Talk:Fréchet algebra

Latest comment: 8 years ago by Lschweitz in topic Properties

Expert needed

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I added Springer EoM as a reference but the definition used there does not match the definition used in the article. The definition used in the article seems to match the definition of m-convex B0-algebra which Springer says is the definition of Fréchet algebra used by "some authors". An expert is needed to add the Springer definition or at least add a redirect to an article that has it.--RDBury (talk) 20:57, 11 February 2010 (UTC)Reply

Michael's problem

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Is it still an open problem? What about the proof by Berit Stensones? http://www.math.lsa.umich.edu/~berit/michaelnov98.ps —Preceding unsigned comment added by Jaan Vajakas (talkcontribs) 23:47, 15 May 2010 (UTC)Reply

A google search shows that the article has not been published but the preprint has been referenced in some works. E. g. http://eom.springer.de/b/b120050.htm tells that "the Michael problem has an affirmative solution" and cites Stensones' paper. http://wwwmath.uni-muenster.de/u/walther.paravicini/files/essay.pdf discusses the problem and says about Stensones' work that "there is hope that the proof is correct". --Jaan Vajakas (talk) 00:20, 16 May 2010 (UTC)Reply

The paper is over 10 years old by now. Hard to believe it would take that long to referee even if it is very technical. So there must be some problems with it. I would regard the conjecture therefore as still open. — Preceding unsigned comment added by 130.75.46.166 (talk) 13:10, 26 October 2011 (UTC)Reply

Today I was told by two professors in the field (W. Želazko and M. Abel) that indeed a gap was found in her proof and the problem is still open. --Jaan Vajakas (talk) 13:33, 30 May 2012 (UTC)Reply



Rough Draft Fréchet Algebra

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In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra   over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplication operation   for   is required to be jointly continuous. If   is an increasing family[1] of seminorms for the topology of  , the joint continuity of multiplication is equivalent to there being a constant   and integer   for each   such that   for all  . Fréchet algebras are also called B0-algebras (Mitiagin et al. 1962).

A Fréchet algebra is  -convex if there exists such a family of semi-norms for which  . In that case, by rescaling the seminorms, we may also take   for each   and the seminorms are said to be submultiplicative:   for all  .  -convex Fréchet algebras may also be called Fréchet algebras (Husain 1991).

A Fréchet algebra may or may not have an identity element  . If   is unital, we do not require that  , as is often done for Banach algebras.

Lschweitz (talk) 22:25, 4 May 2011 (UTC)Reply

Properties

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  1. Continuity of multiplication. Multiplication is separately continuous if   implies   and   for every   and sequence   converging in the Fréchet topology of  . Multiplication is jointly continuous if   and   imply  . Joint continuity of multiplication follows from the definition of a Fréchet algebra. [2] For a Fréchet space with an algebra structure, if the multiplication is separately continuous, then it is automatically jointly continuous (Waelbroeck 1971), Chapter VII, Proposition 1, (Palmer 1994),   2.9.
  2. Group of invertible elements. If   is the set of invertible elements of  , then the inverse map  ,   is continuous if and only if   is a   set (Waelbroeck 1971), Chapter VII, Proposition 2. Unlike for Banach algebras,   may not be an open set. If   is open, then   is called a  -algebra. (If   happens to be non-unital, then we may adjoin a unit to  [3] and work with  , or the set of quasi invertibles[4] may take the place of  .)
  3. Conditions for  -convexity. A Fréchet algebra is  -convex if and only if for every increasing family   of seminorms which topologize  , for each   there exists   and   such that
     
    for all   and   (Mitiagin et al. 1962), Lemma 1.2. A commutative Fréchet  -algebra is  -convex (Żelazko 1965), Theorem 13.17. But there exist examples of non-commutative Fréchet  -algebras which are not  -convex (Żelazko 1994).
  4. Properties of  -convex Fréchet algebras. A Fréchet algebra is  -convex if and only if it is a countable projective limit of Banach algebras (Michael 1952), Theorem 5.1. An element of   is invertible if and only if it's image in each Banach algebra of the projective limit is invertible (Michael 1952), Theorem 5.2.


Lschweitz (talk) 05:44, 1 December 2015 (UTC)Reply

Examples

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  1. Zero multiplication. If   is any Fréchet space, we can make a Fréchet algebra structure by setting   for all  .
  2. Smooth functions on the circle. Let   be the circle group, or 1-sphere. This is a 1-dimensional compact differentiable manifold, with no boundary. Let   be the set of infinitely differentiable complex valued functions on  . This is clearly an algebra over the complex numbers, for pointwise multiplication. (Use the product rule for differentiation.) It is commutative, and the constant function   acts as an identity. Define a countable set of seminorms on   by
     
    where   denotes the supremum of the absolute value of the  th derivative  .[5] Then, by the product rule for differentiation, we have
     
    where   denotes the binomial coefficient  , and  . The primed seminorms are submultiplicative after re-scaling by  .
  3. Sequences on  . Let   be the space of complex-valued sequences on the natural numbers  . Define increasing seminorms by  . With pointwise multiplication,   is a commutative Fréchet algebra. In fact, each seminorm is submultiplicative   for all  . This  -convex Fréchet algebra is unital, since the constant sequence  ,   is in  .
  4. Equipped with the topology of uniform convergence on compact sets, and pointwise multiplication,  , the algebra of all continuous functions on the complex plane  , or to the algebra   of holomorphic functions on  .
  5. Convolution algebra of rapidly vanishing functions on a finitely generated discrete group. Let   be a finitely generated group, with the discrete topology. This means that there exists a set of finitely many elements   such that the union of all products   equals  . Without loss of generality, we may also assume that the identity element   of   is contained in  . Define a function   by
     
    Then  , and  , since we define  . Let   be the  -vector space
     
    where the seminorms   are defined by
     [6]
      is an  -convex Fréchet algebra for the convolution multiplication
     [7]
      is unital because   is discrete, and   is commutative if and only if   is Abelian.
  6. Non  -convex Fréchet algebras. The Aren's algebra   is an example of a commutative non- -convex Fréchet algebra with discontinuous inversion. The topology is given by   norms
     
    and multiplication is given by convolution of functions with respect to Lebesgue measure on   (Fragoulopoulou 2005), Example 6.13 (2).

Lschweitz (talk) 22:23, 4 May 2011 (UTC)Reply

Generalizations

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We can drop the requirement for the algebra to be locally convex, but still a complete metric space. In this case, the underlying space may be called a Fréchet space (Waelbroeck 1971) or an F-space (Rudin 1973, 1.8(e)).

If the requirement that the number of seminorms be countable is dropped, the algebra becomes locally convex (LC) or locally multiplicatively convex (LMC) (Micheal 1952) (Husain 1991). A complete LMC algebra is called an Arens-Michael algebra (Fragoulopoulou 2005, Chapter 1).

Lschweitz (talk) 01:08, 23 November 2015 (UTC)Reply

Open problems

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Perhaps themost famous, still open problem of the theory of topological algebras is whether all linear multiplicative functionals on a Frechet algebra are continuous. The statement that this be the case is known as Michael's Conjecture (Michael 1952).

Lschweitz (talk) 05:36, 30 November 2015 (UTC)Reply

Notes

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  1. ^ An increasing family means that for each  ,  .
  2. ^
     
  3. ^ If   is an algebra over a field  , the unitization   of   is the direct sum  , with multiplication defined as  
  4. ^ If  , then   is a quasi-inverse for   if  .
  5. ^ To see the completeness, let   be a Cauchy sequence. Then each derivative   is a Cauchy sequence in the sup norm on  , and hence converges uniformly to a continuous function   on  . It suffices to check that   is the  th derivative of  . But, using the fundamental theorem of calculus, and taking the limit inside the integral (using uniform convergence), we have  
  6. ^ To see that   is Fréchet space, let   be a Cauchy sequence. Then for each  ,   is a Cauchy sequence in  . Define   to be the limit. Then
     
    where the sum ranges over any finite subset   of  . Let  , and let   be such that   for  . By letting   run, we have
     
    for  . Summing over all of  , we therefore have   for  . By the estimate
     
    we obtain  . Since this holds for each  , we have   and   in the Fréchet topology, so   is complete.
  7. ^
     

Lschweitz (talk) 05:13, 30 November 2015 (UTC)Reply

References

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  • Fragoulopoulou, Maria (2005), Topological Algebras with Involution, North-Holland Mathematics Studies, vol. 200, Amsterdam: Elsevier B.V., doi:10.1016/S0304-0208(05)80031-3, ISBN 978-0-444-52025-8.
  • Husain, Taqdir (1991), Orthogonal Schauder Bases, Pure and Applied Mathematics, vol. 143, New York: Marcel Dekker, Inc., ISBN 0-8247-8508-8.
  • Michael, Ernest A. (1952), Locally Multiplicatively-Convex Topological Algebras, Memoirs of the American Mathematical Society, vol. 11, MR 0051444.
  • Mitiagin, B.; Rolewicz, S.; Żelazko, W. (1962), "Entire functions in B0-algebras", Studia Mathematica, 21: 291–306, MR 0144222.
  • Palmer, T. W. (1994), Banach Algebras and the General Theory of  -algebras, Volume I: Algebras and Banach Algebras, Encyclopedia of Mathematics and its Applications, vol. 49, New York: Cambridge University Press, ISBN 978-0-521-36637-3.
  • Rudin, Walter (1973), Functional Analysis, Series in Higher Mathematics, New York: McGraw-Hill Book Company, ISBN 978-0-070-54236-5.
  • Waelbroeck, Lucien (1971), Topological Vector Spaces and Algebras, Lecture Notes in Mathematics, vol. 230, doi:10.1007/BFb0061234, ISBN 978-3-540-05650-8, MR 0467234.
  • Żelazko, W. (1965), "Metric generalizations of Banach algebras", Rozprawy Mat. (Dissertationes Math.), 47, MR 0193532.
  • Żelazko, W. (1994), "Concerning entire functions in B0-algebras", Studia Mathematica, 110 (3): 283–290, MR 1292849.



Lschweitz (talk) 05:14, 30 November 2015 (UTC)Reply