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Register a new userK-Theory of Approximately Central Projections in the Flip Orbifold
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Authors
Samuel G. Walters
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For an approximately central (AC) Powers-Rieffel projection $e$ in the irrational Flip orbifold C*-algebra $A_theta^Phi,$ where $Phi$ is the Flip automorphism of the rotation C*-algebra $A_theta,$ we compute the Connes-Chern character of the cutdown of any projection by $e$ in terms of K-theoretic invariants of these projections. This result is then applied to computing a complete K-theoretic invariant for the projection $e$ with respect to central equivalence (within the orbifold). Thus, in addition to the canonical trace, there is a $4times6$ K-matrix invariant $K(e)$ arising from unbounded traces of the cutdowns of a canonically constructed basis for $K_0(A_theta^Phi) = mathbb Z^6$. Thanks to a theorem of Kishimoto, this enables us to tell when AC projections in $A_theta^Phi$ are Murray-von Neumann equivalent via an approximately central partial isometry (or unitary) in $A_theta^Phi$. As additional application, we obtain the K-matrix of canonical SL$(2,mathbb Z)$-automorphisms of $e$ and show that there is a subsequence of $e$ such that $e, sigma(e), kappa(e), kappa^2(e), sigmakappa(e), sigmakappa^2(e)$ -- which are the orbit elements of $e$ under the symmetric group $S_3 subset$ SL$(2,mathbb Z)$ -- are pairwise centrally not equivalent, and that each SL$(2,mathbb Z)$ image of $e$ is centrally equivalent to one of these, where $sigma, kappa$ are the Fourier and Cubic transform automorphisms of the rotation algebra.
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It is shown that for any approximately central (AC) projection $e$ in the Flip orbifold $A_theta^Phi$ (of the irrational rotation C*-algebra $A_theta$), and any modular automorphism $alpha$ (arising from SL$(2,mathbb Z)$), the AC projection $alpha(e)$ is centrally Murray-von Neumann equivalent to one of the projections $e, sigma(e), kappa(e), kappa^2(e),$ $sigmakappa(e), sigmakappa^2(e)$ in the $S_3$-orbit of $e,$ where $sigma, kappa$ are the Fourier and Cubic transforms of $A_theta$. (The equivalence being implemented by an approximately central partial isometry in $A_theta^Phi$.) For smooth automorphisms $alpha,beta$ of the Flip orbifold $A_theta^Phi$, it is also shown that if $alpha_*=beta_*$ on $K_0(A_theta^Phi),$ then $alpha(e)$ and $beta(e)$ are centrally equivalent for each AC projection $e$.
Let $A$ be a unital AF-algebra whose Murray-von Neumann order of projections is a lattice. For any two equivalence classes $[p]$ and $[q]$ of projections we write $[p]sqsubseteq [q]$ iff for every primitive ideal $mathfrak p$ of $A$ either $p/mathfrak ppreceq q/mathfrak ppreceq (1- q)/mathfrak p$ or $p/mathfrak psucceq q/mathfrak p succeq (1-q)/mathfrak p.$ We prove that $p$ is central iff $[p]$ is $sqsubseteq$-minimal iff $[p]$ is a characteristic element in $K_0(A)$. If, in addition, $A$ is liminary, then each extremal state of $K_0(A)$ is discrete, $K_0(A)$ has general comparability, and $A$ comes equipped with a centripetal transformation $[p]mapsto [p]^Game$ that moves $p$ towards the center. The number $n(p) $ of $Game$-steps needed by $[p]$ to reach the center has the monotonicity property $[p]sqsubseteq [q]Rightarrow n(p)leq n(q).$ Our proofs combine the $K_0$-theoretic version of Elliotts classification, the categorical equivalence $Gamma$ between MV-algebras and unital $ell$-groups, and L os ultraproduct theorem for first-order logic.
We develop a symbol calculus for normal bimodule maps over a masa that is the natural analogue of the Schur product theory. Using this calculus we are able to easily give a complete description of the ranges of contractive normal bimodule idempotents that avoids the theory of J*-algebras. We prove that if $P$ is a normal bimodule idempotent and $|P| < 2/sqrt{3}$ then $P$ is a contraction. We finish with some attempts at extending the symbol calculus to non-normal maps.
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.
Let $X$ be a compact metric space which is locally absolutely retract and let $phi: C(X)to C(Y, M_n)$ be a unital homomorphism, where $Y$ is a compact metric space with ${rm dim}Yle 2.$ It is proved that there exists a sequence of $n$ continuous maps $alfa_{i,m}: Yto X$ ($i=1,2,...,n$) and a sequence of sets of mutually orthogonal rank one projections ${p_{1, m}, p_{2,m},...,p_{n,m}}subset C(Y, M_n)$ such that $$ lim_{mtoinfty} sum_{i=1}^n f(alfa_{i,m})p_{i,m}=phi(f) for all fin C(X). $$ This is closely related to the Kadison diagonal matrix question. It is also shown that this approximate diagonalization could not hold in general when ${rm dim}Yge 3.$
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