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Using the Evans spectral sequence and its counter-part for real $K$-theory, we compute both the real and complex $K$-theory of several infinite families of $C^*$-algebras based on higher-rank graphs of rank $3$ and $4$. The higher-rank graphs we consider arise from double-covers of cube complexes. By considering the real and complex $K$-theory together, we are able to carry these computations much further than might be possible considering complex $K$-theory alone. As these algebras are classified by $K$-theory, we are able to characterize the isomorphism classes of the graph algebras in terms of the combinatorial and number-theoretic properties of the construction ingredients.

For each odd integer $n \geq 3$, we construct a rank-3 graph $\Lambda_n$ with involution $\gamma_n$ whose real C*-algebra $C^*_\mathbb{R}(\Lambda_n, \gamma_n)$ is stably isomorphic to the exotic Cuntz algebra $\mathcal E_n^\mathbb{R}$. This construction is optimal, as we prove that a rank-2 graph with involution $(\Lambda,\gamma)$ can never satisfy $C^*_\mathbb{R}(\Lambda, \gamma)\sim_{ME} \mathcal E_n^\mathbb{R}$, and the first author reached the same conclusion in previous work. Our construction relies on a rank-1 graph with involution $(\Lambda, \gamma)$ whose real C*-algebra $C^*_\mathbb{R}(\Lambda, \gamma)$ 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 $-2 \leq i \leq 1$.

We initiate the study of real $C^*$-algebras associated to higher-rank graphs $\Lambda$, with a focus on their $K$-theory. Following Kasparov and Evans, we identify a spectral sequence which computes the $\mathcal{CR}$ $K$-theory of $C^*_{\mathbb R} (\Lambda, \gamma)$ for any involution $\gamma$ on $\Lambda$, and show that the $E^2$ page of this spectral sequence can be straightforwardly computed from the combinatorial data of the $k$-graph $\Lambda$ and the involution $\gamma$. We provide a complete description of $K^{CR}(C^*_{\mathbb R}(\Lambda, \gamma))$ for several examples of higher-rank graphs $\Lambda$ with involution.

We prove that all eight KO groups for a real C*-algebra can be constructed from homotopy classes of unitary matrices that respect a variety of symmetries. In this manifestation of the KO groups, all eight boundary maps in the 24-term exact sequence associated to an ideal in a real C*-algebra can be computed as exponential or index maps with formulas that are nearly identical to the complex case., Comment: 79 pages. Reorganization of sections and notational clarifications since previous edition

We give a systematic account of the various pictures of KK-theory for real C*-algebras, proving natural isomorphisms between the groups that arise from each picture. As part of this project, we develop the universal properties of KK-theory, and we use CRT-structures to prove that a natural transformation from F(A) to G(A) between homotopy equivalent, stable, half-exact functors defined on real C*-algebras is an isomorphism provided it is an isomorphism on the smaller class of C*-algebras. Finally, we develop E-theory for real C*-algebras and use that to obtain new negative results regarding the problem of approximating almost commuting real matrices by exactly commuting real matrices., Comment: Version 1: 31 pages. arXiv admin note: text overlap with arXiv:0909.0972 . Version 2: Shorten the paper to 21 pages. Added a section on unitary representations of KO-classes

We establish axiomatic characterizations of $K$-theory and $KK$-theory for real C*-algebras. In particular, let $F$ be an abelian group-valued functor on separable real C*-algebras. We prove that if $F$ is homotopy invariant, stable, and split exact, then $F$ factors through the category $KK$. Also, if $F$ is homotopy invariant, stable, half exact, continuous, and satisfies an appropriate dimension axiom, then there is a natural isomorphism $K(A) \to F(A)$ for a large class of separable real C*-algebras $A$. Furthermore, we prove that a natural transformation $F(A) \to G(A)$ of homotopy invariant, stable, half-exact functors which is an isomorphism when $A$ is complex is necessarily an isomorphism when $A$ is real., Comment: This paper has been replaced by "Pictures of KK-theory for real C*-algebras" which is posted in the arXiv at arXiv:1210.3133. Several different pictures of KK-theory for real C*-algebras is given there and the results are used to obtain new results for the problem of approximating a set of almost commuting matrices over the field of real numbers

We prove that united K-theory is a surjective functor from the category of real simple purely infinite C*-algebras to the cateogry of countable acyclic CRT-modules., Comment: 13 pages

We define united KK-theory for real C*-algebras A and B such that A is separable and B is sigma-unital, extending united K-theory in the sense that KK\crt(\R, B) = K\crt(B). United KK-theory contains real, complex, and self-conjugate KK-theory; but unlike unaugmented real KK-theory, it admits a universal coefficient theorem. For all separable A and B in which the complexification of A is in the bootstrap category, KK\crt(A,B) can be written as the middle term of a short exact sequence whose outer terms involve the united K-theory of A and B. As a corollary, we prove that united K-theory classifies KK-equivalence for real C*-algebras whose complexification is in the bootstrap category., Comment: 42 pages; correction to abstract: the correct hypothesis is that the complexification of A (not B) is in the bootstrap category

We define united K-theory for real C*-algebras, generalizing Bousfield's topological united K-theory. United K-theory incorporates three functors -- real K-theory, complex K-theory, and self-conjugate K-theory -- and the natural transformations among them. The advantage of united K-theory over ordinary K-theory lies in its homological algebraic properties, which allow us to construct a Kunneth-type, non-splitting, short exact sequence whose middle term is the united K-theory of the tensor product of two real C*-algebras A and B which holds as long as the complexification of A is in the bootstrap category. Since united K-theory contains ordinary K-theory, our sequence provides a way to compute the K-theory of the tensor product of two real C*-algebras. As an application, we compute the united K-theory of the tensor product of two real Cuntz algebras. Unlike in the complex case, it turns out that the isomorphism class of the tensor product O_{k+1} otimes O_{l+1} is not determined solely by the greatest common divisor of k and l. Hence we have examples of non-isomorphic, simple, purely infinite, real C*-algebras whose complexifications are isomorphic., Comment: 67 pages, 13 tables, uses xy-pic for commutative diagrams, to appear in "K-theory"