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For each odd integer n \geq 3, we construct a rank-3 graph \Lambda _n with involution \gamma _n whose real C^*-algebra C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n) is stably isomorphic to the exotic Cuntz algebra \mathcal E_n. This construction is optimal, as we prove that a rank-2 graph with involution (\Lambda,\gamma) can never satisfy C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda, \gamma)\sim _{ME} \mathcal E_n, and Boersema reached the same conclusion for rank-1 graphs (directed graphs) in [Münster J. Math. 10 (2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank-1 graph with involution (\Lambda, \gamma) whose real C^*-algebra C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda, \gamma) is stably isomorphic to the suspension S \mathbb {R}. In the Appendix, we show that the i-fold suspension S^i \mathbb {R} is stably isomorphic to a graph algebra iff -2 \leq i \leq 1. [ABSTRACT FROM AUTHOR]

We introduce a real C*-algebra C*R(E;γ ) associated to a graph E with an involution γ. In the case that is the identity, this algebra is isomorphic to the real graph algebra C*R(E) studied in [5]. By allowing nontrivial involutions we obtain much larger class of real C*-algebras. We develop some basic theory, and we prove a Pimsner-Voiculescu type long exact sequence for the real K-theory of C*R(E; γ). We present several examples, some that we analyze directly and others that we analyze by combining K-theory computations with classification theorems. [ABSTRACT FROM AUTHOR]

We prove analogs of the Kaplansky Density Theorem and the Kadison Transitivity Theorem for irreducible representations of a real C*-algebra on a real Hilbert space. Specifically, if a C*-algebra is acting irreducibly on a real Hilbert space, then the Hilbert space has either a real, complex, or quaternionic structure with respect to which the density and transitivity theorems hold. [ABSTRACT FROM AUTHOR]

Abstract: We prove that the united K-theory functor is a surjective functor from the category of real simple separable purely infinite C ∗-algebras to the category of countable acyclic CRT-modules. As a consequence, we show that every complex Kirchberg algebra satisfying the universal coefficient theorem is the complexification of a real -algebra. [Copyright &y& Elsevier]