Mixed limits of some functional spaces
Andrei Novikov, Zohreh Eskandarian, Zamira Kholmatova

TL;DR
This paper introduces the concept of mixed limits of functional spaces, exploring conditions under which upper and lower limits coincide, and examines their properties, especially in relation to non-commutative $L_p$-spaces and Banach algebras.
Contribution
It defines mixed limits of functional spaces and analyzes their properties, including their emergence as limits of non-commutative $L_p$-spaces and Banach algebras.
Findings
Mixed limits generalize inductive and projective limits of functional spaces.
Limit spaces of Banach algebras are (LF)-spaces when convergent.
Conditions identified for upper and lower limits to coincide.
Abstract
In this article we propose a conception of mixed limits of functional spaces as the case, when the upper limit (projective limit of inductive limits) and the lower limit (inductive limit of projective limits) coincide as topological spaces, which are generalization of inductive and projective limits of functional spaces. We show a cases where these mixed limits are naturally obtained as the limit spaces of non-commutative -type spaces associated with the sequence of operators. Also, we obtain results on the properties of limit spaces, we show that limit spaces of Banach algebras are (LF)-spaces, if they converge.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Banach Space Theory · Advanced Operator Algebra Research · Advanced Topics in Algebra
Mixed limits
of some functional spaces
A.A. Novikov∗, Z.Eskandarian∘, Z.Kholmatova†
Abstract
In this article we propose a conception of mixed limits of functional spaces as the case, when the upper limit (projective limit of inductive limits) and the lower limit (inductive limit of projective limits) coincide as topological spaces, which are generalization of inductive and projective limits of functional spaces. We show a cases where these mixed limits are naturally obtained as the limit spaces of non-commutative -type spaces associated with the sequence of operators. Also, we obtain results on the properties of limit spaces, we show that limit spaces of Banach algebras are (LF)-spaces, if they converge.
e-mail: [email protected], Kazan Federal University, Kremlievskaia ul. 18, Kazan, Tatarstan, 420008, Russia
e-mail: [email protected],Kazan Federal University, Kremlievskaia ul. 18, Kazan, Tatarstan, 420008, Russia
e-mail: [email protected], Innopolis University, Universitetskaia ul., 1, Innopolis, Tatarstan, 420500, Russia
subject classification: 46A13, 46B70, 46L10, 46L51, 46L52
keywords: inductive limit, projective limit, power parameter, (LB)-space, (LF)-space, Frechet space, locally convex space, order unit base norm, inductive limit, initial topology, final topology, order unit space, measurable functions, Banach space
Introduction
In the article [16] we have defined the space associated with positive operator affiliated with the von Neumann algebra. Further in [20, 14] we have considered the commutative constructions of the limits spaces , and , and found that they are total for each other in the dualities and In this work we start to apply the same methodology for the noncommutative spaces.
In the result we get the (LF)-spaces (the inductive limits of Frechet spaces), which are studied for example in [3, 8, 7, 10].
1 Definitions and Notation
Let be Banach spaces with the norms and such that . We will write if
[TABLE]
Throughout this paper we adhere to the following notation. By we denote a von Neumann algebra that acts on a Hilbert space with the scalar product . We denote its selfadjoint part by , and the set of all projections in by . Let and , then by we denote the restriction of to (i.e. ), also we denote the reduction of to by . By we denote the center of . By and we denote the predual of and its Hermitian part, respectively. If an operator is affiliated with then we write . We denote the domain of an operator by . The adjoint operator is denoted by . We denote the identity operator, the zero operator and the zero vector by , and , respectively. We use standard notation for multiplication of a functional by an operator , namely, , and denote the linear functionals , and , respectively.
We also consider partial order for positive selfadjoint operators affiliated with . If is affiliated with we denote it as . For positive selfadjoint , we write if and only if
[TABLE]
If for an increasing net of operators affiliated with there exists , then we write For a positive selfadjoint operator we use to denote with . From the Spectral theorem it follows that the mapping is monotone operator-valued function and for all , therefore . For an unbounded and we define as .
From now on stands for a positive selfadjoint operator affiliated with . We consider
[TABLE]
[TABLE]
Note that if operator is bounded, then and We define a seminorm on as
[TABLE]
Also, Theorem 2 from [7] states, that if operator a is bounded, then
[TABLE]
If is a norm, then we call it the -norm. Note that the -norm coincides with the restriction of the standard norm in onto .
definition 1** ([16]).**
By we denote the completion of the real normed space with the norm
[TABLE]
where .
The dual of is (), where
[TABLE]
and . We identify the elements of with the corresponding elements in . Further for an injective operator a we always assume that is equiped with the a-norm.
For we define the sesquilinear form a on by the equality The set of all such sesquilinear forms is denoted by
[TABLE]
We consider partial order on , such that
[TABLE]
if and only if for all . By we denote the seminormed space of sesquilinear forms equiped with the seminorm
definition 2** ([1]).**
Let be the pair of Banach spaces. For and let
[TABLE]
By -method of interpolation we call the construction of the space as the linear subspace of the sum such that
[TABLE]
definition 3**.**
By we denote the noncommutative Lorentz space with which is the interpolation space
definition 4**.**
By we denote the noncommutative Lebesgue space which is interpolation space .
2 Preliminaries
For the equality
[TABLE]
defines the normal functional
If an operator is injective then
[TABLE]
and the latter implies that the mapping is an isometrical isomorphism of onto
For an injective operator the mapping
[TABLE]
is an isometrical isomporphism of onto . Thus, is isometricaly isomorphic to Further we call the isomorphism the canonical isomorphism of onto and identify the corresponding elements. Moreover, the adjoint mapping is an isometrical isomorphism of onto
For an injective operator the continue of the mapping
[TABLE]
is an isometrical isomorphism of onto
Let and be Banach spaces and , then the embedding
[TABLE]
is continuous.
3 Mixed Limits of Banach Spaces
By we denote two-indexed family of Banach spaces. Let be the topology of the norm which is natural norm of the Banach space
Let and , and consider that
[TABLE]
Note that
[TABLE]
with
Consider the limits
[TABLE]
with the topology and the topology , respectively. The topology is the strongest topology on such that the mappings
[TABLE]
are continuous and the topology is the weakest on such that the mappings
[TABLE]
are coninuous.
Lemma 1**.**
Let , then
- (i)
; 2. (ii)
**
Proof.
We have
[TABLE]
Evidently, if , then there exists such that Therefore,
If , then Thus,
[TABLE]
∎
definition 5**.**
We call
[TABLE]
an upper limit.
definition 6**.**
And we call
[TABLE]
a lower limit.
Proposition 1**.**
[TABLE]
Proof.
If , then Therefore,
[TABLE]
If , then Thus,
[TABLE]
∎
definition 7**.**
The family is called converging if .
Lemma 2**.**
Let and
[TABLE]
(i.e. the topology induced by the on ), then
[TABLE]
Proof.
Let , i.e. , where . Then is open in . The embedding
[TABLE]
is continuous, therefore
[TABLE]
is open in for any
The topology is the strongest topology, such that any embedding is continuous. If , then there exists topology
[TABLE]
stronger, then the topology , . We show that for any the preimage is open.
Consider three cases , , ().
If , then is open. 2. 2.
If , then
[TABLE]
[TABLE]
is open, since and are open. 3. 3.
If , then
[TABLE]
[TABLE]
is open, since and are open.
Thus, we get the contradiction with the maximality of the topology , therefore ∎
Lemma 3**.**
Let and
[TABLE]
be the topology induced by the topology on then
[TABLE]
Proof.
The topology is determinded by the family of semi-norms , and the topology is determined by the family Evidently, that ∎
For define the topology the strongest topology, such that any embedding
[TABLE]
is continuous.
Also, determine the topology such that it will be the weakest topology on with the embedings
[TABLE]
being continuous.
Theorem 1**.**
The embedding
[TABLE]
is continuous.
Proof.
Note that
[TABLE]
At the same time
[TABLE]
It is sufficient to prove that for any we have the inclusion
[TABLE]
Thus
[TABLE]
и
[TABLE]
Note, that by reduction we obtain space with the topology:
[TABLE]
But then thus
[TABLE]
∎
4 Limits of noncommutative spaces
We split this section into four parts. Firstly, we consider general properties for spaces and its norms. Secondly, we consider case of bounded , in the third case we consider unbounded such that is bounded, and at last we consider the general case of unbounded .
4.1 Case of bounded operator
Lemma 4**.**
Let (a is affiliated with , ), then for any
Proof.
Let . Evidently, , thus
[TABLE]
[TABLE]
Hence, , and since , are linear spaces, we have .∎
Particularly, if is bounded, then where , if is such that is bounded, then , where .
Proposition 2**.**
Let , , then in is equivalent to , where .
Proof.
Since, is isometrically isomorphic to for we may consider where , then . On the other hand,
[TABLE]
[TABLE]
∎
Particularly, for the bounded we have is equivalent to , where If is bounded and we consider , then is equivalent to .
Now, consider
Lemma 5**.**
For a bounded and , if , then
[TABLE]
Proof.
Let , then we consider where . Thus,
[TABLE]
[TABLE]
Note, that , thus
[TABLE]
∎
The following definition is standard.
definition 8**.**
Let be normed spaces with the norms , respectively. We write if and only if there exists , such that for any the inequality holds.
Lemma 6**.**
For a bounded and . If , then
[TABLE]
Proof.
Let then , where , then
[TABLE]
[TABLE]
∎
Now, let us consider the limit space.
definition 9**.**
For the bounded we define the topological space as the limit space for , where and , , is the topology on of the norm .
The latter definition essentially means, that is the initial topology on the for the family of mapping
[TABLE]
Also, we can describe this topology as the topology on defined by the family of the seminorms . The familiy of the seminorms describes the same topology, thus we get the following theorem.
Theorem 2**.**
For the bounded the space is metriziable locally-convex space (Frechet space).
For the more detailed proof of the latter theorem see [14][Lemma 2].
4.2 Case of unbounded operator with bounded inverse
Now, consider case, when , a is not bounded, but is bounded.
Lemma 7**.**
For a , and . If , then
[TABLE]
Proof.
Let , then we consider where . Thus,
[TABLE]
[TABLE]
Note, that , therefore we have that . ∎
Lemma 8**.**
For , such that is bounded and , such that we have
[TABLE]
Proof.
Let then , where , then
[TABLE]
[TABLE]
[TABLE]
∎
definition 10**.**
For the unbounded with bounded inverse we define the topological space as the limit space for , where and is the strongest topology such that for all the mappings are continuous.
Evidently, is an (LB)-space.
4.3 Case of unbounded operator with unbounded inverse
Now, consider case, when , , and are unbounded, simultaneously. Then we take spectra gecomposition
[TABLE]
and determine
[TABLE]
Note, that
[TABLE]
[TABLE]
[TABLE]
Evidently, , thus we represent as a subalgebra of the algebra of matrices :
[TABLE]
meaning that if acts on , then
[TABLE]
[TABLE]
[TABLE]
Evidently, in this notation is represented as
[TABLE]
where and are defined by the following definitions
definition 11**.**
Let , , then the sesquilinear form
[TABLE]
is defined by the equality
[TABLE]
definition 12**.**
Let . We define the linear space of sesqulinear forms
[TABLE]
endowed with the norm
Let be injective, then is injective bounded operator in acting on and is injective operator affiliated with (acting on ) with bounded inverse. Moreover, is isometrically isomorphic to as well as is isometrically isomorphic to by construction. Also, it is evident, that there exists the isomorphism between the spaces and . Thus, we only need to consider the limits for for
Further we also use the notation
[TABLE]
and
[TABLE]
Let and , , then note, that the following schemes
[TABLE]
As for the norms,
[TABLE]
By we denote the topology of the norm on the space Note, that
[TABLE]
where
[TABLE]
Sine there exists the isomorphism between and , we will only consider one case. Consider the limits
[TABLE]
with the topologies and , respectively. Topology is the strongest topology on , such that all the mappings
[TABLE]
are continuous and the topology is the weakest topology on that all the mappings
[TABLE]
are continuous.
Remark Note, that any is Banach space, thus is an -space.
Remark Note, that is a Frechet space, since its topology is determined by the countable set of the seminorms.
Lemma 9**.**
Let , , then
- (i)
; 2. (ii)
**
Proof.
Since
[TABLE]
it follows that if , then there exists such that , thus
If , then for all thus
[TABLE]
∎
We consider the following constructions:
[TABLE]
Note that
[TABLE]
Proposition 3**.**
[TABLE]
Proof.
Note, that if , then thus
[TABLE]
Note, that if , then thus
[TABLE]
∎
definition 13**.**
We call the family converging if and denote it as
Corollary 1**.**
If the family is converging, then
[TABLE]
Lemma 10**.**
Let and
[TABLE]
i.e. the topology induced by on , then
[TABLE]
Proof.
Let , i.e. with . Then is open in . Note, that the embedding
[TABLE]
is also continuous, thus
[TABLE]
is open in for any
The topology is the strongest topology, such that all of the embeddings are continuous. If , then there exists the toplogy
[TABLE]
such that it is stronger, then , and for any the preimages are open. Further we explain why it is open.
Consider three cases , , ().
-
Let , then evidently is open.
-
Let , then
[TABLE]
[TABLE]
is open, since is open and is open.
- Let , then
[TABLE]
[TABLE]
is open, since is open and is open.
Thus, we get a contradiction with the maximality of , therefore ∎
Lemma 11**.**
Let and
[TABLE]
i.e. the topology induced by on then
[TABLE]
Proof.
The topology is determined with the set of the seminorms and the topology is determined by the system It is sufficient to note, that ∎
For it is natural to define the topology which is determined as the strongest topology such that all of the embeddings
[TABLE]
are continuous.
On the other side, it is natural to define the topology which would be the weakest topology on such that all of the embeddings
[TABLE]
are continuous.
Theorem 3**.**
The embedding
[TABLE]
is continuous.
Proof.
Note, that
[TABLE]
At the same time,
[TABLE]
It is sufficient to prove that for any we have the inclusion
[TABLE]
Thus,
[TABLE]
and
[TABLE]
Note, that the reductions always lead to the space and may be rewrited as
[TABLE]
But then therefore
[TABLE]
∎
Remark Note, that is a topology of -space and is a topology of the projective limit of -spaces.
Remark All the constructions of this section may be applied to the system . We will distinguish such constructions by the notation , , , and so on.
4.4 Bring it all together
Evidently we may consider different limits of spaces, using the constructions above, particularly, we may define
definition 14**.**
Let be a positive operator affiliated with von Neumann algebra acting on the Hilbert space , such that not nor are necessarily bounded. Then we define the lower limit of the spaces as a vector space of sesquilinear forms formally wriiten as
[TABLE]
definition 15**.**
Let be a positive operator affiliated with von Neumann algebra acting on the Hilbert space , such that not nor are necessarily bounded. Then we define the upper limit of the spaces as a vector space of sesquilinear forms formally wriiten as
[TABLE]
Theorem 4**.**
Let , then
[TABLE]
is -space.
Corollary 2**.**
If is affiliated with the center of the von Neumann algebra , then the limits space is an (LF)-space.
5 -spaces
Let be a von Neumann algebra and be a positive bounded injective linear operator. We consider and descirbed previously (also see [16]).
It is easy to see that for any
[TABLE]
We also obtain, that is isometrically isomorphic to .
Note, that for any such that and any pair the inequalities
[TABLE]
[TABLE]
hold.
Thus, we for any bounded the inclusion
[TABLE]
[TABLE]
hold.
Theorem 5**.**
Let be positive injective bounded operator from the von Neumann algebra and .
[TABLE]
[TABLE]
with being the interpolation space .
Proof.
The inclusions are proved analogously. Consider as the example the embedding в First of all note, that
[TABLE]
After that note, that
[TABLE]
[TABLE]
thus if , then At the same time the inequality also implies the continuity. ∎
From the latter Theorem we obtain that we, also, can build the following construction for the fixed .
Let be the inductive limit (by ) of and be the projective limit (by ) of . The projective limit of will be the upper limit and the inductive limit of will be the lower limit . If they coincide tas the topological spaces, then has the mixed limit for converging to infinity.
Acknowledgments
This work was supported by Russian Foundation for Basic Research Grant 18-31-00218.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Bennett, C. Interpolation of operators / C. Bennett, R. Sharpley // Pure and Applied Mathematics, 129, Academic Press, Inc., Boston, MA, pp. xiv+469
- 2[2] Bikchentaev A.M., Sherstnev A.N. Studies on Noncommutative Measure Theory in Kazan University (1968–2018) / A.M. Bikchentaev, A.N. Sherstnev // International Journal of Theoretical Physics – May 2019 – DOI: 10.1007/s 10773-019-04156-x
- 3[3] Esterle, J. Countable inductive limits of Frechet algebras / J. Esterle // Journal d’Analyse Mathématique – 1997 – V. 71, I.1 – pp. 195–204
- 4[4] Falcone, T. The non-commutative flow of weights on a von Neumann algebra / A.J. Falcone, M. Takesaki // J. Funct. Anal. – 2001 – V.182 – №1 – p. 170–206
- 5[5] Falcone, T. Operator valued weights without structure theory / A.J. Falcone, M. Takesaki // Trans. Amer. Math. Soc. – 1999 – V.351 – №1 – p. 323–341
- 6[6] Falcone, T. L 2 subscript 𝐿 2 L_{2} -von Neumann modules, their relative tensor products and the spatial derivative / A.J. Falcone // Illinois J. Math. – 2000 – V.44 – №2 – p. 407-437
- 7[7] Garcia-Lafuente, J. M. On the completion of (LF)-spaces / J. M. Garcia-Lafuente // Monatshefte für Mathematik – 1987 – V.103 – I.2 – p. 115–120
- 8[8] Kunzinger, M. Barrelledness, Baire-like-and (LF)-spaces / - 1993 - Longman Scientific and Technical
