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1. Existentially closed W*-probability spaces

   https://doi.org/10.1007/s00209-022-03038-z  

   Goldbring, Isaac; Houdayer, Cyril (August 2022, Mathematische Zeitschrift)

   Abstract We study several model-theoretic aspects of W $$^*$$ ∗ -probability spaces, that is, $$\sigma $$ σ -finite von Neumann algebras equipped with a faithful normal state. We first study the existentially closed W $$^*$$ ∗ -spaces and prove several structural results about such spaces, including that they are type III $$_1$$ 1 factors that tensorially absorb the Araki–Woods factor $$R_\infty $$ R ∞ . We also study the existentially closed objects in the restricted class of W $$^*$$ ∗ -probability spaces with Kirchberg’s QWEP property, proving that $$R_\infty $$ R ∞ itself is such an existentially closed space in this class. Our results about existentially closed probability spaces imply that the class of type III $$_1$$ 1 factors forms a $$\forall _2$$ ∀ 2 -axiomatizable class. We show that for $$\lambda \in (0,1)$$ λ ∈ ( 0 , 1 ) , the class of III $$_\lambda $$ λ factors is not $$\forall _2$$ ∀ 2 -axiomatizable but is $$\forall _3$$ ∀ 3 -axiomatizable; this latter result uses a version of Keisler’s Sandwich theorem adapted to continuous logic. Finally, we discuss some results around elementary equivalence of III $$_\lambda $$ λ factors. Using a result of Boutonnet, Chifan, and Ioana, we show that, for any $$\lambda \in (0,1)$$ λ ∈ ( 0 , 1 ) , there is a family of pairwise non-elementarily equivalent III $$_\lambda $$ λ factors of size continuum. While we cannot prove the same result for III $$_1$$ 1 factors, we show that there are at least three pairwise non-elementarily equivalent III $$_1$$ 1 factors by showing that the class of full factors is preserved under elementary equivalence. 

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2. Bohr sets in sumsets II: countable abelian groups

   https://doi.org/10.1017/fms.2023.49  

   Griesmer, John T.; Le, Anh N.; Lê, Thái Hoàng (July 2023, Forum of Mathematics, Sigma)

   Abstract We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, lettingGbe a countable discrete abelian group and$$\phi _1, \phi _2, \phi _3: G \to G$$be commuting endomorphisms whose images have finite indices, we show that(1)If$$A \subset G$$has positive upper Banach density and$$\phi _1 + \phi _2 + \phi _3 = 0$$, then$$\phi _1(A) + \phi _2(A) + \phi _3(A)$$contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in$$\mathbb {Z}$$and a recent result of the first author.(2)For any partition$$G = \bigcup _{i=1}^r A_i$$, there exists an$$i \in \{1, \ldots , r\}$$such that$$\phi _1(A_i) + \phi _2(A_i) - \phi _2(A_i)$$contains a Bohr set. This generalizes a result of the second and third authors from$$\mathbb {Z}$$to countable abelian groups.(3)If$$B, C \subset G$$have positive upper Banach density and$$G = \bigcup _{i=1}^r A_i$$is a partition,$$B + C + A_i$$contains a Bohr set for some$$i \in \{1, \ldots , r\}$$. This is a strengthening of a theorem of Bergelson, Furstenberg and Weiss. All results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices$$[G:\phi _j(G)]$$, the upper Banach density ofA(in (1)), or the number of sets in the given partition (in (2) and (3)). 

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3. On realizations of the subalgebra 𝒜^{ℝ}(1) of the ℝ-motivic Steenrod algebra

   https://doi.org/10.1090/btran/114  

   Bhattacharya, P.; Guillou, B.; Li, A. (July 2022, Transactions of the American Mathematical Society, Series B)

   In this paper, we show that the finite subalgebra A R ( 1 ) \mathcal {A}^\mathbb {R}(1) , generated by S q 1 \mathrm {Sq}^1 and S q 2 \mathrm {Sq}^2 , of the R \mathbb {R} -motivic Steenrod algebra A R \mathcal {A}^\mathbb {R} can be given 128 different A R \mathcal {A}^\mathbb {R} -module structures. We also show that all of these A \mathcal {A} -modules can be realized as the cohomology of a 2 2 -local finite R \mathbb {R} -motivic spectrum. The realization results are obtained using an R \mathbb {R} -motivic analogue of the Toda realization theorem. We notice that each realization of A R ( 1 ) \mathcal {A}^\mathbb {R}(1) can be expressed as a cofiber of an R \mathbb {R} -motivic v 1 v_1 -self-map. The C 2 {\mathrm {C}_2} -equivariant analogue of the above results then follows because of the Betti realization functor. We identify a relationship between the R O ( C 2 ) \mathrm {RO}({\mathrm {C}_2}) -graded Steenrod operations on a C 2 {\mathrm {C}_2} -equivariant space and the classical Steenrod operations on both its underlying space and its fixed-points. This technique is then used to identify the geometric fixed-point spectra of the C 2 {\mathrm {C}_2} -equivariant realizations of A C 2 ( 1 ) \mathcal {A}^{\mathrm {C}_2}(1) . We find another application of the R \mathbb {R} -motivic Toda realization theorem: we produce an R \mathbb {R} -motivic, and consequently a C 2 {\mathrm {C}_2} -equivariant, analogue of the Bhattacharya-Egger spectrum Z \mathcal {Z} , which could be of independent interest. 

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4. On the Convolution Inequality f ≥ f ⋆ f

   https://doi.org/10.1093/imrn/rnaa350  

   Carlen, Eric A; Jauslin, Ian; Lieb, Elliott H; Loss, Michael P (January 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We consider the inequality $$f \geqslant f\star f$$ for real functions in $$L^1({\mathbb{R}}^d)$$ where $$f\star f$$ denotes the convolution of $$f$$ with itself. We show that all such functions $$f$$ are nonnegative, which is not the case for the same inequality in $L^p$ for any $$1 < p \leqslant 2$$, for which the convolution is defined. We also show that all solutions in $$L^1({\mathbb{R}}^d)$$ satisfy $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x \leqslant \tfrac 12$$. Moreover, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$, then $$f$$ must decay fairly slowly: $$\int _{{\mathbb{R}}^{\textrm{d}}}|x| f(x)\ \textrm{d}x = \infty $$, and this is sharp since for all $r< 1$, there are solutions with $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$ and $$\int _{{\mathbb{R}}^{\textrm{d}}}|x|^r f(x)\ \textrm{d}x <\infty $$. However, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x =: a < \tfrac 12$$, the decay at infinity can be much more rapid: we show that for all $$a<\tfrac 12$$, there are solutions such that for some $$\varepsilon>0$$, $$\int _{{\mathbb{R}}^{\textrm{d}}}e^{\varepsilon |x|}f(x)\ \textrm{d}x < \infty $$. 

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5. Doping evolution of the Mott–Hubbard landscape in infinite-layer nickelates

   https://doi.org/10.1073/pnas.2007683118  

   Goodge, Berit H.; Li, Danfeng; Lee, Kyuho; Osada, Motoki; Wang, Bai Yang; Sawatzky, George A.; Hwang, Harold Y.; Kourkoutis, Lena F. (January 2021, Proceedings of the National Academy of Sciences)

   null (Ed.)

   The recent observation of superconductivity in N d 0.8 S r 0.2 N i O 2 has raised fundamental questions about the hierarchy of the underlying electronic structure. Calculations suggest that this system falls in the Mott–Hubbard regime, rather than the charge-transfer configuration of other nickel oxides and the superconducting cuprates. Here, we use state-of-the-art, locally resolved electron energy-loss spectroscopy to directly probe the Mott–Hubbard character of N d 1 − x S r x N i O 2 . Upon doping, we observe emergent hybridization reminiscent of the Zhang–Rice singlet via the oxygen-projected states, modification of the Nd 5d states, and the systematic evolution of Ni 3d hybridization and filling. These experimental data provide direct evidence for the multiband electronic structure of the superconducting infinite-layer nickelates, particularly via the effects of hole doping on not only the oxygen but also nickel and rare-earth bands. 

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