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)