International Journal of Mathematics

When a unital C*-algebra A has topological stable rank one (write tsr(A) = 1), we know that tsr(pAp) = 1 for a non-zero projection p ∈ A. When, however, tsr(A) ≥ 2, it is generally false. We prove that if a unital C*-algebra A has a simple unital C*-subalgebra D of A with common unit such that D has Property (SP) and sup_{p ∈ P(D)}tsr(pAp) < ∞, then tsr(A) ≤ 2. As an application let A be a simple unital C*-algebra with tsr(A) = 1 and Property (SP), finite groups, α_k actions from G_k to Aut((⋯((A × α₁ G₁) ×_α2 G₂)⋯) ×_αk-1 G_k-1). (G₀ = {1}). Then

We give a new and detailed proof of the variation formulas for the equivariant Ray–Singer metric, which are originally due to J. M. Bismut and W. Zhang.

We introduce and study the definition, main properties and applications of iterated twisted tensor products of algebras, motivated by the problem of defining a suitable representative for the product of spaces in noncommutative geometry. We find conditions for constructing an iterated product of three factors and prove that they are enough for building an iterated product of any number of factors. As an example of the geometrical aspects of our construction, we show how to construct differential forms and involutions on iterated products starting from the corresponding structures on the factors and give some examples of algebras that can be described within our theory. We prove a certain result (called "invariance under twisting") for a twisted tensor product of two algebras, stating that the twisted tensor product does not change when we apply certain kind of deformation. Under certain conditions, this invariance can be iterated, containing as particular cases a number of independent and previously unrelated results from Hopf algebra theory.

In this paper, we continue the investigation of coherent systems of type (n, d, k) on the projective line which are stable with respect to some value of a parameter α. We consider the case k = 1 and study the variation of the moduli spaces with α. We determine inductively the first and last moduli spaces and the flip loci, and give an explicit description for ranks 2 and 3. We also determine the Hodge polynomials explicitly for ranks 2 and 3 and in certain cases for arbitrary rank.

We consider a class of non-symmetric Cantor sets C determined by C = a₀C ∪ (a₁C + 1 - a₁), a₀, a₁ ∈ (0, 1). Under certain conditions on a₀ and a₁, the upper and lower densities are obtained explicitly for each x ∈ C. It extends the results for symmetric Cantor sets obtained by Feng, Hua and Wen (The pointwise densities of the Cantor measure, J. Math. Anal. Appl.250 (2000) 692–705).

Let G be a finite group acting on {1,…,n}. For any C*-algebra , this defines an action α of G on . We show that if tensorially absorbs a UHF algebra of infinite type, the Jiang–Su algebra, or is approximately divisible, then has the corresponding property as well.