Farey stellar subdivisions, ultrasimplicial groups, and \(K_ 0\) of AF \(C^*\)-algebras.

*(English)*Zbl 0678.06008The ideas and techniques in this paper range very far and wide. The first part develops Sylvester’s insights concerning generalized Farey series. In particular, it is shown that every finite triangulation K of \([0,1]^ n\) with rational vertices has a finite subdivision L such that the lattice associated with each n-simplex of L obeys Sylvester’s generalized law of unit-determinants. L is obtained from K via Alexander’s starring using Blichfeldt’s theorem in the geometry of numbers.

The author uses the \(\ell\)-groups \(M_ n\) of McNaughton’s functions over \([0,1]^ n\) to prove that every abelian \(\ell\)-group is ultrasimplicial, that is, order-isomorphic to the inductive limit of an upward directed system of simplicial groups with positive 1-1 homomorphisms. This strengthens both an earlier result of the author [J. Algebra 105, 236-241 (1987; Zbl 0605.06014)] and also some earlier results in the \(K_ 0\)- theory of AF \(C^*\)-algebras.

The author defines the nonsimple unital AF \(C^*\)-algebra \({\mathfrak M}_ 1\) by \((K_ 0({\mathfrak M}_ 1),[1_{{\mathfrak M}_ 1}])\cong (M_ 1,1)\), where \(M_ 1\) is the \(\ell\)-group of one variable McNaughton functions and \(\cong\) means isomorphism in the category \({\mathbb{A}}\) of abelian \(\ell\)-groups with strong unit and unit preserving \(\ell\)-homomorphisms. The AF \(C^*\)-algebra \({\mathfrak M}\) is similarly defined, replacing \(M_ 1\) with M (the McNaughton functions defined on the Hilbert cube). In the paper cited, the author proved every AF \(C^*\)-algebra may be embedded in a quotient of \({\mathfrak M}\). The author here proves that \({\mathfrak M}_ 1\) has analogous properties. Define the Effros-Shen unital AF \(C^*\)- algebra \({\mathfrak F}_{\theta}\) by \((K_ 0({\mathfrak F}_{\theta})[1_{{\mathfrak F}_{\theta}}])\cong ({\mathbb{Z}}+{\mathbb{Z}}\theta,1)\subset {\mathbb{R}}\), where \(0<\theta <1\), \(\theta\) irrational. Then, up to isomorphism, the \(C^*\)-algebras \({\mathfrak F}_{\theta}\) are precisely the infinite dimensional simple quotients of \({\mathfrak M}_ 1\), and each irrational rotation \(C^*\)-algebra may be embedded in a simple quotient of \({\mathfrak M}_ 1\). Further, \(K_ 0({\mathfrak M})=K_ 0({\mathfrak M}_ 1)\sqcup K_ 0({\mathfrak M}_ 1)\sqcup..\). (\(\omega\) copies), and \(K_ 0({\mathfrak M})\cong_{\to}({\mathbb{Z}}^{n(m)},A_ m)\). The positive 1-1 homomorphisms \(A_ m\) are recursively given as a sequence of (0,1)- matrices.

The author uses the \(\ell\)-groups \(M_ n\) of McNaughton’s functions over \([0,1]^ n\) to prove that every abelian \(\ell\)-group is ultrasimplicial, that is, order-isomorphic to the inductive limit of an upward directed system of simplicial groups with positive 1-1 homomorphisms. This strengthens both an earlier result of the author [J. Algebra 105, 236-241 (1987; Zbl 0605.06014)] and also some earlier results in the \(K_ 0\)- theory of AF \(C^*\)-algebras.

The author defines the nonsimple unital AF \(C^*\)-algebra \({\mathfrak M}_ 1\) by \((K_ 0({\mathfrak M}_ 1),[1_{{\mathfrak M}_ 1}])\cong (M_ 1,1)\), where \(M_ 1\) is the \(\ell\)-group of one variable McNaughton functions and \(\cong\) means isomorphism in the category \({\mathbb{A}}\) of abelian \(\ell\)-groups with strong unit and unit preserving \(\ell\)-homomorphisms. The AF \(C^*\)-algebra \({\mathfrak M}\) is similarly defined, replacing \(M_ 1\) with M (the McNaughton functions defined on the Hilbert cube). In the paper cited, the author proved every AF \(C^*\)-algebra may be embedded in a quotient of \({\mathfrak M}\). The author here proves that \({\mathfrak M}_ 1\) has analogous properties. Define the Effros-Shen unital AF \(C^*\)- algebra \({\mathfrak F}_{\theta}\) by \((K_ 0({\mathfrak F}_{\theta})[1_{{\mathfrak F}_{\theta}}])\cong ({\mathbb{Z}}+{\mathbb{Z}}\theta,1)\subset {\mathbb{R}}\), where \(0<\theta <1\), \(\theta\) irrational. Then, up to isomorphism, the \(C^*\)-algebras \({\mathfrak F}_{\theta}\) are precisely the infinite dimensional simple quotients of \({\mathfrak M}_ 1\), and each irrational rotation \(C^*\)-algebra may be embedded in a simple quotient of \({\mathfrak M}_ 1\). Further, \(K_ 0({\mathfrak M})=K_ 0({\mathfrak M}_ 1)\sqcup K_ 0({\mathfrak M}_ 1)\sqcup..\). (\(\omega\) copies), and \(K_ 0({\mathfrak M})\cong_{\to}({\mathbb{Z}}^{n(m)},A_ m)\). The positive 1-1 homomorphisms \(A_ m\) are recursively given as a sequence of (0,1)- matrices.

Reviewer: S.Hurd

##### MSC:

06F20 | Ordered abelian groups, Riesz groups, ordered linear spaces |

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |

11H06 | Lattices and convex bodies (number-theoretic aspects) |

##### Keywords:

ultrasimplicial groups; generalized Farey series; abelian \(\ell \)-group; \(K_ 0\)-theory of AF \(C^*\)-algebras
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DOI

##### References:

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