Lexicographic TAF Algebras

Peters, Justin; Poon, Yiu

doi:10.1090/S0002-9947-97-02040-0

Lexicographic TAF Algebras
HTML articles powered by AMS MathViewer

by Justin R. Peters and Yiu Tung Poon PDF

Trans. Amer. Math. Soc. 349 (1997), 4825-4855 Request permission

Abstract:

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 $\Omega$.

References

Allan P. Donsig, Semisimple triangular AF algebras, J. Funct. Anal. 111 (1993), no. 2, 323–349. MR 1203457, DOI 10.1006/jfan.1993.1016
Allan P. Donsig and Alan Hopenwasser, Order preservation in limit algebras, J. Funct. Anal. 133 (1995), no. 2, 342–394. MR 1354035, DOI 10.1006/jfan.1995.1129
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. MR 1194074, DOI 10.1142/S0129167X92000382
Sze-tsen Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959. MR 0106454
Paul S. Muhly and Baruch Solel, Subalgebras of groupoid $C^*$-algebras, J. Reine Angew. Math. 402 (1989), 41–75. MR 1022793, DOI 10.1515/crll.1989.402.41
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. MR 1191255
J. R. Peters, Y. T. Poon, and B. H. Wagner, Triangular AF algebras, J. Operator Theory 23 (1990), no. 1, 81–114. MR 1054818
Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94
Y. T. Poon and B. H. Wagner, $\textbf {Z}$-analytic TAF algebras and dynamical systems, Houston J. Math. 19 (1993), no. 2, 181–199. MR 1225457
S. C. Power, On the outer automorphism groups of triangular alternation limit algebras, J. Funct. Anal. 113 (1993), no. 2, 462–471. MR 1218104, DOI 10.1006/jfan.1993.1058
S. C. Power, Infinite lexicographic products of triangular algebras, Bull. London Math. Soc. 27 (1995), no. 3, 273–277. MR 1328704, DOI 10.1112/blms/27.3.273
S. C. Power, Lexicographic semigroupoids, Ergodic Theory Dynam. Systems 16 (1996), no. 2, 365–377. MR 1389629, DOI 10.1017/S0143385700008853
Joseph G. Rosenstein, Linear orderings, Pure and Applied Mathematics, vol. 98, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. MR 662564
B. Wagner, Triangular AF algebras induced by lexicographic orders, preliminary report.