The ideal structure of certain nonselfadjoint operator algebras

Author:
Justin Peters

Journal:
Trans. Amer. Math. Soc. **305** (1988), 333-352

MSC:
Primary 47D25; Secondary 46H20, 46L55, 46M99

DOI:
https://doi.org/10.1090/S0002-9947-1988-0920162-6

MathSciNet review:
920162

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a locally compact dynamical system, and the norm-closed subalgebra of the crossed product generated by the nonnegative powers of in case is a homeomorphism. If is just a continuous map, can still be defined by a crossed product type construction. The ideal structure of these algebras is determined in case acts freely. A class of strictly transitive Banach modules is described, indicating that for the nonselfadjoint operator algebras considered here, not all irreducible representations are on Hilbert space. Finally in a special case, the family of all invariant maximal right ideals is given.

**[1]**William Arveson,*Ten lectures on operator algebras*, CBMS Regional Conference Series in Mathematics, vol. 55, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1984. MR**762819****[2]**William B. Arveson and Keith B. Josephson,*Operator algebras and measure preserving automorphisms. II*, J. Functional Analysis**4**(1969), 100–134. MR**0250081****[3]**James R. Brown,*Ergodic theory and topological dynamics*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 70. MR**0492177****[4]**W. F. Eberlein,*Abstract ergodic theorems and weak almost periodic functions*, Trans. Amer. Math. Soc.**67**(1949), 217–240. MR**0036455**, https://doi.org/10.1090/S0002-9947-1949-0036455-9**[5]**Edward G. Effros and Frank Hahn,*Locally compact transformation groups and 𝐶*- algebras*, Memoirs of the American Mathematical Society, No. 75, American Mathematical Society, Providence, R.I., 1967. MR**0227310****[6]**H. Furstenberg,*Recurrence in ergodic theory and combinatorial number theory*, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR**603625****[7]**Walter Helbig Gottschalk and Gustav Arnold Hedlund,*Topological dynamics*, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R. I., 1955. MR**0074810****[8]**I. N. Herstein,*Noncommutative rings*, The Carus Mathematical Monographs, No. 15, Published by The Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968. MR**0227205****[9]**Kenneth Hoffman,*Banach spaces of analytic functions*, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR**0133008****[10]**Paul S. Muhly,*Radicals, crossed products, and flows*, Ann. Polon. Math.**43**(1983), no. 1, 35–42. MR**727885**, https://doi.org/10.4064/ap-43-1-35-42**[11]**J. Peters,*Semi-crossed products of*-*algebras*, J. Funct. Anal.**599**(1984), 498-534.**[12]**Charles E. Rickart,*General theory of Banach algebras*, The University Series in Higher Mathematics, D. van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0115101****[13]**Baruch Solel,*Irreducible triangular algebras*, Mem. Amer. Math. Soc.**47**(1984), no. 290, iv+87. MR**727817**, https://doi.org/10.1090/memo/0290**[14]**Peter Walters,*Ergodic theory—introductory lectures*, Lecture Notes in Mathematics, Vol. 458, Springer-Verlag, Berlin-New York, 1975. MR**0480949**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0920162-6

Keywords:
Dynamical system,
nonselfadjoint operator algebra,
free action,
Banach module

Article copyright:
© Copyright 1988
American Mathematical Society