A$generalized polynomial$is a real-valued function which is obtained from conventional polynomials by the use of the operations of addition, multiplication, and taking the integer part; a$generalized polynomial mapping$is a vector-valued mapping whose coordinates are generalized polynomials. We show that any bounded generalized polynomial mapping$u$:$Z$^{$d$}→$R$^{$l$}has a representation$u$($n$) =$f$($ϕ$($n$)$x$),$n$∈$Z$^{$d$}, where$f$is a piecewise polynomial function on a compact nilmanifold$X$,$x$∈$X$, and$ϕ$is an ergodic$Z$^{$d$}-action by translations on$X$. This fact is used to show that the sequence$u$($n$),$n$∈$Z$^{$d$}, is well distributed on a piecewise polynomial surface $\mathcal{S}\subset\mathbf{R}^{l}$ (with respect to the Borel measure on $\mathcal{S}$ that is the image of the Lebesgue measure under the piecewise polynomial function defining $\mathcal{S}$ ). As corollaries we also obtain a von Neumann-type ergodic theorem along generalized polynomials and a result on Diophantine approximations extending the work of van der Corput and of Furstenberg–Weiss.

The product ϕ_{λ}^{(α,β)}(t_{1})ϕ_{λ}^{(α,β)}(t_{2}) of two Jacobi functions is expressed as an integral in terms of ϕ_{λ}^{(α,β)}(t_{3}) with explicit non-negative kernel, when α≧β≧−1/2. The resulting convolution structure for Jacobi function expansions is studied. For special values of α and β the results are known from the theory of symmetric spaces.

Let A 1, A 2 be standard operator algebras on complex Banach spaces X 1, X 2, respectively. For k 2, let (i 1,, i m) be a sequence with terms chosen from {1,, k}, and define the generalized Jordan product T 1 T k= T i 1 T i m+ T i m T i 1 on elements in A i. This includes the usual Jordan product A 1 A 2= A 1 A 2+ A 2 A 1, and the triple {A 1, A 2, A 3}= A 1 A 2 A 3+ A 3 A 2 A 1. Assume that at least one of the terms in (i 1,, i m) appears exactly once. Let a map : A 1 A 2 satisfy that ( (A 1) (A k))= (A 1 A k) whenever any one of A 1,, A k has rank at most one. It is shown in this paper that if the range of contains all operators of rank at most three, then must be a Jordan isomorphism multiplied by an m th root of unity. Similar results for maps between self-adjoint operators acting on Hilbert spaces are also obtained.

This is the third of a series of papers on a new equivariant cohomology that takes values in a vertex algebra, and contains and generalizes the classical equivariant cohomology of a manifold with a Lie group action a la H. Cartan. In this paper, we compute this cohomology for spheres and show that for any simple connected group G, there is a sphere with infinitely many actions of G which have distinct chiral equivariant cohomology, but identical classical equivariant cohomology. Unlike the classical case, the description of the chiral equivariant cohomology of spheres requires a substantial amount of new structural theory, which we fully develop in this paper. This includes a quasi-conformal structure, equivariant homotopy invariance, and the values of this cohomology on homogeneous spaces. These results rely on crucial features of the underlying vertex algebra valued complex that have no classical analogues. ,

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