Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Separable topological algebras. I


Author: Michael J. Liddell
Journal: Trans. Amer. Math. Soc. 195 (1974), 31-59
MSC: Primary 46H05; Secondary 46M20
MathSciNet review: 0352985
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let A be a complete topological algebra with identity and B a subalgebra of the center of A. A notion of relative topological tensor product $ {\hat \otimes _B}$ for topological A modules and the resultant relative homology theory are introduced. Algebras of bidimension zero in this sense are called separable relative to B. Structure theorems are proved for such algebras under various topological assumptions on the algebra and its maximal ideal space.


References [Enhancements On Off] (What's this?)

  • [1] M. Auslander and O. Goldman, The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367-409. MR 22 #12130. MR 0121392 (22:12130)
  • [2] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 17, 1040. MR 0077480 (17:1040e)
  • [3] F. DeMeyer and E. Ingraham, Separable algebras over commutative rings, Lecture Notes in Math., vol. 181, Springer-Verlag, New York and Berlin, 1971. MR 43 #6199. MR 0280479 (43:6199)
  • [4] A. G. Dors, On the spectrum of an F-algebra, Ph.D. thesis, University of Utah, Salt Lake City, Utah, 1970.
  • [5] P. Enflo, A counterexample to the approximation property, Acta. Math. (to appear). MR 0402468 (53:6288)
  • [6] J. M. G. Fell, The structure of algebras of operator fields, Acta Math. 106 (1961), 233-280. MR 29 #1547. MR 0164248 (29:1547)
  • [7] A. Grothendieck, Produit tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955). MR 17, 763. MR 0075539 (17:763c)
  • [8] I. Kaplansky, The structure of certain operator algebras, Trans. Amer. Math. Soc. 70 (1951), 219-255. MR 13, 48. MR 0042066 (13:48a)
  • [9] -, Algebraic and analytic aspects of operator algebras, Regional Conference Series in Math., no. 1, Amer. Math. Soc., Providence, R. I., 1969.
  • [10] J. Lambek, Lectures on rings and modules, Blaisdell, Waltham, Mass., 1966. MR 34 #5857. MR 0206032 (34:5857)
  • [11] S. Mac Lane, Homology, Die Grundlehren der math. Wissenschaften, Band 114, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #122. MR 0349792 (50:2285)
  • [12] A. Mallios, On the spectra of topological algebras, J. Functional Analysis 3 (1969), 301-309. MR 39 #777. MR 0239420 (39:777)
  • [13] E. A. Michael, Locally multiplicatively convex topological algebras, Mem. Amer. Math. Soc. No. 11 (1952). MR 14, 482. MR 0051444 (14:482a)
  • [14] H. Schaefer, Topological vector spaces, Macmillan, New York, 1966. MR 33 #1689. MR 0193469 (33:1689)
  • [15] M. Takesaki and J. Tomiyama, Applications of fibre bundles to a certain class of $ {C^\ast}$ algebras, Tôhoku Math J. (2) 13 (1961), 498-522. MR 0139025 (25:2465)
  • [16] J. L. Taylor, Homology and cohomology for topological algebras, Advances in Math. (9) 2 (1972), 137-182. MR 0328624 (48:6966)
  • [17] -, A general framework for a multi-operator functional calculus, Advances in Math. (9) 2 (1972), 183-252. MR 0328625 (48:6967)
  • [18] F. Treves, Topological vector spaces, distributions, and kernels, Academic Press, New York, 1967. MR 37 #726. MR 0225131 (37:726)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46H05, 46M20

Retrieve articles in all journals with MSC: 46H05, 46M20


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0352985-9
PII: S 0002-9947(1974)0352985-9
Keywords: Splitting idempotent, separability, homogeneity, matrix-modular algebras, l.m.c. algebras, nuclearity, fully complete spaces, WSD spaces, projective tensor product, relative topological tensor product, split exact sequence
Article copyright: © Copyright 1974 American Mathematical Society