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Funktsional. Anal. i Prilozhen., 2001, Volume 35, Issue 4, Pages 20–25 (Mi faa269)  

This article is cited in 2 scientific papers (total in 3 papers)

A Constructive Proof of the Generalized Gelfand Isomorphism

V. M. Buchstabera, E. G. Reesb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b University of Edinburgh

Abstract: Using an analog of the classical Frobenius recursion, we define the notion of a Frobenius $n$-homomorphism. For $n=1$, this is an ordinary ring homomorphism. We give a constructive proof of the following theorem. Let $X$ be a compact Hausdorff space, $\operatorname{Sym}^n(X)$ the $n$th symmetric power of $X$, and $\mathbb{C}(X)$ the algebra of continuous complex-valued functions on $X$ with the sup-norm; then the evaluation map $\mathcal{E}\colon\operatorname{Sym}^n(X)\to\operatorname{Hom}(\mathbb{C}(X),\mathbb{C})$ defined by the formula $[x_1,…,x_n]\to(g\to\sum g(x_k))$ identifies the space $\operatorname{Sym}^n(X)$ with the space of all Frobenius $n$-homomorphisms of the algebra $\mathbb{C}(X)$ into $\mathbb{C}$ with the weak topology.

DOI: https://doi.org/10.4213/faa269

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English version:
Functional Analysis and Its Applications, 2001, 35:4, 257–260

Bibliographic databases:

Document Type: Article
UDC: 517.5
Received: 10.09.2001

Citation: V. M. Buchstaber, E. G. Rees, “A Constructive Proof of the Generalized Gelfand Isomorphism”, Funktsional. Anal. i Prilozhen., 35:4 (2001), 20–25; Funct. Anal. Appl., 35:4 (2001), 257–260

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. A. A. Bolibrukh, A. P. Veselov, A. B. Zhizhchenko, I. M. Krichever, A. A. Mal'tsev, S. P. Novikov, T. E. Panov, Yu. M. Smirnov, “Viktor Matveevich Buchstaber (on his 60th birthday)”, Russian Math. Surveys, 58:3 (2003), 627–635  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. V. M. Buchstaber, E. G. Rees, “Rings of continuous functions, symmetric products, and Frobenius algebras”, Russian Math. Surveys, 59:1 (2004), 125–145  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. D. V. Gugnin, “Topological applications of graded Frobenius $n$-homomorphisms”, Trans. Moscow Math. Soc., 72 (2011), 97–142  mathnet  crossref  mathscinet  zmath  elib
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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