The classification of real purely infinite simple C*-algebras

Abstract

We classify real Kirchberg algebras using united Ktheory. Precisely, let A and B be real simple separable nuclear purely infinite C*-algebras that satisfy the universal coefficient theorem such that A C and B C are also simple. In the stable case, A and B are isomorphic if and only if K CRT (A) ∼ = K CRT (B). In the unital case, A and B are isomorphic if and only ifWe also prove that the complexification of such a real C*-algebra is purely infinite, resolving a question left open from [43]. Thus the real C*-algebras classified here are exactly those real C*-algebras whose complexification falls under the classification result of Kirchberg [26] and Phillips [35]. As an application, we find all real forms of the complex Cuntz algebras O n for 2 ≤ n ≤ ∞.

L‐theory of C∗$C^*$‐algebras

“…Using that the map of Proposition 5.1 is to be compatible with external products, we immediately get the desired multiples. This proves cases (1), (4), and (10). By Theorem B and the remark following Theorem B in the introduction, we have natural surjections K 1 (𝐴) → L 1 (𝐴) and K 0 (𝐴 ℂ ) → L 2 (𝐴).…”

The Stable Exotic Cuntz Algebras are Higher-Rank Graph Algebras

“…As |Λ 0 n | = ∞, C * R (Λ n , γ n ) is a stable, simple, purely infinite, real C * -algebra, thanks to Proposition 4.1 and [6, Example 6.2]. We also know that K R ⊗ R E 2n+1 is a a stable, simple, purely infinite, real C * -algebra, because its complexification K⊗O n is simple and purely infinite (see Theorem 3.9 of [8]). Thus the first statement of the theorem follows by the classification of real Kirchberg algebras, [8, Theorem 10.2, Part (1)].…”

“…In our work, we will use the full united K-theory K CRT (A) (introduced in [3]) as well as the abbreviated variation K CR (A) which contains just the real and complex parts. Theorem 10.2 of [8] shows that the category of real purely infinite simple C * -algebras, whose complexifications are simple and in the UCT class, is classified up to isomorphism by either of these invariants. We tend to use K CR (A) since it is simpler and usually sufficient, but we will also need to use K CRT (A) on occasion since that is the context in which we have the Künneth formula.…”

“…For every odd integer n ≥ 3, there are two real C * -algebras whose complexification is O n : the real Cuntz algebra O R n and the exotic Cuntz algebra E n . While the existence of E n follows from the classification of simple purely infinite real C * -algebras [4,8], the non-constructive nature of the existence portion of this classification theorem [4, Theorem 1] means that we know very little about E n beyond its K-theory. In particular, until now there has been no description of E n in terms of familiar C * -algebraic objects.…”

“…We describe the graph Λ and involution γ in Proposition 4.1, and show that C * R (Λ, γ) is a real Kirchberg algebra which is KK-equivalent to the suspension algebra SR ∼ = C 0 ((0, 1), R). The Künneth Theorem for real C * -algebras [3] and the classification of real Kirchberg algebras [8]…”