Extensions Of Classical Theorems Research Articles

On a theorem about projectivities

The main result of this paper is a theorem about projectivities in then-dimensional complex projective spacePn (n ≥2). Puttingn = 2, we showed in [3] that the theorem of Desargues inPn is a special case of this theorem. And not only the theorem of Desargues but also the converse of the theorem of Pascal, the theorem of Pappus-Pascal, the theorem of Miquel, the Newton line, the Brocard points and a lot of lesser known results in the projective, the affine and the Euchdean plane were obtained from this theorem as special cases without any further proof. Many extensions of classical theorems in the projective, affine and Euclidean plane to higher dimensions can be found in the literature and probably some of these are special cases of this theorem inPn. We only give a few examples and leave it as an open problem which other special cases could be found.

𝐶*-algebras generated by Fourier-Stieltjes transforms

For G a locally compact group and G ^ \hat G its dual, let M d ( G ^ ) {\mathcal {M}_d}(\hat G) be the C ∗ {C^ \ast } -algebra generated by the Fourier-Stieltjes transforms of the discrete measures on G. We show that the canonical trace on M d ( G ^ ) {\mathcal {M}_d}(\hat G) is faithful if and only if G is amenable as a discrete group. We further show that if G is nondiscrete and amenable as a discrete group, then the only measures in M d ( G ^ ) {\mathcal {M}_d}(\hat G) are the discrete measures, and also the sup and lim sup norms are identical on M d ( G ^ ) {\mathcal {M}_d}(\hat G) . These results are extensions of classical theorems on almost periodic functions on locally compact abelian groups.