Let Formula omitted be a locally compact quantum group with dual Formula omitted. Suppose that the left Haar weight Formula omitted and the dual left Haar weight Formula omitted are tracial, e.g. ...Formula omitted is a unimodular Kac algebra. We prove that for Formula omitted, the Fourier multiplier Formula omitted is bounded from Formula omitted to Formula omitted whenever the symbol x lies in Formula omitted, where Formula omitted. Moreover, we have ||mx:Lp(G^,phi^)right arrowLq(G^,phi^)||less than or equal tocp,q||x||Lr,infinity(G,phi),where Formula omitted is a constant depending only on p and q. This was first proved by Hörmander (Acta Math 104:93-140, 1960) for Formula omitted, and was recently extended to more general groups and quantum groups. Our work covers all these results and the proof is simpler. In particular, this also yields a family of Formula omitted-Fourier multipliers over discrete group von Neumann algebras. A similar result for Formula omitted- Formula omitted Schur multipliers is also proved.
We compute the numerical index of the two-dimensional real Lp space for 65⩽p⩽1+α0 and α1⩽p⩽6, where α0 is the root of f(x)=1+x−2−(x−1x+x1x) and 11+α0+1α1=1. This, together with the previous results ...in Merí and Quero On the numerical index of absolute symmetric norms on the plane. Linear Multilinear Algebra. 2021;69(5):971–979 and Monika and Zheng The numerical index of ℓp2. Linear Multilinear Algebra. 2022;1–6. Published online DOI:10.1080/03081087.2022.2043818, gives the numerical index of the two-dimensional real Lp space for 65⩽p⩽6.
In this paper, we present a candidate for Formula omitted extended higher-spin Formula omitted supergravity with the most general boundary conditions discussed by Grumiller and Riegler recently. We ...show that the asymptotic symmetry algebra consists of two copies of the Formula omitted affine algebra in the presence of the most general boundary conditions. Furthermore, we impose some certain restrictions on gauge fields on the most general boundary conditions and that leads us to the supersymmetric extension of the Brown-Henneaux boundary conditions. We eventually see that the asymptotic symmetry algebra reduces to two copies of the Formula omitted algebra for Formula omitted extended higher-spin supergravity.
In this paper, we investigate the necessary and sufficient conditions for solving a dual split quaternion matrix equation AXB=C, and present the general solution expression when the solvability ...conditions are met. As an application, we delve into the necessary and sufficient conditions for the existence of a Hermitian solution to this equation by using a newly defined real representation method. Furthermore, we obtain the solutions for the dual split quaternion matrix equations AX=C and XB=C. Finally, we provide a numerical example to demonstrate the findings of this paper.
In this paper, for each commutative and integral quantale, we give the stratified Sierpinski $L$-algebraic closure space and a sobrification of stratified $L$-algebraic closure spaces. Furthermore, ...we show that $\bf{S}$$L$-$\bf{AC}_0$---the category of stratified $S_0$-$L$-algebraic closure spaces is epireflective in $\bf{S}$$L$-$\bf{AC}$---the category of stratified $L$-algebraic closure spaces, and $\bf{Sob}$$L$-$\bf{AC}$---the category of sober $L$-algebraic closure spaces is epireflective and $\mathcal{E}$-firm epireflective in the category $\bf{S}$$L$-$\bf{AC}_0$.
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