A simple, monotracial, stably projectionless C*-algebra

Fundamental Group of Simple C*-algebras with Unique Trace III

“…Remark 4.19. Recently, Jacelon [15] construct a simple, nuclear, stably projectionless C * -algebra W with a unique normalized trace, which shares some of the important properties of the Cuntz algebra O 2 . This C * -algebra is an inductive limit of building blocks A(n, m).…”

Reduction of the dimension of nuclear C*-algebras

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We show that for a large class of C*-algebras A, containing arbitrary direct limits of separable type I C*-algebras, the following statement holds: If A ∈ A and B is a simple projectionless C*-algebra with trivial K-groups that can be written as a direct limit of a system of (nonunital) recursive subhomogeneous algebras with no dimension growth then the stable rank of A ⊗ B is one. As a consequence we show that if A ∈ A and W is the C*-algebra constructed in [12] then the stable rank of A ⊗ W is one. We also prove the following stronger result: If A is separable C*-algebra that can be written as a direct limit of C*-algebras of the form C0(X) ⊗ Mn, where X is locally compact and Hausdorff, then A ⊗ W can be written as a direct limit of a sequence of 1-dimensional noncommutative CW-complexes.

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“…An important example of a simple stably projectionless C*-algebra with trivial K-groups is the C*-algebra W constructed in [12]. This C*-algebra is an inductive limit of RSH 0 -algebra; thus, the first part of Theorem 3.6 applies to W. In other words, the following statement hold: Corollary 3.8.…”

“…In this paper we study different notions of dimension for certain C*algebras of the form A ⊗ B, where B is simple, nuclear, either projectionless or unital with no nonzero projections but its unit, and with a specified direct limit decomposition. We are particularly interested in two cases: the first case is when B is the C*-algebra W constructed in [12] and the second is when B is the Jiang-Su algebra Z. It was shown in [27] that if A is a Date: August 16, 2018. simple finite C*-algebra then the stable rank of A ⊗ Z is one.…”

Picard groups of certain stably projectionless C*-algebras

“…Since there exists a unital simple Z-stable algebra A with a unique tracial state such that Out(A) is not a normal subgroup of Pic(A), Z-stable stably projectionless C * -algebras are in this sense more well-behaved than unital stably finite Z-stable C * -algebras. Let W 2 be the Razak-Jacelon algebra studied in [13], [37], which has trivial K-groups and a unique tracial state and no unbounded trace. Then W 2 is Z-stable, and hence the sequence above is exact in this case.…”