An extension of compact operators by compact operators with no nontrivial multipliers

Abstract

We construct a nonhomogeneous, separably represented, type I and approximately finite dimensional C * -algebra such that its multiplier algebra is equal to its unitization. This algebra is an essential extension of the algebra K(ℓ 2 (c)) of compact operators on a nonseparable Hilbert space by the algebra K(ℓ 2 ) of compact operators on a separable Hilbert space, where c denotes the cardinality of continuum. Although both K(ℓ 2 (c)) and K(ℓ 2 ) are stable, our algebra is not. This sheds light on the permanence properties of the stability in the nonseparable setting. Namely, unlike in the separable case, an extension of a stable nonseparable C * -algebra by K(ℓ 2 ) does not have to be stable. Our construction can be considered as a noncommutative version of Mrówka's Ψ-space; a space whose one point compactification is equal to itš Cech-Stone compactification and is induced by a special uncountable family of almost disjoint subsets of N.

Extending representations of Banach algebras to their biduals

“…Indeed, there exist non-unital C * -algebras whose multiplier algebras agree with their minimal unitizations, and every C * -algebra is complemented in its minimal unitization. For examples and a further discussion, we refer to [GK16].…”

Noncommutative Cantor–Bendixson derivatives and scattered C⁎-algebras

“…As in the case of a thin-tall C * -algebra the above Ψ-algebra may or may not be stable. An example exhibiting very strong nonstability (its multiplier algebra is isomorphic to the minimal unitization) is constructed in [25].…”

“…One can also wonder how interesting such constructions are from the point of view of C * -algebras. We found two more examples, one of a thin-tall algebra and the other of a Ψ-algebra, which exhibit extraordinary behaviors, however, because of their complexity they are presented in separate papers [25,26].…”

“…All the above C * -algebras Aαs and A satisfy the following version of the definition of an AF algebra: any finite subset can be approximated from a finite-dimensional subalgebra. Two more complex constructions based on the language developed in this paper are presented in separate papers [25,26].Proof. We prove J α = I α ⊗ K(ℓ 2 ) by induction on α ≤ ht(A).…”

“…Through the Stone duality, they are quite topological, referring to the level of K. We aim at three goals: a) to present the usefulness of a natural notion of Cantor-Bendixson derivative of a C * -algebra by obtaining elementary and parallel (to the commutative case) new proofs of many results concerning scattered C * -algebras; b) to motivate the involution and talk about a it as a C * -algebra, by C(K) we always mean the C * -algebra of complex valued continuous functions. a collection of problems and constructions of C * -algebras parallel to the commutative well established topic of cardinal sequences of superatomic Boolean algebras or scattered (locally) compact Hausdorff spaces (more constructions can be found in our two subsequent papers [25] and [26]); c) to illustrate the accessibility of the noncommutative realm to the reasonings of set-theoretic topological nature.…”