Systems of derivations
HTML articles powered by AMS MathViewer
- by Frances Gulick PDF
- Trans. Amer. Math. Soc. 149 (1970), 465-488 Request permission
Abstract:
Let $A$ and $B$ be two complex algebras. A system of derivations of order $m$ from $A$ into $B$ is a set of $m + 1$ linear operators ${D_k}:A \to B(k = 0,1, \ldots ,m)$ such that for $x$, $y \in A$ and $k = 0$, 1, 2, …, $m$, \[ D_k(xy) = \sum _{j = 0}^k \binom {k}{j} (D_j x)(D_{k-j} y).\] If $A$ is a commutative, regular, semisimple $F$-algebra with an identity, $B$ the algebra of continuous functions on the closed maximal ideal space of $A$ and $(D_0$, $D_1$, …, $D_m)$ a system of derivations from $A$ into $B$ with $D_0$ the Gelfand mapping, then each $D_k$ is continuous. The continuity of the operators in a system of derivations from $C^n(U)$ into $C(U)(U \subset R \;\mathrm {open})$ is used to obtain a formula for $D_k f, f \in {C^n}(U)$, in terms of the ordinary derivatives of $f$ and functions in $C(U)$. Each system of derivations from $A$ into $B$ and each multiplicative seminorm on $B$ determine a multiplicative seminorm on $A$. Let $U$ be a subset of $C$ and $(D_0$, $D_1$, …, $D_m)$ a system of derivations from the algebra $P(x)$ of polynomials on $U$ into $C(U)$ with ${D_0}$ the identity operator. Then the system of derivations determines a Hausdorff topology on $P(x)$. If $U$ is open in $\mathbf {R}$ and $D_1 x(t) \ne 0$ for $t \in U(x(t) = t)$, then the completion of $P(x)$ in this topology is $C^m(U)$. If $U$ is open in $C$, then the completion of $P(x)$ in this topology is the algebra of functions analytic on $U$.References
- W. G. Bade and P. C. Curtis Jr., Homomorphisms of commutative Banach algebras, Amer. J. Math. 82 (1960), 589–608. MR 117577, DOI 10.2307/2372972
- R. M. Brooks, The structure space of a commutative locally convex algebra, Pacific J. Math. 25 (1968), 443–454. MR 231203
- Andrew Browder, Point derivations on function algebras, J. Functional Analysis 1 (1967), 22–27. MR 0211262, DOI 10.1016/0022-1236(67)90024-9
- Philip C. Curtis Jr., Derivations of commutative Banach algebras, Bull. Amer. Math. Soc. 67 (1961), 271–273. MR 126179, DOI 10.1090/S0002-9904-1961-10579-X
- Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
- Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Company, Boston, Mass.-New York-Toronto, Ont., 1962. MR 0201608
- Nathan Jacobson, Lectures in abstract algebra. Vol III: Theory of fields and Galois theory, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London-New York, 1964. MR 0172871
- B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math. 90 (1968), 1067–1073. MR 239419, DOI 10.2307/2373290
- Irving Kaplansky, Functional analysis, Some aspects of analysis and probability, Surveys in Applied Mathematics. Vol. 4, John Wiley & Sons, Inc., New York, N.Y.; Chapman & Hall, Ltd., London, 1958, pp. 1–34. MR 0101475
- Richard J. Loy, A note on the preceding paper by J. B. Miller, Acta Sci. Math. (Szeged) 28 (1967), 233–236. MR 213398
- Ernest A. Michael, Locally multiplicatively-convex topological algebras, Mem. Amer. Math. Soc. 11 (1952), 79. MR 51444
- John Boris Miller, Homomorphisms, higher derivations, and derivations on associative algebras, Acta Sci. Math. (Szeged) 28 (1967), 221–231. MR 212052 M. Naimark, Normed rings, GITTL, Moscow, 1956; English transl., Noordhoff, Groningen, 1964. MR 19, 870; MR 34 #4928.
- Howard Osborn, Modules of differentials. I, Math. Ann. 170 (1967), 221–244. MR 213987, DOI 10.1007/BF01350153
- 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
- Melvin Rosenfeld, Commutative $F$-algebras, Pacific J. Math. 16 (1966), 159–166. MR 190786
- Walter Rudin, Principles of mathematical analysis, 2nd ed., McGraw-Hill Book Co., New York, 1964. MR 0166310
- Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210528
- Donald R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. 111 (1964), 240–272. MR 161177, DOI 10.1090/S0002-9947-1964-0161177-1
- John Wermer, Bounded point derivations on certain Banach algebras, J. Functional Analysis 1 (1967), 28–36. MR 0215105, DOI 10.1016/0022-1236(67)90025-0
- Leopoldo Nachbin, Algebras of finite differential order and the operational calculus, Ann. of Math. (2) 70 (1959), 413–437. MR 108730, DOI 10.2307/1970323
- Leopoldo Nachbin, Sur les algèbres denses de fonctions différentiables sur une variété, C. R. Acad. Sci. Paris 228 (1949), 1549–1551 (French). MR 30590
Additional Information
- © Copyright 1970 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 149 (1970), 465-488
- MSC: Primary 46.55
- DOI: https://doi.org/10.1090/S0002-9947-1970-0275170-4
- MathSciNet review: 0275170