AF-embeddings into [formula omitted]-algebras of real rank zero

Let Ext ( C ( X ), A ) be the set of unitarily equivalence classes of essential C *-algebra extensions of the following form: 0 → A → E → C ( X ) → 0, where A is a nonunital separable simple C *-algebra of real rank zero, stable rank one with unique normalized trace and X is a finite CW complex. We show that there is a bijection [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]: Ext ( C ( X ), A ) → KK ( C ( X ), M ( A )/ A ), where M ( A ) is the multiplier algebra of A . In particular, we determine when an extension is actually splitting. We also, in a more general setting, give a condition when an essential extension is quasidiagonal.

A unital C*-algebra A is said to have cancellation of projections if the semigroup D(A) of Murray–von Neumann equivalence classes of projections in matrices over A is cancellative. It has long been known that stable rank one implies cancellation for any A, and some partial converses have been established. In this paper it is proved that cancellation does not imply stable rank one for simple, stably finite C*-algebras. 2000 Mathematics Subject Classification 46L80 (primary), 46L85 (secondary).

We prove a number of fundamental facts about the canonical order on projections in C*-algebras of real rank zero. Specifically, we show that this order is separative and that arbitrary countable collections have equivalent (in terms of their lower bounds) decreasing sequences. Under the further assumption that the order is countably downwards closed, we show how to characterize greatest lower bounds of finite collections of projections, and their existence, using the norm and spectrum of simple product expressions of the projections in question. We also characterize the points at which the canonical homomorphism to the Calkin algebra preserves least upper bounds of countable collections of projections, namely that this occurs precisely when the span of the corresponding subspaces is closed.

We obtain an order sharp estimate for the distance from a given bounded operator A A on a Hilbert space to the set of normal operators in terms of ‖ [ A , A ∗ ] ‖ \|[A,A^*]\| and the distance to the set of invertible operators. A slightly modified estimate holds in a general C ∗ C^* -algebra of real rank zero.

Almost Commuting Unitaries and Classification of Purely Infinite SimpleC*-Algebras

On the Range of Certain ASH Algebras of Real Rank Zero

Classification of Homomorphisms from C (X) to Simple C *-Algebras of Real Rank Zero

On the Classification of C ∗ -Algebras of Real Rank Zero, II

Distance between unitary orbits of unitary elements in $$C^*$$-algebras of real rank zero

Let H and K be two complex Hilbert spaces and 𝒜 ⊂ B(H) and ℬ ⊂ B(K) be two unital C*-algebras. It is shown that if φ : 𝒜 → ℬ is an additive surjective mapping satisfying φ(|A|) = |φ(A)| for every A ∈ 𝒜 and φ(I) is a projection, then the restriction of mapping φ to both 𝒜s and 𝒜sk is a Jordan *-homomorphism onto corresponding set in ℬ, where 𝒜s and 𝒜sk denote the set of all self-adjoint and skew-self-adjoint elements, respectively. Furthermore, if ℬ is a C*-algebra of real-rank zero then φ is a ℂ-linear or ℂ-antilinear *-homomorphism.

We investigate properties of group gradings on matrix rings M n ( R ) , where R is an associative unital ring and n is a positive integer. More precisely, we introduce very good gradings and show that any very good grading on M n ( R ) is necessarily epsilon-strong. We also identify a condition that is sufficient to guarantee that M n ( R ) is an epsilon-crossed product, i.e. isomorphic to a crossed product associated with a unital twisted partial action. In the case where R has IBN, we provide a characterization of when M n ( R ) is an epsilon-crossed product. Our results are illustrated by several examples.

Commutator relations and structure constants for rank 2 Kac–Moody algebras

A nonzero ring R is said to be uniformly strongly prime (of bound n) if n is the smallest positive integer such that for some n-element subset X of R we have xXy ≠ 0 whenever 0 ≠ x, y ∈ R. The study of uniformly strongly prime rings reduces to that of orders in matrix rings over division rings, except in the case n = 1. This paper is devoted primarily to an investigation of uniform bounds of primeness in matrix rings over fields. It is shown that the existence of certain n-dimensional nonassociative algebras over a field F decides the uniform bound of the n × n matrix ring over F.

Let R be a ring, K a field, and n a natural number. We will be concerned with the following type of questions: (a) Classify the representations of R in (hO, (or n dimensional representations). (b) Classify n dimensional representations up to the natural equivalence. If ~bt, ~b2 : R ~ (K), we say ~bx is equivalent to ~b 2 if there is a K automorphism 7 of (K), such that 7q~1 = 4)2. (c) Classify equivalence classes of irreducible and semisimple representations. For the sake of simplicity we will restrict ourselves mainly to the case that K is algebraically closed and R is a finitely generated K algebra. In fact, for most of this paper, we will assume that R = K{xa, ..., x,) is a free algebra; at the end we will deduce from the results in this ease the more general theorems for not necessarily free algebras. Before describing the theorems that we will obtain, let us digress in order to motivate the use that we will make of the theory of rings with polynomial identities as a tool for attacking the problems stated before. Let us recall, therefore, some of the basic structural results of this theory and interpret them in terms of representation theory.

A polynomial in noncommuting variables is vanishing, nil or central in a ring, R R , if its value under every substitution from R R is 0, nilpotent or a central element of R R , respectively. THEOREM. If R R has no nonvanishing multilinear nil polynomials then neither has the matrix ring R n {R_n} . THEOREM. Let R R be a ring satisfying a polynomial identity modulo its nil radical N N , and let f f be a multilinear polynomial. If f f is nil in R R then f f is vanishing in R / N R/N . Applied to the polynomial x y − y x xy - yx , this establishes the validity of a conjecture of Herstein’s, in the presence of polynomial identity. THEOREM. Let m m be a positive integer and let F F be a field containing no m m th roots of unity other than 1. If f f is a multilinear polynomial such that for some n > 2 f m n > 2{f^m} is central in F n {F_n} , then f f is central in F n {F_n} . This is related to the (non)existence of noncrossed products among p 2 {p^2} -dimensional central division rings.