View source for Fréchet algebra

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f1101702.png" />-algebra, algebra of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f1101704.png" />''

A completely metrizable [[Topological algebra|topological algebra]]. The joint continuity of multiplication need not be demanded since it follows from the separate continuity (see [[Fréchet topology|Fréchet topology]]). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f1101705.png" />-algebras can be classified similarly as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f1101706.png" />-spaces (see [[Fréchet topology|Fréchet topology]]), so one can speak about complete locally bounded algebras, algebras of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f1101708.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017010.png" />-algebras), and locally pseudo-convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017012.png" />-algebras, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017013.png" />-algebras whose underlying topological vector space is a locally bounded space, etc.

==Locally bounded algebras of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017014.png" />.==
These are also called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017016.png" />-algebras. The topology of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017017.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017018.png" /> can be given by means of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017019.png" />-homogeneous [[Norm|norm]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017020.png" />, satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017022.png" /> (the submultiplicativity condition) and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017023.png" /> has a unity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017025.png" />. The theory of these algebras is analogous to that of Banach algebras (cf. [[Banach algebra|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017027.png" />-algebras see [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]].

==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017028.png" />-algebras.==
The topology of such an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017029.png" /> can be given by means of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017030.png" /> of semi-norms (cf. [[Semi-norm|Semi-norm]]) satisfying

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017032.png" /> has a unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017034.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017035.png" />. Such an algebra is said to be multiplicatively-convex (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017039.png" />-convex) if its topology can be given by means of semi-norms satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017040.png" /> instead of (a1) (some authors give the name  "Fréchet algebra"  to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017041.png" />-convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017042.png" />-algebras). Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017043.png" />-convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017044.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017045.png" />-algebras. A Gelfand–Mazur-type theorem holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017047.png" />-algebras; however, completeness is essential, and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017048.png" />-algebra can contain a dense subalgebra isomorphic to the field of all rational functions. The latter is impossible for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017049.png" />-convex algebras. The operation of taking an inverse is not continuous on arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017050.png" />-algebras, but it is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017051.png" />-convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017052.png" />-algebras (the operation of taking an inverse is continuous for a general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017053.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017054.png" /> if and only if the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017055.png" /> of its invertible elements is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017057.png" />-set). A commutative unital <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017058.png" />-algebra can have dense maximal ideals of infinite codimension also if it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017059.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017060.png" />-convex algebra has a functional calculus of several complex variables, but in the non-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017061.png" />-convex case it is possible that there operate only the polynomials. If a commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017062.png" />-algebra is such that its set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017063.png" /> of invertible elements is open, then it must be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017064.png" />-convex. This fails in the non-commutative case, so that a non-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017065.png" />-convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017066.png" />-algebra can have all its commutative subalgebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017067.png" />-convex. Also, a non-Banach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017068.png" />-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 [[#References|[a2]]], [[#References|[a4]]]) is the question whether for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017069.png" />-convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017070.png" />-algebra all its multiplicative linear functionals are continuous (this question is also open for algebras of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017072.png" />). This famous Michael–Mazur problem has been positively solved by R. Arens for finitely generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017073.png" />-convex algebras. For more on these algebras see [[#References|[a1]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]].

==Locally pseudo-convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017074.png" />-algebras.==
These are analogous to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017075.png" />-algebras, but with semi-norms replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017076.png" />-homogeneous semi-norms, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017077.png" />. Their theories are similar; however, there are some differences. E.g., a commutative locally pseudo-convex algebra of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017078.png" /> with open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017079.png" /> need not be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017080.png" />-pseudo-convex. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017081.png" />-pseudo-convex algebra of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017082.png" /> is an inverse limit of a sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017083.png" />-algebras. For more details see [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]].

Not much is known about general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017084.png" />-algebras; e.g., a Gelfand–Mazur-type theorem is still (1996) unknown. One result, however, has to be noted. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017085.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017086.png" />-algebra with a continuous involution. Then each positive (i.e. satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017087.png" />) functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017088.png" /> is continuous [[#References|[a2]]]. Every complete topological algebra is an inverse limit of a directed system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017089.png" />-algebras.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Beckenstein,   L. Narici,   C. Suffel,   "Topological algebras" , Amsterdam  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Husain,   ",Multiplicative functionals on topological algebras" , London  (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Mallios,   "Topological algebras. Selected topics" , Amsterdam  (1986)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Michael,   "Locally multiplicatively-convex topological algebras" , ''Memoirs'' , '''11''' , Amer. Math. Soc.  (1952)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L. Waelbroeck,   "Topological vector spaces and algebras" , ''Lecture Notes in Mathematics'' , '''230''' , Springer  (1971)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  W. Zelazko,   "Metric generalizations of Banach algebras"  ''Dissert. Math.'' , '''47'''  (1965)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  W. Zelazko,   "Selected topics in topological algebras" , ''Lecture Notes'' , '''31''' , Aarhus Univ.  (1971)</TD></TR></table>