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Farey stellar subdivisions, ultrasimplicial groups, and \(K_ 0\) of AF \(C^*\)-algebras. (English) Zbl 0678.06008
The 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.
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)
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[1] Alexander, J.W, The combinatorial theory of complexes, Ann. of math., 31, 292-320, (1930) · JFM 56.0497.02
[2] Blichfeldt, H.F, A new principle in the geometry of numbers, with some applications, Trans. amer. math. soc., 15, 227-235, (1914) · JFM 45.0314.01
[3] Cassels, J.W.S, An introduction to the geometry of numbers, () · Zbl 0866.11041
[4] Cauchy, A.L, Démonstration d’un thèoréme curieux sur LES nombres, Bull. sc. soc. philomatique Paris, Exercices math., Oeuvres, 6, 2, 146-148, (1887), Reproduced in
[5] Dieudonné, J, Present trends in pure mathematics, Advances in math., 27, 235-255, (1978) · Zbl 0382.00009
[6] Effros, E.G, Dimensions and C∗-algebras, (), No. 46 · Zbl 0152.33203
[7] Effros, E.G; Shen, C.L, Approximately finite C∗-algebras and continued fractions, Indiana univ. math. J., 29, 191-204, (1980) · Zbl 0457.46046
[8] Effros, E.G; Handelman, D.E; Shen, C.L, Dimension groups and their affine representations, Amer. J. math., 102, 385-407, (1980) · Zbl 0457.46047
[9] Elliott, G.A, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. algebra, 38, 29-44, (1976) · Zbl 0323.46063
[10] Elliott, G.A, On totally ordered groups, and K0, (), 1-49
[11] Fuchs, L, Partially ordered algebraic systems, (1963), Pergamon Oxford · Zbl 0137.02001
[12] Handelman, D.E, Ultrasimplicial dimension groups, Arch. math. (basel), 40, 109-115, (1983) · Zbl 0513.46049
[13] Hardy, G.H; Wright, E.M, An introduction to the theory of numbers, (1979), Oxford Univ. Press (Clarendon) London · Zbl 0423.10001
[14] Lekkerkerker, C.G, Geometry of numbers, (1969), Wolters-Noordhoff Groningen, North-Holland, Amsterdam · Zbl 0198.38002
[15] McNaughton, R, A theorem about infinite-valued sentential logic, J. symbolic logic, 16, 1-13, (1951) · Zbl 0043.00901
[16] Monk, J.D, Mathematical logic, (1976), Springer-Verlag New York · Zbl 0354.02002
[17] Mundici, D, Interpretation of AF C∗-algebras in łukasiewicz sentential calculus, J. funct. analy., 65, 15-63, (1986) · Zbl 0597.46059
[18] Mundici, D, Every abelian l-group with two positive generators is ultrasimplicial, J. algebra, 105, 236-241, (1987) · Zbl 0605.06014
[19] Pierce, R.S, Amalgamations of lattice ordered groups, Trans. amer. math. soc., 172, 249-260, (1972) · Zbl 0259.06017
[20] Pimsner, M; Voiculescu, D, Imbedding the irrational rotation C∗-algebra into an AF algebra, J. operator theory, 4, 201-210, (1980) · Zbl 0525.46031
[21] Riedel, N, Classification of dimension groups and iterating systems, Math. scand., 48, 226-234, (1981) · Zbl 0498.46057
[22] Riedel, N, A counterexample to the unimodular conjecture on finitely generated dimension groups, (), 11-15 · Zbl 0474.06017
[23] Rieffel, M.A, C∗-algebras associated with irrational rotations, Pacific J. math., 93, 415-429, (1981) · Zbl 0499.46039
[24] Rourke, C.P; Sanderson, B.J, Introduction to piecewise-linear topology, () · Zbl 0254.57010
[25] Semadeni, Z, Schauder bases in Banach spaces of continuous functions, () · Zbl 0124.31703
[26] Shen, C.L, On the classification of the ordered groups associated with the approximately finite dimensional C∗-algebras, Duke math. J., 46, 613-633, (1979) · Zbl 0422.46046
[27] Sylvester, J.J, Intuitional exegesis of generalized Farey series, Amer. J. math., Collected math. papers, IV, 78-81, (1912)
[28] Tarski, A, Some model-theoretic results concerning weak second-order logic, Notices amer. math. soc., 5, 550, (1958)
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