A relative bicommutant theorem: The stable case of Pedersen's question

Abstract

In 1976, D. Voiculescu proved that every separable unital sub-C*algebra of the Calkin algebra is equal to its (relative) bicommutant. In his minicourse (see [17]), G. Pedersen asked in 1988 if Voiculescu's theorem can be extended to a simple corona algebra of a σ-unital C*-algebra. In this note, we answer Pedersen's question for a stable σ-unital C*-algebra.

Double relative commutants in coronas of separable C*-algebras

“…The following lemma is similar to [17], Lemma 3.3. It applies to, for example, separable subalgebras of the range of the trivial extension …”

“…Giordano and Ng [17,Corollary 3.5] proved: Theorem 4. Let B be a stable separable C * -algebra and suppose that either B is the compact operators or B is simple and purely infinite.…”

“…At a conference in 1988, Pedersen posed the problem of generalizing Voiculescu’s theorem to the setting of general corona algebras [38], and this note provides a partial answer to his problem. We use some ideas from Kadison–Singer [25], a theorem from KK -theory [13], and previous double commutant theorems [14, 17] in our proof. In [14] is the following double relative commutant theorem for hereditary subalgebras in C * -algebras of the form with B simple and separable:…”

The Calkin algebra, Kazhdan's property (T), and strongly self‐absorbing C∗$\mathrm{C}^*$‐algebras

“…They are simple and purely infinite: By [45, Theorem 3.2] or by [35, Definition 2.5 and Theorem 2.8] and [36], if $$A$$ is unital, then $$\mathcal {Q}(A\otimes \mathcal {K})$$ is simple if and only if $$A=M_n({\mathbb {C}})$$ or $$A$$ is simple and purely infinite. Every separable, unital $$\mathrm{C}^*$$‐subalgebra is equal to its double commutant (this is a consequence of Voiculescu's theorem for $$\mathcal {Q}(H)$$, and proved in [28, Theorem B] using the Elliott–Kucerovsky theory of absorbing extensions, [20] for $$\mathcal {Q}(\mathcal {O}_\infty \otimes \mathcal {K})$$; see also [40]). Many general properties of the Ext functor resemble the ones familiar from the BDF theory ([8, 9], also [15]) by [42].…”