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Register a new userA note on weak-star and norm Borel sets in the dual of the space of continuous functions
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S. Ferrari
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Let $Bo(T,tau)$ be the Borel $sigma$-algebra generated by the topology $tau$ on $T$. In this paper we show that if $K$ is a Hausdorff compact space, then every subset of $K$ is a Borel set if, and only if, $$Bo(C^*(K),w^*)=Bo(C^*(K),|cdot|);$$ where $w^*$ denotes the weak-star topology and $|cdot|$ is the dual norm with respect to the sup-norm on the space of real-valued continuous functions $C(K)$. Furthermore we study the topological properties of the Hausdorff compact spaces $K$ such that every subset is a Borel set. In particular we show that, if the axiom of choice holds true, then $K$ is scattered.
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$C_p(X)$ denotes the space of continuous real-valued functions on a Tychonoff space $X$ endowed with the topology of pointwise convergence. A Banach space $E$ equipped with the weak topology is denoted by $E_{w}$. It is unknown whether $C_p(K)$ and $C(L)_{w}$ can be homeomorphic for infinite compact spaces $K$ and $L$ cite{Krupski-1}, cite{Krupski-2}. In this paper we deal with a more general question: what are the Banach spaces $E$ which admit certain continuous surjective mappings $T: C_p(X) to E_{w}$ for an infinite Tychonoff space $X$? First, we prove that if $T$ is linear and sequentially continuous, then the Banach space $E$ must be finite-dimensional, thereby resolving an open problem posed in cite{Kakol-Leiderman}. Second, we show that if there exists a homeomorphism $T: C_p(X) to E_{w}$ for some infinite Tychonoff space $X$ and a Banach space $E$, then (a) $X$ is a countable union of compact sets $X_n, n in omega$, where at least one component $X_n$ is non-scattered; (b) $E$ necessarily contains an isomorphic copy of the Banach space $ell_{1}$.
The proximinality of certain subspaces of spaces of bounded affine functions is proved. The results presented here are some line
We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Our main interest is to obtain preasymptotic estimates for the corresponding sampling numbers. We obtain results in terms of the decay of related singular numbers of the compact embedding into $L_2(D,varrho_D)$ multiplied with the supremum of the Christoffel function of the subspace spanned by the first $m$ singular functions. Here the measure $varrho_D$ is at our disposal. As an application we obtain near optimal upper bounds for the sampling numbers for periodic Sobolev type spaces with general smoothness weight. Those can be bounded in terms of the corresponding benchmark approximation number in the uniform norm, which allows for preasymptotic bounds. By applying a recently introduced sub-sampling technique related to Weavers conjecture we mostly lose a $sqrt{log n}$ and sometimes even less. Finally we point out a relation to the corresponding Kolmogorov numbers.
The paper is concerned with the problem of identifying the norm attaining operators in the von Neumann algebra generated by two orthogonal projections on a Hilbert space. This algebra contains every skew projection on that Hilbert space and hence the results of the paper also describe functions of skew projections and their adjoints that attain the norm.
Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article Periods and factors of weak model sets we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moodys work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets.
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