Abstract. We consider derivations in the image of the canonical contraction θ A from the Haagerup tensor product of a C * -algebra A with itself to the space of completely bounded maps on A . We show that such derivations are necessarily inner if A is prime or if A is central. We also provide an example of a C * -algebra which has an outer derivation implemented by an elementary operator.Mathematics subject classification (2010): 46L05, 46L07, 47B47.

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 $A$.

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 ⊗F i Ai, where Fi is a field extension of F , Ci is a commutative Fi-algebra, and Ai is a central simple Fi-algebra.

Let A be a C*-algebra and let ΘA be the canonical contraction form the Haagerup tensor product of M(A) with itself to the space of completely bounded maps on A. In this paper we consider the following conditions on A: (a) A is a finitely generated module over the centre of M(A); (b) the image of ΘA is equal to the set E(A) of all elementary operators on A; and (c) the lengths of elementary operators on A are uniformly bounded. We show that A satisfies (a) if and only if it is a finite direct sum of unital homogeneous C*-algebras. We also show that if a separable A satisfies (b) or (c), then A is necessarily subhomogeneous and the C*-bundles corresponding to the homogeneous subquotients of A must be of finite type.

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