In 1967 Coburn showed that C ∗ -algebras generated by a single non-unitary isometry on a Hilbert space don’t depend on the particular choice of the isometry. And R. G. Douglas proved that the C ∗ - algebra AΓ generated by a non-unitary one-parameter semigroup of isometries is canonically unique for a subgroup Γ of the real number group R. A. Nica called it the uniqueness property, which means to some extend that C ∗ -algebras generated by non-unitary isometries on a Hilbert space don’t depend on the particular choice of the isometries. Since Coburn, many operator algebraists have extended Coburn’s result cosistently. Toeplitz algebra, Cuntz algebra, Wiener-Hopf C ∗ -algebra W(G, M ) for a discrete group G with a semigroup M are their outcomes. We can see that if the Wiener-Hopf C ∗ -algebra W(G, M ) of a partially ordered group G with the positive cone M has the uniqueness property, then (G, M ) is weakly unperforated. We also can see that the extented Coburn’s result of the Wiener-Hopf C ∗ -algebra W(G, M ) depends on the order structure of the semigroup M .

We give an introduction to the theory of Chow groups and explain classical and new results obtained by considering the Ceresa cycle.

A way to study the geometry of a homogeneous variety under a semi-simple algebraic group is to investigate its Chow group of algebraic cycles modulo the rational equivalence relation. In general, the problem of determining the Chow group of a projective homogeneous variety reduces to computing the torsion. In this talk, we discuss the latter problem including the cases of Severi-Brauer varieties and Spin-flags. 

TBA