Let A be an algebra and let X be an A-bimodule. We call a linear subspace Y of X a Jordan A-submodule of X if Ay + yAY for all A A and yY (if X = A, then this coincides with the classical concept of a Jordan ideal). When is a Jordan A-submodule a submodule? We give a thorough analysis of this question in both algebraic and analytic context. In the first part of the paper, we consider general algebras and general Banach algebras. In the second part, we treat some more specific topics, such as symmetrically normed Jordan A-submodules. Some of our results are of interest also in the classical situation; in particular, we show that there exist C*-algebras having Jordan ideals that are not ideals.
Let = kQ/I be a Koszul algebra over a field k, where Q is a finite quiver. An algorithmic method for finding a minimal projective resolution F of the graded simple modules over is given in [E. L. Green and Ø. Solberg, An algorithmic approach to resolutions, J. Symbolic Comput., 42 (2007), 1012–1033]. This resolution is shown to have a ‘comultiplicative’ structure in [E. L. Green, G. Hartman, E. N. Marcos and Ø. Solberg, Resolutions over Koszul algebras, Arch. Math.85 (2005), 118–127.], and this is used to find a minimal projective resolution P of over the enveloping algebra e. Using these results, we show that the multiplication in the Hochschild cohomology ring of relative to the resolution P is given as a cup product and also provide a description of this product. This comultiplicative structure also yields the structure constants of the Koszul dual of with respect to a canonical basis over k associated to the resolution F. The natural map from the Hochschild cohomology to the Koszul dual of is shown to be surjective onto the graded centre of the Koszul dual.
We study the relation between the polynomial numerical indices of a complex vector-valued function space and the ones of its range space. It is proved that the spaces C(K, X) and L(µ, X) have the same polynomial numerical index as the complex Banach space X for every compact Hausdorff space K and every -finite measure µ, which does not hold any more in the real case. We give an example of a complex Banach space X such that, for every k ≥ 2, the polynomial numerical index of order k of X is the greatest possible, namely 1, while the one of X** is the least possible, namely kk/(1–k). We also give new examples of Banach spaces with the polynomial Daugavet property, namely L(µ, X) when µ is atomless, and Cw(K, X), Cw*(K, X*) when K is perfect.
In this note, we show that the set of n such that the arithmetic mean of the first n primes is an integer is of asymptotic density zero. We use the same method to show that the set of n such that the sum of the first n primes is a square is also of asymptotic density zero. We also prove that both the arithmetic mean of the first n primes as well as the square root of the sum of the first n primes are well distributed modulo 1.