesMp(Λ)if theFnare the spans of arbitrary finitely many monomials over[0,1].

Canonical seeds and Prikry trees

Applying the seed concept to Prikry tree forcing ℙμ, I investigate how well ℙμ preserves the maximality property of ordinary Prikry forcing and prove that ℙμ, Prikry sequences are maximal exactly when μ admits no non-canonical seeds via a finite iteration. In particular, I conclude that if μ is a strongly normal supercompactness measure, then ℙμ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. Hugh Woodin's.

On a Maximality Property of Partition Regular Systems of Equations

In this note we will study the following problem. For a given partition regular system of equations, which equations can be added to this system without introducing new variables, such that the new augmented system is again partition regular. It turns that the Hindman system on finite sums as well as the Deuber-Hindman system on finite sums of (m, p, c)-sets are maximal in this sense.

Applications of the asymptotic range to analytic subalgebras of groupoid C*-algebras

For a 1-cocycle c on a principal r-discrete groupoid G, that vanishes only on the unit space of G, we show that the asymptotic range of c, , is an invariant for the total order c−1([0, ∞]). It follows that is also an invariant (with respect to isometric isomorphisms) of the triangular analytic algebra supported on c−1([0, ∞]). We also prove that if and only if the analytic algebra has a certain maximality property.