Completely bounded mappings and simplicial complex structure in the primitive ideal space of a $C^*$-algebra

Abstract

Abstract. We consider the natural contraction from the central Haagerup tensor product of a C*-algebra A with itself to the space of completely bounded maps CB(A) on A and investigate those A where there exists an inverse map with finite norm L(A). We show that a stabilised version L (A) = sup n L(M n (A)) depends only on the primitive ideal space Prim(A). The dependence is via simplicial complex structures (defined from primal intersections) on finite sets of primitive ideals that contain a Glimm ideal of A., with K(H) the compact operators, which requires us to develop the theory in the context of C*-algebras that are not necessarily unital.

The mathematical legacy of Richard Timoney

“…Theorem 22 is proved in [28] and extended in a highly technical fashion to the non-unital case in [31].…”

Separation properties in the primitive ideal space of a multiplier algebra

“…Furthermore J A ⊆ ker θ, so θ induces a contractive map θ Z : A ⊗ Z,h A → CB(A) where θ Z (u + J A ) = θ(u) (u ∈ A ⊗ h A). It was shown in [14,Theorem 3.8] that θ Z is injective if and only if every Glimm ideal of A is 2-primal and that θ Z is isometric if and only if every Glimm ideal of A is primal.…”

On derivations and elementary operators onC*-algebras

“…If A is not prime, one considers the central [31]). in [29,Theorem 4] and [7, Theorem 7] (see also [8]); θ Z A is isometric if and only if each Glimm ideal of A is primal. in [29,Theorem 4] and [7, Theorem 7] (see also [8]); θ Z A is isometric if and only if each Glimm ideal of A is primal.…”

“…The problem of when θ Z A is isometric has been recently completely solved by Archbold et al . in [29,Theorem 4] and [7,Theorem 7] (see also [8]); θ Z A is isometric if and only if each Glimm ideal of A is primal. As an easy consequence of this result, we obtain the following.…”