$\mathcal{Z}$ is universal

Abstract

We use order zero maps to express the Jiang-Su algebra Z as a universal C -algebra on countably many generators and relations, and we show that a natural deformation of these relations yields the stably projectionless algebra W studied by Kishimoto, Kumjian and others. Our presentation is entirely explicit and involves only -polynomial and order relations.

Rigidity Conjectures

“…This is, on the other hand, open for some of the most relevant stably finite algebras, such as Jiang and Su's Z ( [20]). It is known that Z is universal ( [18]), but since it is not weakly semiprojective, a finite set of weakly stable relations cannot be found. Again, if one works in Kadison-Kastler perturbation theory, it is known (see [17,Theorem 2.3]) that if A and B are separable C * -algebras on the same Hilbert space, and A is nuclear, then the fact that the unit ball of A is ǫ-contained in the unit ball of B for a sufficiently small ǫ (independent of A and B) is enough to prove that A embeds into B.…”

The Jiang-Su algebra is strongly self-absorbing revisited

“…This also confirms that the difficulty of showing that Z is strongly selfabsorbing lies between that for UHF-algebras of infinite type and O 2 or O ∞ (cf. [JW14,5.2]). Let us also mention that very recently yet another proof of the strong self-absorption of Z has been found in [Gha19], using the theory of Fraïssé limits.…”

“…To do so, we use a different picture of Z. Today, there are many descriptions and characterizations of Z, for example as universal C * -algebra ( [JW14]) or as the initial object in the category of strongly self-absorbing C * -algebras ( [Win11]). For us however, a construction of Z as an inductive limit of generalized dimension drop algebras will be most suitable.…”

Strongly self-absorbing $C^*$-algebras and Fraïssé limits

“…Since then many different characterizations of Z are given (cf. [17], [2] and [8]). It has been shown, already in [9], that Z is strongly self-absorbing.…”