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''<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" />''
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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.
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==Locally bounded algebras of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017014.png" />.==
+
'' $  F $-
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]]].
+
algebra, algebra of type $  F $''
  
==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017028.png" />-algebras.==
+
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  $  F $-
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
+
algebras can be classified similarly as the  $  F $-
 +
spaces (see [[Fréchet topology|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.
  
<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>
+
==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|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|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 [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]].
  
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]]].
+
== $ 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|Semi-norm]]) satisfying
  
==Locally pseudo-convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017074.png" />-algebras.==
+
$$ \tag{a1 }
These are analogous to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017075.png" />-algebras, but with semi-norms replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017076.png" />-homogeneous semi-norms, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017077.png" />. Their theories are similar; however, there are some differences. E.g., a commutative locally pseudo-convex algebra of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017078.png" /> with open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017079.png" /> need not be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017080.png" />-pseudo-convex. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017081.png" />-pseudo-convex algebra of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017082.png" /> is an inverse limit of a sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110170/f11017083.png" />-algebras. For more details see [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]].
+
\left \| {xy } \right \| _ {i} \leq  \left \| x \right \| _ {i + 1 }  \left \| y \right \| _ {i + 1 }  , \quad i = 1,2 \dots
 +
$$
  
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.
+
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 [[#References|[a2]]], [[#References|[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 [[#References|[a1]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[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 [[#References|[a5]]], [[#References|[a6]]], [[#References|[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 [[#References|[a2]]]. Every complete topological algebra is an inverse limit of a directed system of $  F $-
 +
algebras.
  
 
====References====
 
====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>
 
<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>

Revision as of 19:40, 5 June 2020


$ 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=23278
This article was adapted from an original article by W. Zelazko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article