Stereotype algebra

In functional analysis and related areas of mathematics, stereotype
algebras are topological algebras defined as stereotype spaces with a
(unital and associative) multiplication satisfying the condition of
separate uniform continuity on compact sets. Contents 1 Definition 2
Examples 3 Stereotype modules 4 Applications 5 See also 6 Notes 7
References Definition[edit] There are two equivalent ways to define
stereotype algebra. An abstract definition: a (projective[1])
stereotype algebra[2][3] is a monoid in the monoidal category
(Ste,[math]\displaystyle{ \circledast }[/math],[math]\displaystyle{
{\mathbb C} }[/math]) of stereotype spaces with the tensor product
[math]\displaystyle{ \circledast }[/math]. An analytic definition: let
[math]\displaystyle{ A }[/math] be a stereotype space (over the field
[math]\displaystyle{ \mathbb C }[/math] of complex numbers[4]) and
suppose it is endowed with a bilinear operation [math]\displaystyle{
(x,y)\in A\times A\mapsto x\cdot y\in A }[/math] that turns
[math]\displaystyle{ A }[/math] into a unital and associative algebra
(over the field [math]\displaystyle{ \mathbb C }[/math]). Then
[math]\displaystyle{ A }[/math] is called a (projective) stereotype
algebra[2][5][6][7] if the multiplication [math]\displaystyle{
(x,y)\in A\times A\to A }[/math] is a continuous bilinear map in the
stereotype sense, i.e. if for each compact set [math]\displaystyle{
S\subseteq A }[/math] and for each neighbourhood of zero
[math]\displaystyle{ U\subseteq A }[/math] there is a neighbourhood of
zero [math]\displaystyle{ V\subseteq A }[/math] such that
[math]\displaystyle{ S\cdot V\subseteq U\quad \&\quad V\cdot
S\subseteq U. }[/math][8] A morphism of stereotype algebras
[math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] is
a continuous unital[9] homomorphism [math]\displaystyle{ \varphi:A\to
B }[/math]. Stereotype algebras form a subclass in the class of
separately continuous topological algebras. Examples[edit] 1. All
Fréchet algebras are stereotype. In particular, all Banach algebras
are stereotype.[10] 2. For each stereotype space [math]\displaystyle{
X }[/math] the stereotype space of operators [math]\displaystyle{
X\oslash X }[/math], [math]\displaystyle{ \varphi:X\to X }[/math], is
a stereotype algebra with respect to the operation of composition.[10]
3. The main examples of functional algebras[10] in Analysis and
Geometry are stereotype algebras: the algebra [math]\displaystyle{
{\mathcal C}(M) }[/math] of continuous functions on a paracompact
locally compact topological space [math]\displaystyle{ M }[/math] with
the topology of uniform convergence on compact sets
[math]\displaystyle{ T\subseteq M }[/math], the algebra
[math]\displaystyle{ {\mathcal E}(M) }[/math] of smooth functions on a
smooth manifold [math]\displaystyle{ M }[/math] with the topology of
uniform convergence with all derivatives on compact sets
[math]\displaystyle{ T\subseteq M }[/math], the algebra
[math]\displaystyle{ {\mathcal O}(M) }[/math] of holomorphic functions
on a Stein manifold [math]\displaystyle{ M }[/math] with the topology
of uniform convergence on compact sets [math]\displaystyle{ T\subseteq
M }[/math], the algebra [math]\displaystyle{ {\mathcal P}(M) }[/math]
of polynomials (regular functions) on an affine algebraic variety
[math]\displaystyle{ M }[/math] with the strongest locally convex
topology. 4. Stereotype group algebras. In the case when the space
[math]\displaystyle{ M }[/math] in the previous examples has a
structure of a topological group, the dual stereotype spaces to the
functional algebras on [math]\displaystyle{ M }[/math] are stereotype
algebras with respect to the operation of convolution:[10] the algebra
[math]\displaystyle{ {\mathcal C}^\star(G) }[/math] of Radon measures
with compact support on a locally compact group [math]\displaystyle{ G
}[/math],[11] the algebra [math]\displaystyle{ {\mathcal E}^\star(G)
}[/math] of distributions with compact support on a real Lie group
[math]\displaystyle{ G }[/math], the algebra [math]\displaystyle{
{\mathcal O}^\star(G) }[/math] of holomorphic functionals on a Stein
group[12] [math]\displaystyle{ G }[/math], the algebra
[math]\displaystyle{ {\mathcal P}^\star(G) }[/math] of currents on an
affine algebraic group [math]\displaystyle{ G }[/math]. In contrast to
the other models of group algebras in functional analysis, the
stereotype group algebras possess the universal property[13][14] and
the structure of Hopf algebra[15]. Stereotype modules[edit] A
stereotype space [math]\displaystyle{ X }[/math] is called a left
stereotype module over a stereotype algebra [math]\displaystyle{ A
}[/math], if a bilinear operation [math]\displaystyle{ (a,x)\in
A\times X\mapsto a\cdot x\in X }[/math] is defined that turns
[math]\displaystyle{ X }[/math] into a left module over
[math]\displaystyle{ A }[/math], and this operation is a continuous
bilinear map in the stereotype sense: for each compact set
[math]\displaystyle{ S\subseteq A }[/math] and for each neighbourhood
of zero [math]\displaystyle{ U\subseteq X }[/math] there is a
neighbourhood of zero [math]\displaystyle{ V\subseteq X }[/math] such
that [math]\displaystyle{ S\cdot V\subseteq U, }[/math] and at the
same time for each compact set [math]\displaystyle{ T\subseteq X
}[/math] and for each neighbourhood of zero [math]\displaystyle{
U\subseteq X }[/math] there is a neighbourhood of zero
[math]\displaystyle{ W\subseteq A }[/math] such that
[math]\displaystyle{ W\cdot T\subseteq U. }[/math] The notion of a
right stereotype module over a stereotype algebra [math]\displaystyle{
A }[/math] is defined by analogy. The theory of stereotype algebras
allows to simplify the formulations in the theory of topological
algebras and to bring this field closer to abstract algebra. The
following two results (which do not hold for general jointly or
separately continuous algebras) are typical illustrations.[16]
Theorem.[17] A stereotype space [math]\displaystyle{ X }[/math] with a
structure of left (right) module over a stereotype algebra
[math]\displaystyle{ A }[/math] is a stereotype module over
[math]\displaystyle{ A }[/math] if and only if the operation of
multiplication [math]\displaystyle{ (a,x)\in A\times X\mapsto a\cdot
x\in X }[/math] (respectively, [math]\displaystyle{ (x,a)\in X\times
A\mapsto x\cdot a\in X }[/math]) defines a continuous map (a
representation) [math]\displaystyle{ A\to X\oslash X }[/math].[18]
Theorem.[19] For each stereotype algebra [math]\displaystyle{ A
}[/math] the categories [math]\displaystyle{ A }[/math]Ste and
Ste[math]\displaystyle{ A }[/math] of left and right stereotype
modules over [math]\displaystyle{ A }[/math] are enriched categories
over the monoidal category (Ste,[math]\displaystyle{ \circledast
}[/math],[math]\displaystyle{ {\mathbb C} }[/math]) of stereotype
spaces. Applications[edit] The notion of stereotype algebra is used in
the generalizations[20] of the Pontryagin duality theory to the
classes of non-commutative groups based on the notion of 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, differential geometry, and topology – for
certain classes of (not necessarily commutative) topological groups
considered in these disciplines (affine algebraic groups, and some
classes of Lie groups and Moore groups).[21][22][20][23] See
also[edit] Stereotype space Topological algebra Fréchet algebra
Banach algebra C*-algebra Operator algebra Notes[edit] ↑ There is
another notion, an injective stereotype algebra, which is defined as a
monoid in the monoidal category (Ste,[math]\displaystyle{ \odot
}[/math],[math]\displaystyle{ {\mathbb C} }[/math]) of stereotype
spaces with the dual tensor product, [math]\displaystyle{ \odot
}[/math]. The class of these algebras is much narrower. ↑ 2.0 2.1
Akbarov 2003, p. 271. ↑ Akbarov 2005. ↑ ...or the field
[math]\displaystyle{ \mathbb R }[/math] of real numbers ↑ Akbarov
2009, p. 483. ↑ Akbarov 2016, p. 147. ↑ Akbarov 2017, p. 594.
↑ The requirement on [math]\displaystyle{ A }[/math] in the analytic
definition of stereotype algebra can be equivalently expressed as
follows: for each compact set [math]\displaystyle{ S\subseteq A
}[/math] and for each net [math]\displaystyle{ y_i }[/math] tending to
zero in [math]\displaystyle{ A }[/math], [math]\displaystyle{
y_i\overset{A}{\underset{i\to\infty}{\longrightarrow}}0 }[/math] the
nets [math]\displaystyle{ x\cdot y_i }[/math] and [math]\displaystyle{
y_i\cdot x }[/math] tend to zero in [math]\displaystyle{ A }[/math]
uniformly on [math]\displaystyle{ x\in S }[/math]:
[math]\displaystyle{ 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] ↑ A
homomorphism [math]\displaystyle{ \varphi:A\to B }[/math] is said to
be unital if it preserves unit: [math]\displaystyle{ \varphi(1_A)=1_B
}[/math]. ↑ 10.0 10.1 10.2 10.3 Akbarov 2003, p. 272. ↑ If
[math]\displaystyle{ G }[/math] is an infinite locally compact group
then the algebra [math]\displaystyle{ {\mathcal C}^\star(G) }[/math]
of measures on [math]\displaystyle{ G }[/math] is not a Fréchet
algebra. In the case when [math]\displaystyle{ G }[/math] is compact,
[math]\displaystyle{ {\mathcal C}^\star(G) }[/math] is a Smith space.
If [math]\displaystyle{ G }[/math] is [math]\displaystyle{ \sigma
}[/math]-compact, then [math]\displaystyle{ {\mathcal C}^\star(G)
}[/math] is a Brauner space. ↑ A Stein group is a complex Lie group
[math]\displaystyle{ G }[/math] which is a Stein manifold. ↑ I.e.
the homomorphisms of stereotype algebras [math]\displaystyle{
{\mathcal C}^\star(G)\to A }[/math] are in one-to-one correspondence
with the continuous representations [math]\displaystyle{ G\to A
}[/math] of the group [math]\displaystyle{ G }[/math], and the same is
true for the algebras [math]\displaystyle{ {\mathcal E}^\star(G)
}[/math], [math]\displaystyle{ {\mathcal O}^\star(G) }[/math] and
[math]\displaystyle{ {\mathcal P}^\star(G) }[/math] with the smooth,
holomorphic and polynomial representations of the corresponding
groups. ↑ Akbarov 2003, p. 275. ↑ Akbarov 2009, p. 507. ↑
Other illustrations are the already mentioned universality property
for the stereotype group algebrasAkbarov 2003, p. 275 and the
structure of Hopf algebra on themAkbarov 2009, p. 507. ↑ Akbarov
2003, p. 283. ↑ Here [math]\displaystyle{ X\oslash X }[/math] is
the stereotype space of operators [math]\displaystyle{ \varphi:X\to X
}[/math]. ↑ Akbarov 2003, p. 289. ↑ 20.0 20.1 Akbarov 2017. ↑
Akbarov 2009. ↑ Akbarov 2016. ↑ Kuznetsova 2013. References[edit]
Akbarov, S.S. (2022). Stereotype spaces and algebras. De Gruyter
Expositions in Mathematics. De Gruyter. ISBN 9783110780918.
https://www.degruyter.com/document/isbn/9783110780918/html.  Akbarov,
S.S. (2003). "Pontryagin duality in the theory of topological vector
spaces and in topological algebra". Journal of Mathematical Sciences
113 (2): 179–349. doi:10.1023/A:1020929201133.  Akbarov, S.S.
(2005). "Pontryagin duality and topological algebras". Banach Center
Publications 67: 55–71. doi:10.4064/bc67-0-5.
http://webmail.impan.gov.pl/cgi-bin/bc/pdf?bc67-0-05.  Akbarov, S.S.
(2009). "Holomorphic functions of exponential type and duality for
Stein groups with algebraic connected component of identity". Journal
of Mathematical Sciences 162 (4): 459–586.
doi:10.1007/s10958-009-9646-1.  Akbarov, S.S. (2016). "Envelopes and
refinements in categories, with applications to functional analysis".
Dissertationes Mathematicae 513: 1–188. doi:10.4064/dm702-12-2015.
https://www.impan.pl/en/publishing-house/journals-and-series/dissertationes-mathematicae/all/513. 
Akbarov, S.S. (2017). "Continuous and smooth envelopes of topological
algebras. Part 1". Journal of Mathematical Sciences 227 (5):
531–668. doi:10.1007/s10958-017-3599-6.  Akbarov, S.S. (2017).
"Continuous and smooth envelopes of topological algebras. Part 2".
Journal of Mathematical Sciences 227 (6): 669–789.
doi:10.1007/s10958-017-3600-4.  Kuznetsova, Y. (2013). "A duality for
Moore groups". Journal of Operator Theory 69 (2): 101–130.
doi:10.7900/jot.2011mar17.1920. Bibcode: 2009arXiv0907.1409K. 
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