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In [[Functional analysis|functional analysis]] and related areas of [[Mathematics|mathematics]], '''stereotype algebras''' are [[Topological algebra|topological algebra]]s defined as [[Stereotype space|stereotype space]]s with a (unital and associative) multiplication satisfying the condition of separate uniform continuity on compact sets.
== Definition ==
There are two equivalent ways to define stereotype algebra. 

* '''An abstract definition''': a (projective<ref name=projectivealgebra>There is another notion, an '''injective stereotype algebra''', which is defined as a [[Monoid (category theory)|monoid]] in the [[Stereotype space#Category of stereotype spaces|monoidal category ('''Ste''',<math>\odot</math>,<math>{\mathbb C}</math>)]] of stereotype spaces with the dual tensor product, <math>\odot</math>. The class of these algebras is much narrower.</ref>) '''stereotype algebra'''{{sfn|Akbarov|2003|p=271}}{{sfn|Akbarov|2005}} is a [[Monoid (category theory)|monoid]] in the [[Stereotype space#Category of stereotype spaces|monoidal category ('''Ste''',<math>\circledast</math>,<math>{\mathbb C}</math>)]] of stereotype spaces with the tensor product <math>\circledast</math>.

* '''An analytic definition''': let <math>A</math> be a [[Stereotype space|stereotype space]] (over the field <math>\mathbb C</math> of complex numbers<ref>...or the field <math>\mathbb R</math> of real numbers</ref>) and suppose it is endowed with a bilinear operation <math>(x,y)\in A\times A\mapsto x\cdot y\in A</math> that turns <math>A</math> into a unital and [[Associative algebra|associative algebra]] (over the field <math>\mathbb C</math>). Then <math>A</math> is called a (projective) '''stereotype algebra'''{{sfn|Akbarov|2003|p=271}}{{sfn|Akbarov|2009|p=483}}{{sfn|Akbarov|2016|p=147}}{{sfn|Akbarov|2017|p=594}} if the multiplication <math>(x,y)\in A\times A\to A</math> is a [[Stereotype space#Universality of tensor product|continuous bilinear map in the stereotype sense]], i.e. if for each compact set <math>S\subseteq A</math> and for each neighbourhood of zero <math>U\subseteq A</math> there is a neighbourhood of zero <math>V\subseteq A</math> such that <math>S\cdot V\subseteq U\quad \&\quad V\cdot S\subseteq U.
</math><ref>The requirement on <math>A</math> in the analytic definition of stereotype algebra can be equivalently expressed as follows: for each compact set <math>S\subseteq A</math> and for each [[Net (mathematics)|net]] <math>y_i</math> tending to zero in <math>A</math>,
<math>
y_i\overset{A}{\underset{i\to\infty}{\longrightarrow}}0
</math> 
the nets <math>x\cdot y_i</math> and <math>y_i\cdot x</math> tend to zero in <math>A</math> uniformly on <math>x\in S</math>:
<math>
x\cdot y_i\overset{A}{\underset{x\in S}{\underset{i\to\infty}{\rightrightarrows}}}0
\quad \&\quad y_i\cdot x\overset{A}{\underset{x\in S}{\underset{i\to\infty}{\rightrightarrows}}}0.
</math></ref>

A ''morphism of stereotype algebras'' <math>A</math> and <math>B</math> is a continuous unital<ref>A homomorphism <math>\varphi:A\to B</math> is said to be ''unital'' if it preserves unit: <math>\varphi(1_A)=1_B</math>.</ref> homomorphism <math>\varphi:A\to B</math>.

Stereotype algebras form a subclass in the class of ''separately continuous'' [[Topological algebra|topological algebra]]s.

== Examples ==

:1. All Fréchet algebras are stereotype. In particular, all [[Banach algebra]]s are stereotype.{{sfn|Akbarov|2003|p=272}}  
:2. For each [[Stereotype space|stereotype space]] <math>X</math> the stereotype space of operators <math>X\oslash X</math>, <math>\varphi:X\to X</math>, is a stereotype algebra with respect to the operation of composition.{{sfn|Akbarov|2003|p=272}}
:3. The main examples of ''functional algebras''{{sfn|Akbarov|2003|p=272}} in Analysis and Geometry are stereotype algebras: 
::* the algebra <math>{\mathcal C}(M)</math> of [[Continuous function|continuous function]]s on a [[Paracompact space|paracompact]] locally compact topological space <math>M</math> with the topology of uniform convergence on compact sets <math>T\subseteq M</math>, 
::* the algebra <math>{\mathcal E}(M)</math> of smooth functions on a smooth manifold <math>M</math> with the topology of uniform convergence with all derivatives on compact sets <math>T\subseteq M</math>, 
::* the algebra <math>{\mathcal O}(M)</math> of [[Holomorphic function|holomorphic function]]s on a [[Stein manifold]] <math>M</math> with the topology of uniform convergence on compact sets <math>T\subseteq M</math>, 
::* the algebra <math>{\mathcal P}(M)</math> of polynomials (regular functions) on an affine algebraic variety <math>M</math> with the strongest locally convex topology. 
:4. [[Group algebra#Stereotype group algebras|Stereotype group algebras]]. In the case when the space <math>M</math> in the previous examples has a structure of a [[Topological group|topological group]], the [[Stereotype space#Definition|dual stereotype spaces]] to the functional algebras on <math>M</math> are stereotype algebras with respect to the operation of [[Convolution#Convolution of measures|convolution]]:{{sfn|Akbarov|2003|p=272}} 
::* the algebra <math>{\mathcal C}^\star(G)</math> of [[Radon measure]]s with compact support on a [[Locally compact group|locally compact group]] <math>G</math>,<ref>If <math>G</math> is an infinite locally compact group then the algebra <math>{\mathcal C}^\star(G)</math> of measures on <math>G</math> is not a Fréchet algebra. In the case when <math>G</math> is compact, <math>{\mathcal C}^\star(G)</math> is a [[Smith space]]. If <math>G</math> is <math>\sigma</math>-compact, then <math>{\mathcal C}^\star(G)</math> is a [[Brauner space]].</ref> 
::* the algebra <math>{\mathcal E}^\star(G)</math> of [[Distribution (mathematics)|distribution]]s with compact support on a real Lie group <math>G</math>, 
::* the algebra <math>{\mathcal O}^\star(G)</math> of holomorphic functionals on a Stein group<ref name=Stein-group>A ''Stein group'' is a [[Complex Lie group|complex Lie group]] <math>G</math> which is a [[Stein manifold]].</ref> <math>G</math>, 
::* the algebra <math>{\mathcal P}^\star(G)</math> of currents on an affine algebraic group <math>G</math>.
:In contrast to the other models of group algebras in functional analysis, the stereotype group algebras possess the [[Group algebra#Stereotype group algebras|universal property]]<ref>I.e. the homomorphisms of stereotype algebras <math>{\mathcal C}^\star(G)\to A</math> are in one-to-one correspondence with the continuous representations <math>G\to A</math> of the group <math>G</math>, and the same is true for the algebras <math>{\mathcal E}^\star(G)</math>, <math>{\mathcal O}^\star(G)</math> and <math>{\mathcal P}^\star(G)</math> with the smooth, holomorphic and polynomial representations of the corresponding groups.</ref>{{sfn|Akbarov|2003|p=275}} and the structure of [[Hopf algebra#Hopf algebras in braided monoidal categories|Hopf algebra]]{{sfn|Akbarov|2009|p=507}}.

== Stereotype modules ==

A stereotype space <math>X</math> is called a ''left stereotype module'' over a stereotype algebra <math>A</math>, if a bilinear operation <math>(a,x)\in A\times X\mapsto a\cdot x\in X</math> is defined that turns <math>X</math> into a left [[Module (mathematics)|module]] over <math>A</math>, and this operation is a [[Stereotype space#Universality of tensor product|continuous bilinear map in the stereotype sense]]: for each compact set <math>S\subseteq A</math> and for each neighbourhood of zero <math>U\subseteq X</math> there is a neighbourhood of zero <math>V\subseteq X</math> such that 

<math>
S\cdot V\subseteq U,
</math>

and at the same time for each compact set <math>T\subseteq X</math> and for each neighbourhood of zero <math>U\subseteq X</math> there is a neighbourhood of zero <math>W\subseteq A</math> such that 

<math>
W\cdot T\subseteq U.
</math>

The notion of a ''right stereotype module'' over a stereotype algebra <math>A</math> is defined by analogy.

The theory of stereotype algebras allows to simplify the formulations in the theory of [[Topological algebra|topological algebra]]s and to bring this field closer to [[Abstract algebra|abstract algebra]]. The following two results (which do not hold for general jointly or separately continuous algebras) are typical illustrations.<ref>Other illustrations are the already mentioned [[Group algebra#Stereotype group algebras|universality property for the stereotype group algebras]]{{harvnb|Akbarov|2003|p=275}} and the structure of [[Hopf algebra]] on them{{harvnb|Akbarov|2009|p=507}}.</ref>

'''Theorem'''.{{sfn|Akbarov|2003|p=283}} ''A stereotype space <math>X</math> with a structure of left (right) module over a stereotype algebra <math>A</math> is a stereotype module over <math>A</math> if and only if the operation of multiplication <math>(a,x)\in A\times X\mapsto a\cdot x\in X</math> (respectively, <math>(x,a)\in X\times A\mapsto x\cdot a\in X</math>) defines a continuous map (a representation) <math>A\to X\oslash X</math>.''<ref>Here <math>X\oslash X</math> is the [[Stereotype space#Ste as a *-autonomous category|stereotype space of operators <math>\varphi:X\to X</math>]].</ref>

'''Theorem'''.{{sfn|Akbarov|2003|p=289}} ''For each stereotype algebra <math>A</math> the categories <sub><math>A</math></sub>'''Ste''' and '''Ste'''<sub><math>A</math></sub> of left and right stereotype modules over <math>A</math> are [[Enriched category|enriched categories]] over the [[Stereotype space#Ste as a *-autonomous category|monoidal category ('''Ste''',<math>\circledast</math>,<math>{\mathbb C}</math>)]] of stereotype spaces.''

== Applications ==

The notion of stereotype algebra is used in the generalizations{{sfn|Akbarov|2017}} of the Pontryagin  duality theory to the classes of non-commutative groups based on the notion of [[Envelope (category theory)|envelope]]: the holomorphic, the smooth and the continuous envelopes of stereotype algebras lead respectively to the constructions of the holomorphic, the smooth and the continuous dualities in ''big geometric disciplines'' – [[Complex geometry|complex geometry]], [[Differential geometry|differential geometry]], and [[Topology|topology]] – for certain classes of (not necessarily commutative) topological groups considered in these disciplines (affine algebraic groups, and some classes of [[Lie group]]s and Moore groups).{{sfn|Akbarov|2009}}{{sfn|Akbarov|2016}}{{sfn|Akbarov|2017}}{{sfn|Kuznetsova|2013}}

==See also==

* [[Stereotype space]]
* [[Topological algebra]]
* Fréchet algebra
* [[Banach algebra]]
* [[C*-algebra]]
* [[Operator algebra]]

==Notes==
{{reflist}}

== References ==
*{{cite book|last=Akbarov|first=S.S.|title=Stereotype spaces and algebras|year=2022|publisher=De Gruyter| series= De Gruyter Expositions in Mathematics |isbn=9783110780918 |url=https://www.degruyter.com/document/isbn/9783110780918/html| ref = harv}}
*{{cite journal|last=Akbarov|first=S.S.|title=Pontryagin duality in the theory of topological vector spaces and in topological algebra|journal=Journal of Mathematical Sciences|year=2003|volume=113|issue=2|pages=179–349|doi=10.1023/A:1020929201133| ref = harv}}
*{{cite journal|last=Akbarov|first=S.S.|title=Pontryagin duality and topological algebras|journal=Banach Center Publications|year=2005|volume=67|pages=55–71|url=http://webmail.impan.gov.pl/cgi-bin/bc/pdf?bc67-0-05|doi=10.4064/bc67-0-5| ref = harv}}
*{{cite journal|last=Akbarov|first=S.S.|title=Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity|journal=Journal of Mathematical Sciences|year=2009|volume=162|issue=4|pages=459–586|arxiv=0806.3205|doi=10.1007/s10958-009-9646-1| ref = harv}}
*{{cite journal|last=Akbarov|first=S.S.|title=Envelopes and refinements in categories, with applications to functional analysis|url=https://www.impan.pl/en/publishing-house/journals-and-series/dissertationes-mathematicae/all/513|journal=Dissertationes Mathematicae|year=2016|volume=513|pages=1–188|arxiv=1110.2013|doi=10.4064/dm702-12-2015| ref = harv}}
*{{cite journal|last=Akbarov|first=S.S.|title=Continuous and smooth envelopes of topological algebras. Part 1|journal=Journal of Mathematical Sciences|year=2017|volume=227|issue=5|pages=531–668|arxiv=1303.2424|doi=10.1007/s10958-017-3599-6| ref = harv}}
*{{cite journal|last=Akbarov|first=S.S.|title=Continuous and smooth envelopes of topological algebras. Part 2|journal=Journal of Mathematical Sciences|year=2017|volume=227|issue=6|pages=669–789|arxiv=1303.2424|doi=10.1007/s10958-017-3600-4| ref = harv}}
*{{cite journal|last=Kuznetsova|first=Y.|title=A duality for Moore groups|journal=Journal of Operator Theory|year=2013|volume=69|issue=2|pages=101–130|arxiv=0907.1409|bibcode=2009arXiv0907.1409K|doi=10.7900/jot.2011mar17.1920| ref = harv}}


{{Functional Analysis}}
{{Category theory}}

[[Category:Category theory]]
[[Category:Functional analysis]]
[[Category:Algebra]]
[[Category:Duality theories]]

{{Sourceattribution|Stereotype algebra|1}}

{{Author|Sergei Akbarov|Serge}}