We consider the Bochner–Riesz means S^α _R associated with the sub-Laplacian ℒ for the quaternion Heisenberg group and we prove that S^α _R are uniformly bounded on L^p(𝒬×ℝ^3) when α > α (p)=9| 1/p-1/2| with 1 ≤ p ≤ 2.
We provide a characterization of semifiniteness for von Neumann algebras in terms of positivity-preserving (partial) embedding(s) of the GNS Hilbert space into the von Neumann algebra and vice versa.
Let ℓ ^1(ℤ^2_max,ω ) be the weighted semigroup algebra over the abelian semigroup ℤ^2_max with the pointwise maximum operation. We show that ℓ ^1(ℤ^2_max, ω ) is a BED-algebra for any weight ω . If the weight additionally satisfies ω (-n,-n) ≤ K for some K > 0 and all n ∈ℕ , we prove that ℓ ^1(ℤ^2_max, ω ) is a BSE-algebra. We also present a counterexample demonstrating that this condition is sharp and cannot be omitted.
Let G=PSL(n,ℂ) . Let T be a maximal torus of G. Let ω _r denote the rth fundamental weight. Let ℒ(nω _r) denote the line bundle on the Grassmannian G_r,n associated to the character nω _r of T. In (Kannan and Sardar, Proc. Indian Acad. Sci. (Math. Sci.) 119(1) (2009) 81–100), it is proved that there is a unique minimal dimensional Schubert variety X(w_r,n) in G_r,n admitting semistable points for the T-linearized ample line bundle ℒ(nω _r) . Assume that n=rq+1 , where r,q∈ℕ and q≥ 2 . In this article, we study the GIT quotient of X(w_r,n) modulo a subtorus T_J_r of T generated by the one parameter subgroups of T corresponding to the peaks of w_r,n (see [Perrin, Compos. Math. 143(5) (2007) 1255–1312, Definition 4.6]). We prove that the GIT quotient of X(w_r,n) modulo T_J_r is isomorphic to the total space of the rth stage of an iterated projective space bundle over ℙ^q-1.
