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, $\circledast$ , ${\mathbb {C} }$ ) of stereotype spaces with the tensor product $\circledast$ .

An analytic definition: let $A$ be a stereotype space (over the field $\mathbb {C}$ of complex numbers^[4]) and suppose it is endowed with a bilinear operation $(x,y)\in A\times A\mapsto x\cdot y\in A$ that turns $A$ into a unital and associative algebra (over the field $\mathbb {C}$ ). Then $A$ is called a (projective) stereotype algebra^[2]^[5]^[6]^[7] if the multiplication $(x,y)\in A\times A\to A$ is a continuous bilinear map in the stereotype sense, i.e. if for each compact set $S\subseteq A$ and for each neighbourhood of zero $U\subseteq A$ there is a neighbourhood of zero $V\subseteq A$ such that $S\cdot V\subseteq U\quad \&\quad V\cdot S\subseteq U.$ ^[8]

A morphism of stereotype algebras $A$ and $B$ is a continuous unital^[9] homomorphism $\varphi :A\to B$ .

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

X

the stereotype space of operators

X\oslash X

,

\varphi :X\to X

, 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 ${\mathcal {C}}(M)$ of continuous functions on a paracompact locally compact topological space $M$ with the topology of uniform convergence on compact sets $T\subseteq M$ ,
the algebra ${\mathcal {E}}(M)$ of smooth functions on a smooth manifold $M$ with the topology of uniform convergence with all derivatives on compact sets $T\subseteq M$ ,
the algebra ${\mathcal {O}}(M)$ of holomorphic functions on a Stein manifold $M$ with the topology of uniform convergence on compact sets $T\subseteq M$ ,
the algebra ${\mathcal {P}}(M)$ of polynomials (regular functions) on an affine algebraic variety $M$ with the strongest locally convex topology.

The algebras

{\mathcal {C}}(M)

,

{\mathcal {E}}(M)

, and

{\mathcal {O}}(M)

are Fréchet spaces, while the algebra

{\mathcal {P}}(M)

is a Brauner space.

4. Stereotype group algebras. In the case when the space

M

in the previous examples has a structure of a topological group, the dual stereotype spaces to the functional algebras on

M

are stereotype algebras with respect to the operation of convolution:^[10]

the algebra ${\mathcal {C}}^{\star }(G)$ of Radon measures with compact support on a locally compact group $G$ ,^[11]
the algebra ${\mathcal {E}}^{\star }(G)$ of distributions with compact support on a real Lie group $G$ ,
the algebra ${\mathcal {O}}^{\star }(G)$ of holomorphic functionals on a Stein group^[12] $G$ ,
the algebra ${\mathcal {P}}^{\star }(G)$ of currents on an affine algebraic group $G$ .

The algebras

{\mathcal {C}}^{\star }(G)

,

{\mathcal {E}}^{\star }(G)

, and

{\mathcal {O}}^{\star }(G)

are Brauner spaces, while the algebra

{\mathcal {P}}^{\star }(G)

is a Fréchet space. 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 $X$ is called a left stereotype module over a stereotype algebra $A$ , if a bilinear operation $(a,x)\in A\times X\mapsto a\cdot x\in X$ is defined that turns $X$ into a left module over $A$ , and this operation is a continuous bilinear map in the stereotype sense: for each compact set $S\subseteq A$ and for each neighbourhood of zero $U\subseteq X$ there is a neighbourhood of zero $V\subseteq X$ such that

$S\cdot V\subseteq U,$

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

$W\cdot T\subseteq U.$

The notion of a right stereotype module over a stereotype algebra $A$ 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 $X$ with a structure of left (right) module over a stereotype algebra $A$ is a stereotype module over $A$ if and only if the operation of multiplication $(a,x)\in A\times X\mapsto a\cdot x\in X$ (respectively, $(x,a)\in X\times A\mapsto x\cdot a\in X$ ) defines a continuous map (a representation) $A\to X\oslash X$ .^[18]

Theorem.^[19] For each stereotype algebra $A$ the categories _$A$Ste and Ste_$A$ of left and right stereotype modules over $A$ are enriched categories over the monoidal category (Ste, $\circledast$ , ${\mathbb {C} }$ ) 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]

Notes

^ There is another notion, an injective stereotype algebra, which is defined as a monoid in the monoidal category (Ste, $\odot$ , ${\mathbb {C} }$ ) of stereotype spaces with the dual tensor product, $\odot$ . The class of these algebras is much narrower.
^ ^a ^b Akbarov 2003, p. 271.
^ Akbarov 2005.
^ ...or the field $\mathbb {R}$ of real numbers
^ Akbarov 2009, p. 483.
^ Akbarov 2016, p. 147.
^ Akbarov 2017, p. 594. sfn error: multiple targets (2×): CITEREFAkbarov2017 (help)
^ The requirement on $A$ in the analytic definition of stereotype algebra can be equivalently expressed as follows: for each compact set $S\subseteq A$ and for each net $y_{i}$ tending to zero in $A$ , $y_{i}{\overset {A}{\underset {i\to \infty }{\longrightarrow }}}0$ the nets $x\cdot y_{i}$ and $y_{i}\cdot x$ tend to zero in $A$ uniformly on $x\in S$ : $x\cdot y_{i}{\overset {A}{\underset {x\in S}{\underset {i\to \infty }{\rightrightarrows }}}}0\quad \&\quad y_{i}\cdot x{\overset {A}{\underset {x\in S}{\underset {i\to \infty }{\rightrightarrows }}}}0.$
^ A homomorphism $\varphi :A\to B$ is said to be unital if it preserves unit: $\varphi (1_{A})=1_{B}$ .
^ ^a ^b ^c ^d Akbarov 2003, p. 272.
^ If $G$ is an infinite locally compact group then the algebra ${\mathcal {C}}^{\star }(G)$ of measures on $G$ is not a Fréchet algebra. In the case when $G$ is compact, ${\mathcal {C}}^{\star }(G)$ is a Smith space. If $G$ is $\sigma$ -compact, then ${\mathcal {C}}^{\star }(G)$ is a Brauner space.
^ A Stein group is a complex Lie group $G$ which is a Stein manifold.
^ I.e. the homomorphisms of stereotype algebras ${\mathcal {C}}^{\star }(G)\to A$ are in one-to-one correspondence with the continuous representations $G\to A$ of the group $G$ , and the same is true for the algebras ${\mathcal {E}}^{\star }(G)$ , ${\mathcal {O}}^{\star }(G)$ and ${\mathcal {P}}^{\star }(G)$ 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 algebras Akbarov 2003, p. 275 and the structure of Hopf algebra on themAkbarov 2009, p. 507.
^ Akbarov 2003, p. 283.
^ Here $X\oslash X$ is the stereotype space of operators $\varphi :X\to X$ .
^ Akbarov 2003, p. 289.
^ ^a ^b Akbarov 2017. sfn error: multiple targets (2×): CITEREFAkbarov2017 (help)
^ Akbarov 2009.
^ Akbarov 2016.
^ Kuznetsova 2013.

References

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. {{cite journal}}: Invalid |ref=harv (help)
Akbarov, S.S. (2005). "Pontryagin duality and topological algebras". Banach Center Publications. 67: 55–71. doi:10.4064/bc67-0-5. {{cite journal}}: Invalid |ref=harv (help)
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. arXiv:0806.3205. doi:10.1007/s10958-009-9646-1. {{cite journal}}: Invalid |ref=harv (help)
Akbarov, S.S. (2016). "Envelopes and refinements in categories, with applications to functional analysis". Dissertationes Mathematicae. 513: 1–188. arXiv:1110.2013. doi:10.4064/dm702-12-2015. {{cite journal}}: Invalid |ref=harv (help)
Akbarov, S.S. (2017). "Continuous and smooth envelopes of topological algebras. Part 1". Journal of Mathematical Sciences. 227 (5): 531–668. arXiv:1303.2424. doi:10.1007/s10958-017-3599-6. {{cite journal}}: Invalid |ref=harv (help)
Akbarov, S.S. (2017). "Continuous and smooth envelopes of topological algebras. Part 2". Journal of Mathematical Sciences. 227 (6): 669–789. arXiv:1303.2424. doi:10.1007/s10958-017-3600-4. {{cite journal}}: Invalid |ref=harv (help)
Kuznetsova, Y. (2013). "A duality for Moore groups". Journal of Operator Theory. 69 (2): 101–130. arXiv:0907.1409. Bibcode:2009arXiv0907.1409K. doi:10.7900/jot.2011mar17.1920. {{cite journal}}: Invalid |ref=harv (help)

[projectivealgebra-1] There is another notion, an injective stereotype algebra, which is defined as a monoid in the monoidal category (Ste, $\odot$ , ${\mathbb {C} }$ ) of stereotype spaces with the dual tensor product, $\odot$ . The class of these algebras is much narrower.

[FOOTNOTEAkbarov2003271-2] Akbarov 2003, p. 271.

[FOOTNOTEAkbarov2005-3] Akbarov 2005.

[4] ...or the field $\mathbb {R}$ of real numbers

[FOOTNOTEAkbarov2009483-5] Akbarov 2009, p. 483.

[FOOTNOTEAkbarov2016147-6] Akbarov 2016, p. 147.

[FOOTNOTEAkbarov2017594-7] Akbarov 2017, p. 594. sfn error: multiple targets (2×): CITEREFAkbarov2017 (help)

[8] The requirement on $A$ in the analytic definition of stereotype algebra can be equivalently expressed as follows: for each compact set $S\subseteq A$ and for each net $y_{i}$ tending to zero in $A$ , $y_{i}{\overset {A}{\underset {i\to \infty }{\longrightarrow }}}0$ the nets $x\cdot y_{i}$ and $y_{i}\cdot x$ tend to zero in $A$ uniformly on $x\in S$ : $x\cdot y_{i}{\overset {A}{\underset {x\in S}{\underset {i\to \infty }{\rightrightarrows }}}}0\quad \&\quad y_{i}\cdot x{\overset {A}{\underset {x\in S}{\underset {i\to \infty }{\rightrightarrows }}}}0.$

[9] A homomorphism $\varphi :A\to B$ is said to be unital if it preserves unit: $\varphi (1_{A})=1_{B}$ .

[FOOTNOTEAkbarov2003272-10] Akbarov 2003, p. 272.

[11] If $G$ is an infinite locally compact group then the algebra ${\mathcal {C}}^{\star }(G)$ of measures on $G$ is not a Fréchet algebra. In the case when $G$ is compact, ${\mathcal {C}}^{\star }(G)$ is a Smith space. If $G$ is $\sigma$ -compact, then ${\mathcal {C}}^{\star }(G)$ is a Brauner space.

[Stein-group-12] A Stein group is a complex Lie group $G$ which is a Stein manifold.

[13] I.e. the homomorphisms of stereotype algebras ${\mathcal {C}}^{\star }(G)\to A$ are in one-to-one correspondence with the continuous representations $G\to A$ of the group $G$ , and the same is true for the algebras ${\mathcal {E}}^{\star }(G)$ , ${\mathcal {O}}^{\star }(G)$ and ${\mathcal {P}}^{\star }(G)$ with the smooth, holomorphic and polynomial representations of the corresponding groups.

[FOOTNOTEAkbarov2003275-14] Akbarov 2003, p. 275.

[FOOTNOTEAkbarov2009507-15] Akbarov 2009, p. 507.

[16] Other illustrations are the already mentioned universality property for the stereotype group algebras Akbarov 2003, p. 275 and the structure of Hopf algebra on themAkbarov 2009, p. 507.

[FOOTNOTEAkbarov2003283-17] Akbarov 2003, p. 283.

[18] Here $X\oslash X$ is the stereotype space of operators $\varphi :X\to X$ .

[FOOTNOTEAkbarov2003289-19] Akbarov 2003, p. 289.

[FOOTNOTEAkbarov2017-20] Akbarov 2017. sfn error: multiple targets (2×): CITEREFAkbarov2017 (help)

[FOOTNOTEAkbarov2009-21] Akbarov 2009.

[FOOTNOTEAkbarov2016-22] Akbarov 2016.

[FOOTNOTEKuznetsova2013-23] Kuznetsova 2013.

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