Finite sums of nilpotent elements in properly infinite $C^{*}$-algebras

K1-injectivity for Properly Infinite C ∗-algebras

One of the main tools to classify C∗-algebras is the study of its projections and its unitaries. It was proved by Cuntz in [Cun81] that if A is a purely infinite simple C∗-algebra, then the kernel of the natural map for the unitary group U(A) to the Ktheory groupK1(A) is reduced to the connected component U(A), i.e. A isK1-injective (see §3). We study in this note a finitely generated C∗-algebr...

Finitely Generated Nilpotent Group C*-algebras Have Finite Nuclear Dimension

We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra A generated by an irreducible representation of such a group has decomposition rank at most 3. If, in addition, A satisfies the universal coefficient theorem, another string of deep results shows it is classifiable by its Ellio...

The centralisers of nilpotent elements in the classical Lie algebras

The definition of index goes back to Dixmier [3, 11.1.6]. This notion is important in Representation Theory and also in Invariant Theory. By Rosenlicht’s theorem [12], generic orbits of an arbitrary action of a linear algebraic group on an irreducible algebraic variety are separated by rational invariants; in particular, ind g = tr.degK(g). The index of a reductive algebra equals its rank. Comp...

NILPOTENT GRAPHS OF MATRIX ALGEBRAS

Let $R$ be a ring with unity. The undirected nilpotent graph of $R$, denoted by $Gamma_N(R)$, is a graph with vertex set ~$Z_N(R)^* = {0neq x in R | xy in N(R) for some y in R^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in N(R)$, or equivalently, $yx in N(R)$, where $N(R)$ denoted the nilpotent elements of $R$. Recently, it has been proved that if $R$ is a left A...

Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup