Let A be a ring and
a ring endomorphism. A generalized skew (or σ-)derivation of A is an additive map
for which there exists a map
such that
for all
. If A is a prime
-algebra and σ is surjective, we ...determine the structure of generalized σ-derivations of A that belong to the cb-norm closure of elementary operators
on A; all such maps are of the form
for suitable elements a,b,c of the multiplier algebra
. As a consequence, if an epimorphism
lies in the cb-norm closure of
, then σ must be an inner automorphism. We also show that these results cannot be extended even to relatively well-behaved non-prime
-algebras like
.
We define a unital algebra A over a field
to be nearly simple if A contains a unique non-trivial ideal I
A
such that
If A and B are two nearly simple algebras over
, we consider the ideal structure ...of their tensor product
The obvious non-trivial ideals of
are:
The purpose of this article is to characterize when are all non-trivial ideals of
of the above form.
We describe, up to isomorphism, the local multiplier algebra Mloc(A) of a C∗-algebra A which admits only finite dimensional irreducible representations: Mloc(A) in this case is a finite or countable ...direct product of C∗-algebras of the form C(Xn)⊗Mn, where each space Xn is Stonean. In particular, Mloc(A) is an AW∗-algebra of type I, so it admits only inner derivations and it coincides with the injective envelope I(A) of A.
Abstract
We give a number of equivalent conditions (including weak centrality) for a general $C^*$-algebra to have the centre-quotient property. We show that every $C^*$-algebra $A$ has a largest ...weakly central ideal $J_{wc}(A)$. For an ideal $I$ of a unital $C^*$-algebra $A$, we find a necessary and sufficient condition for a central element of $A/I$ to lift to a central element of $A$. This leads to a characterisation of the set $V_A$ of elements of an arbitrary $C^*$-algebra $A$, which prevent $A$ from having the centre-quotient property. The complement $\textrm{CQ}(A):= A \setminus V_A$ always contains $Z(A)+J_{wc}(A)$ (where $Z(A)$ is the centre of $A$), with equality if and only if $A/J_{wc}(A)$ is abelian. Otherwise, $\textrm{CQ}(A)$ fails spectacularly to be a $C^*$-subalgebra of $A$.
Let X be a right Hilbert module over a C⁎-algebra A equipped with the canonical operator space structure. We define an elementary operator on X as a map ϕ:X→X for which there exists a finite number ...of elements ui in the C⁎-algebra B(X) of adjointable operators on X and vi in the multiplier algebra M(A) of A such that ϕ(x)=∑iuixvi for x∈X. If X=A this notion agrees with the standard notion of an elementary operator on A. In this paper we extend Mathieu's theorem for elementary operators on prime C⁎-algebras by showing that the completely bounded norm of each elementary operator on a non-zero Hilbert A-module X agrees with the Haagerup norm of its corresponding tensor in B(X)⊗M(A) if and only if A is a prime C⁎-algebra.
For a Hilbert C(X)-module V, where X is a compact metrizable space, we show that the following conditions are equivalent: (i) V is topologically finitely generated, (ii) there exists K∈N such that ...every algebraically finitely generated submodule of V can be generated with k≤K generators, (iii) V is canonically isomorphic to the Hilbert C(X)-module Γ(E) of all continuous sections of an (F) Hilbert bundle E=(p,E,X) over X, whose fibres Ex have uniformly finite dimensions, and each restriction bundle of E over a set where dimEx is constant is of finite type, (iv) there exists N∈N such that for every Banach C(X)-module W, each tensor in the C(X)-projective tensor product V⊗πC(X)W is of (finite) rank at most N.
We define an algebra A to be centrally stable if, for every epimorphism φ from A to another algebra B, the center Z(B) of B is equal to φ(Z(A)), the image of the center of A. After providing some ...examples and basic observations, we consider in somewhat greater detail central stability in tensor products of algebras, and finally establish our main result which states that a finite-dimensional unital algebra A over a perfect field F is centrally stable if and only if A is isomorphic to a direct product of algebras of the form Ci⊗FiAi, where Fi is a field extension of F, Ci is a commutative Fi-algebra, and Ai is a central simple Fi-algebra.
Abstract
We study variants of the Dixmier property that apply to elements of a unital C*-algebra, rather than to the C*-algebra itself. By a Dixmier element in a C*-algebra we understand one that can ...be averaged into a central element by means of a sequence of unitary mixing operators. Examples include all self-commutators and all quasinilpotent elements. We do a parallel study of an element-wise version of weak centrality, where the averaging to the centre is done using unital completely positive elementary operators (as in Magajna’s characterization of weak centrality). We also obtain complete descriptions of more tractable sets of elements, where the corresponding averaging can be done arbitrarily close to the centre. This is achieved through several “spectral conditions”, involving numerical ranges and tracial states.
Let A be a subhomogeneous C*-algebra. Then A contains an essential closed ideal J with the property that for every derivation δ of A there exists a multiplier a ∈ M(J) such that δ = ad(a) and ||δ|| = ...2||a||.
Let X be a compact Hausdorff space and let A be a unital C(X)-algebra, where C(X) is embedded as a unital
-subalgebra of the centre of A. We consider the problem of characterizing the existence of a ...conditional expectation
of finite index in terms of the associated
-bundle of A over X. More precisely, we show that if A admits a C(X)-valued conditional expectation of finite index, then A is necessarily a continuous C(X)-algebra, and there exists a positive integer N such that every fibre
of A is finite-dimensional, with
. We also give some sufficient conditions on A that ensure the existence of a C(X)-valued conditional expectation of finite index.
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