Authors:Volodymyr Mazorchuk, Vanessa Miemietz, Xiaoting Zhang Pages: 161 - 178 Abstract: We describe the structure of bimodules (over finite dimensional algebras) which have the property that the functor of tensoring with such a bimodule sends any module to a projective module. The main result is that all such bimodules are $\Bbbk $-split in the sense that they factor (inside the tensor category of bimodules) over $\Bbbk $-vector spaces. As one application, we show that any simple $2$-category has a faithful $2$-representation inside the $2$-category of $\Bbbk $-split bimodules. As another application, we classify simple transitive $2$-representations of the $2$-category of projective bimodules over the algebra $\Bbbk [x,y]/(x^2,y^2,xy)$. PubDate: 2019-06-17 DOI: 10.7146/math.scand.a-111146 Issue No:Vol. 124, No. 2 (2019)
Authors:Becky Hooper, Adam Van Tuyl Pages: 179 - 187 Abstract: Ehrenborg, Govindaiah, Park, and Readdy recently introduced the van der Waerden complex, a pure simplicial complex whose facets correspond to arithmetic progressions. Using techniques from combinatorial commutative algebra, we classify when these pure simplicial complexes are vertex decomposable or not Cohen-Macaulay. As a corollary, we classify the van der Waerden complexes that are shellable. PubDate: 2019-06-17 DOI: 10.7146/math.scand.a-111923 Issue No:Vol. 124, No. 2 (2019)
Authors:Cleto B. Miranda-Neto Pages: 188 - 202 Abstract: A difficult issue in modern commutative algebra asks for examples of modules (more interestingly, reflexive vector bundles) having prescribed reduction number $r\geq 1$. The problem is even subtler if in addition we are interested in good properties for the Rees algebra. In this note we consider the case $r=1$. Precisely, we show that the module of logarithmic vector fields of the Fermat divisor of any degree in projective $3$-space is a reflexive vector bundle of reduction number $1$ and Gorenstein Rees ring. PubDate: 2019-06-17 DOI: 10.7146/math.scand.a-111889 Issue No:Vol. 124, No. 2 (2019)
Authors:A. Cabot, A. Jourani, L. Thibault, D. Zagrodny Pages: 203 - 246 Abstract: The attainment set of the $\varphi$-envelope of a function at a given point is investigated. The inclusion of the attainment set of the $\varphi$-envelope of the closed convex hull of a function into the attainment set of the function is preserved in sufficiently general settings to encompass the case $\varphi$ being a norm in a power not less than $1$. The non-emptiness of the attainment set is guaranteed on generic subsets of a given space, in several fundamental cases. PubDate: 2019-06-17 DOI: 10.7146/math.scand.a-110766 Issue No:Vol. 124, No. 2 (2019)
Authors:Eduardo Brandani da Silva, Dicesar Lass Fernandez Pages: 247 - 262 Abstract: The behavior of bilinear operators acting on the interpolation of Banach spaces in relation to compactness is analyzed, and an one-sided compactness theorem is obtained for bilinear operators interpolated by the ρ interpolation method. PubDate: 2019-06-17 DOI: 10.7146/math.scand.a-111424 Issue No:Vol. 124, No. 2 (2019)
Authors:Dariusz Cichoń, Jan Stochel, Franciszek Hugon Szafraniec Pages: 263 - 288 Abstract: Complex moment sequences are exactly those which admit positive definite extensions on the integer lattice points of the upper diagonal half-plane. Here we prove that the aforesaid extension is unique provided the complex moment sequence is determinate and its only representing measure has no atom at $0$. The question of converting the relation is posed as an open problem. A partial solution to this problem is established when at least one of representing measures is supported in a plane algebraic curve whose intersection with every straight line passing through $0$ is at most one point set. Further study concerns representing measures whose supports are Zariski dense in $\mathbb{C} $ as well as complex moment sequences which are constant on a family of parallel “Diophantine lines”. All this is supported by a bunch of illustrative examples. PubDate: 2019-06-17 DOI: 10.7146/math.scand.a-112091 Issue No:Vol. 124, No. 2 (2019)
Authors:Galatia Cleanthous, Athanasios G. Georgiadis, Morten Nielsen Pages: 289 - 304 Abstract: A new general Hörmander type condition involving anisotropies and mixed norms is introduced, and boundedness results for Fourier multipliers on anisotropic Besov and Triebel-Lizorkin spaces of distributions with mixed Lebesgue norms are obtained. As an application, the continuity of such operators is established on mixed Sobolev and Lebesgue spaces too. Some lifting properties and equivalent norms are obtained as well. PubDate: 2019-06-17 DOI: 10.7146/math.scand.a-113031 Issue No:Vol. 124, No. 2 (2019)
Authors:Samuel G. Walters Pages: 305 - 319 Abstract: The noncommutative Fourier transform $\sigma (U)=V^{-1}$, $\sigma (V)=U$ of the irrational rotation C*-algebra $A_\theta $ (generated by canonical unitaries $U$, $V$ satisfying $VU = e^{2\pi i\theta } UV$) is shown to have the following K-inductive structure (for a concrete class of irrational parameters, containing dense $G_\delta $'s). There are approximately central matrix projections $e_1$, $e_2$, $f$ that are σ-invariant and which form a partition of unity in $K_0$ of the fixed-point orbifold $A_\theta ^\sigma $, where $f$ has the form $f = g+\sigma (g) +\sigma ^2(g)+\sigma ^3(g)$, and where $g$ is an approximately central matrix projection as well. PubDate: 2019-06-17 DOI: 10.7146/math.scand.a-114723 Issue No:Vol. 124, No. 2 (2019)
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