# Fréchet algebra

$F$-algebra, algebra of type $F$

A completely metrizable topological algebra. The joint continuity of multiplication need not be demanded since it follows from the separate continuity (see Fréchet topology). The $F$-algebras can be classified similarly as the $F$-spaces (see Fréchet topology), so one can speak about complete locally bounded algebras, algebras of type $B _ {o}$ ($B _ {o}$-algebras), and locally pseudo-convex $F$-algebras, i.e. $F$-algebras whose underlying topological vector space is a locally bounded space, etc.

## Locally bounded algebras of type $F$.

These are also called $LB$-algebras. The topology of an $LB$-algebra $A$ can be given by means of a $p$-homogeneous norm, $0 < p \leq 1$, satisfying $\| {xy } \| \leq \| x \| \| y \|$, $x, y \in A$ (the submultiplicativity condition) and, if $A$ has a unity $e$, $\| e \| = 1$. The theory of these algebras is analogous to that of Banach algebras (cf. Banach algebra). In particular, a Gel'fand–Mazur-type theorem holds (even if completeness and joint continuity of multiplication is not assumed). For commutative complex algebras there is a Gelfand-type theory (including a holomorphic functional calculus). That means that local boundedness, and not local convexity, is responsible for the theory of Banach algebras. For more information on $LB$-algebras see [a5], [a6], [a7].

## $B _ {o}$-algebras.

The topology of such an algebra $A$ can be given by means of a sequence $\| x \| _ {1} \leq \| x \| _ {2} \leq \dots$ of semi-norms (cf. Semi-norm) satisfying

$$\tag{a1 } \left \| {xy } \right \| _ {i} \leq \left \| x \right \| _ {i + 1 } \left \| y \right \| _ {i + 1 } , \quad i = 1,2 \dots$$

and, if $A$ has a unit $e$, $\| e \| _ {i} = 1$ for all $i$. Such an algebra is said to be multiplicatively-convex ( $m$-convex) if its topology can be given by means of semi-norms satisfying $\| {xy } \| _ {i} \leq \| x \| _ {i} \| y \| _ {i}$ instead of (a1) (some authors give the name "Fréchet algebra" to $m$-convex $B _ {o}$-algebras). Each $m$-convex $B _ {o}$-algebra is an inverse (projective) limit of a sequence of Banach algebras. Each complete locally convex topological algebra is an inverse limit of a directed system of $B _ {o}$-algebras. A Gelfand–Mazur-type theorem holds for $B _ {o}$-algebras; however, completeness is essential, and a $B _ {o}$-algebra can contain a dense subalgebra isomorphic to the field of all rational functions. The latter is impossible for $m$-convex algebras. The operation of taking an inverse is not continuous on arbitrary $B _ {o}$-algebras, but it is continuous on $m$-convex $B _ {o}$-algebras (the operation of taking an inverse is continuous for a general $F$-algebra $A$ if and only if the group $G ( A )$ of its invertible elements is a $G _ \delta$-set). A commutative unital $B _ {o}$-algebra can have dense maximal ideals of infinite codimension also if it is $m$-convex, but in the latter case there is always a closed maximal ideal of codimension one, which is the kernel of a multiplicative linear functional (this is not true in general). Every $m$-convex algebra has a functional calculus of several complex variables, but in the non- $m$-convex case it is possible that there operate only the polynomials. If a commutative $B _ {o}$-algebra is such that its set $G ( A )$ of invertible elements is open, then it must be $m$-convex. This fails in the non-commutative case, so that a non- $m$-convex $B _ {o}$-algebra can have all its commutative subalgebras $m$-convex. Also, a non-Banach $m$-convex algebra can have the property that all its closed commutative subalgebras are Banach algebras. One of most challenging open questions in the theory of topological algebras (see [a2], [a4]) is the question whether for an $m$-convex $B _ {o}$-algebra all its multiplicative linear functionals are continuous (this question is also open for algebras of type $B _ {o}$ and $F$). This famous Michael–Mazur problem has been positively solved by R. Arens for finitely generated $m$-convex algebras. For more on these algebras see [a1], [a3], [a4], [a5], [a6], [a7].

## Locally pseudo-convex $F$-algebras.

These are analogous to $B _ {o}$-algebras, but with semi-norms replaced by $p$-homogeneous semi-norms, $0 < p \leq 1$. Their theories are similar; however, there are some differences. E.g., a commutative locally pseudo-convex algebra of type $F$ with open set $G ( A )$ need not be $m$-pseudo-convex. Every $m$-pseudo-convex algebra of type $F$ is an inverse limit of a sequence of $LB$-algebras. For more details see [a5], [a6], [a7].

Not much is known about general $F$-algebras; e.g., a Gelfand–Mazur-type theorem is still (1996) unknown. One result, however, has to be noted. Let $A$ be an $F$-algebra with a continuous involution. Then each positive (i.e. satisfying $f ( x ^ {*} x ) \geq 0$) functional on $A$ is continuous [a2]. Every complete topological algebra is an inverse limit of a directed system of $F$-algebras.

#### References

 [a1] E. Beckenstein, L. Narici, C. Suffel, "Topological algebras" , Amsterdam (1977) [a2] T. Husain, ",Multiplicative functionals on topological algebras" , London (1983) [a3] A. Mallios, "Topological algebras. Selected topics" , Amsterdam (1986) [a4] E. Michael, "Locally multiplicatively-convex topological algebras" , Memoirs , 11 , Amer. Math. Soc. (1952) [a5] L. Waelbroeck, "Topological vector spaces and algebras" , Lecture Notes in Mathematics , 230 , Springer (1971) [a6] W. Zelazko, "Metric generalizations of Banach algebras" Dissert. Math. , 47 (1965) [a7] W. Zelazko, "Selected topics in topological algebras" , Lecture Notes , 31 , Aarhus Univ. (1971)
How to Cite This Entry:
Fréchet algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_algebra&oldid=52380
This article was adapted from an original article by W. Zelazko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article