Centrally stable algebras

Abstract

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.

The centre-quotient property and weak centrality for $C^*$-algebras

“…The following result was obtained in [4, Proposition 2.2.4] but we give a shorter argument in one direction by using the method of [16,Proposition 2.15]. Proposition 3.4.…”

The Centre-Quotient Property and Weak Centrality forC*-Algebras

“…We begin this section with the following C * -algebraic version of [16,Proposition 2.1] in which for a ∈ A,…”

“…Motivated by [16], and with a view to identifying the individual elements which prevent the CQ-property, we now introduce a local version of the CQ-property. The CQ-property and weak centrality 21…”

“…Recently, Robert, Tikuisis and the first-named author found the exact gap between weak centrality and the Dixmier property for unital C * -algebras [8,Theorem 2.6] and showed that a postliminal C * -algebra has the (singleton) Dixmier property if and only if it has the CQ-property [8,Theorem 2.12]. Also, in a recent paper [16], Brešar and the second-named author studied an analogue of the CQ-property in a wider algebraic setting (so called 'centrally stable algebras').…”

“…By a well-known result of Vesterstrøm [24] a unital C * -algebra A is weakly central if and only if it satisfies the centre-quotient property (the CQ-property), that is for any closed twosided ideal I of A, (Z(A) + I)/I = Z(A/I). Motivated by results in [9], where the setting is purely algebraic, the first two authors defined in [5] a local version of the CQ-property as follows. An element a ∈ A is called a CQ-element if for every closed two-sided ideal I of A, a + I ∈ Z(A/I) implies a ∈ Z(A) + I ( [5,Definition 4.1]).…”