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Peters, Justin R.; Poon, Yiu Tung

Lexicographic TAF Algebras. (English) Zbl 0992.46046

Trans. Am. Math. Soc. 349, No. 12, 4825-4855 (1997).
Summary: Lexicographic TAF algebras constitute a class of triangular AF algebras which are determined by a countable ordered set \(\Omega\), a dimension function, and a third parameter. While some of the important examples of TAF algebras belong to the class, most algebras in this class have not been studied. The semigroupoid of the algebra, the lattice of invariant projections, the Jacobson radical, and for some cases the automorphism group are computed. Necessary and sufficient conditions for analyticity are given. The results often involve the order properties of the set \(\Omega\).

Cited in 3 Documents

MSC:

46L35 Classifications of \(C^*\)-algebras
47L30 Abstract operator algebras on Hilbert spaces
46M40 Inductive and projective limits in functional analysis
06F25 Ordered rings, algebras, modules

Keywords:

lexicographic TAF algebras; triangular AF algebras; semigroupoid; lattice of invariant projections; Jacobson radical; automorphism group
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References:

[1] Allan P. Donsig, Semisimple triangular AF algebras, J. Funct. Anal. 111 (1993), no. 2, 323 – 349. · Zbl 0808.47031 · doi:10.1006/jfan.1993.1016
[2] Allan P. Donsig and Alan Hopenwasser, Order preservation in limit algebras, J. Funct. Anal. 133 (1995), no. 2, 342 – 394. · Zbl 0891.47031 · doi:10.1006/jfan.1995.1129
[3] Richard H. Herman, Ian F. Putnam, and Christian F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3 (1992), no. 6, 827 – 864. · Zbl 0786.46053 · doi:10.1142/S0129167X92000382
[4] Sze-tsen Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959. · Zbl 0088.38803
[5] Paul S. Muhly and Baruch Solel, Subalgebras of groupoid \?*-algebras, J. Reine Angew. Math. 402 (1989), 41 – 75. · Zbl 0673.46037 · doi:10.1515/crll.1989.402.41
[6] J. R. Peters and B. H. Wagner, Triangular AF algebras and nest subalgebras of UHF algebras, J. Operator Theory 25 (1991), no. 1, 79 – 123. · Zbl 0783.46030
[7] J. R. Peters, Y. T. Poon, and B. H. Wagner, Triangular AF algebras, J. Operator Theory 23 (1990), no. 1, 81 – 114. · Zbl 0717.46050
[8] J.R. Peters, Y.T. Poon and B.H. Wagner, Analytic TAF algebras, Can. J. Math., Vol. 45 (5), (1993), 1009-1031. (Correction: Vol. 46 (2) (1994), 395-6.). · Zbl 0810.46046
[9] Y. T. Poon and B. H. Wagner, \?-analytic TAF algebras and dynamical systems, Houston J. Math. 19 (1993), no. 2, 181 – 199. · Zbl 0816.46040
[10] S. C. Power, On the outer automorphism groups of triangular alternation limit algebras, J. Funct. Anal. 113 (1993), no. 2, 462 – 471. · Zbl 0812.46056 · doi:10.1006/jfan.1993.1058
[11] S. C. Power, Infinite lexicographic products of triangular algebras, Bull. London Math. Soc. 27 (1995), no. 3, 273 – 277. · Zbl 0829.47034 · doi:10.1112/blms/27.3.273
[12] S. C. Power, Lexicographic semigroupoids, Ergodic Theory Dynam. Systems 16 (1996), no. 2, 365 – 377. · Zbl 0849.46048 · doi:10.1017/S0143385700008853
[13] Joseph G. Rosenstein, Linear orderings, Pure and Applied Mathematics, vol. 98, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. · Zbl 0488.04002
[14] B. Wagner, Triangular AF algebras induced by lexicographic orders, preliminary report.
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