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.

Definition

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

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

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

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

• Stereotype space
• Topological algebra
• Fréchet algebra
• Banach algebra
• C*-algebra
• Operator algebra

Notes

References

• 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. 

0.00 (0 votes)