Separable topological algebras. I

Liddell, Michael J.

doi:10.1090/S0002-9947-1974-0352985-9

Separable topological algebras. I
HTML articles powered by AMS MathViewer

by Michael J. Liddell PDF

Trans. Amer. Math. Soc. 195 (1974), 31-59 Request permission

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

Maurice Auslander and Oscar Goldman, The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367–409. MR 121392, DOI 10.1090/S0002-9947-1960-0121392-6
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
Frank DeMeyer and Edward Ingraham, Separable algebras over commutative rings, Lecture Notes in Mathematics, Vol. 181, Springer-Verlag, Berlin-New York, 1971. MR 0280479

On the spectrum of an F-algebra

Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317. MR 402468, DOI 10.1007/BF02392270
J. M. G. Fell, The structure of algebras of operator fields, Acta Math. 106 (1961), 233–280. MR 164248, DOI 10.1007/BF02545788
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
Irving Kaplansky, The structure of certain operator algebras, Trans. Amer. Math. Soc. 70 (1951), 219–255. MR 42066, DOI 10.1090/S0002-9947-1951-0042066-0

Algebraic and analytic aspects of operator algebras

Joachim Lambek, Lectures on rings and modules, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. With an appendix by Ian G. Connell. MR 0206032
Saunders MacLane, Homology, 1st ed., Die Grundlehren der mathematischen Wissenschaften, Band 114, Springer-Verlag, Berlin-New York, 1967. MR 0349792
Anastasios Mallios, On the spectra of topological algebras, J. Functional Analysis 3 (1969), 301–309. MR 0239420, DOI 10.1016/0022-1236(69)90044-5
Ernest A. Michael, Locally multiplicatively-convex topological algebras, Mem. Amer. Math. Soc. 11 (1952), 79. MR 51444
Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
Jun Tomiyama and Masamichi Takesaki, Applications of fibre bundles to the certain class of $C^{\ast }$-algebras, Tohoku Math. J. (2) 13 (1961), 498–522. MR 139025, DOI 10.2748/tmj/1178244253
Joseph L. Taylor, Homology and cohomology for topological algebras, Advances in Math. 9 (1972), 137–182. MR 328624, DOI 10.1016/0001-8708(72)90016-3
Joseph L. Taylor, A general framework for a multi-operator functional calculus, Advances in Math. 9 (1972), 183–252. MR 328625, DOI 10.1016/0001-8708(72)90017-5
François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131