Mixed limits of some functional spaces

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 $L_p$-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.

Equations

$$\exists C\in {\mathbb{R}}^{+}\forall x\in {\mathcal{B}}_1\Vert x{\Vert }_2\le C\Vert x{\Vert }_1.$$

$$D(y^{\frac{1}{2}})\subset D(x^{\frac{1}{2}})\text{ and }\Vert x^{\frac{1}{2}}f{\Vert }^2\le \Vert y^{\frac{1}{2}}f{\Vert }^2\text{ for all }f\in D(y^{\frac{1}{2}}).$$

$${\mathfrak{D}}_a^{+}\equiv \{\phi \in {\mathcal{M}}_{\ast }|\phi (a)<+\infty \},$$

$${\mathfrak{D}}_a^h\equiv {\mathfrak{D}}_a^{+}-{\mathfrak{D}}_a^{+}\text{ and }{\mathfrak{D}}_a\equiv {\mathrm{lin}}_{\mathbb{C}}{\mathfrak{D}}_a^h.$$

$$\Vert \phi {\Vert }_a:=\inf \{{\phi }_1(a)+{\phi }_2(a)|\phi ={\phi }_1-{\phi }_2;{\phi }_1,{\phi }_2\in {\mathfrak{D}}^{+}\}.$$

$$\Vert \phi {\Vert }_a=\Vert a^{\frac{1}{2}}\phi a^{\frac{1}{2}}\Vert .$$

$$r_a(\phi )=\inf \{{\phi }_1(a)+{\phi }_2(a)|\phi ={\phi }_1-{\phi }_2,{\phi }_1,{\phi }_2\in {\mathfrak{D}}_a^{+}\},$$

$$L_{\infty }(a)\equiv \{x\in ({\mathfrak{D}}_a{)}^{\mathrm{al}}|\lambda \in \mathbb{R},-\lambda a\le x\le \lambda a\}$$

$${\mathcal{S}}_a(\mathcal{M})\equiv \{\widehat{a^{\frac{1}{2}}x{a}^{\frac{1}{2}}}|x\in \mathcal{M}\}.$$

$$\widehat{a^{\frac{1}{2}}x{a}^{\frac{1}{2}}}\le \widehat{a^{\frac{1}{2}}y{a}^{\frac{1}{2}}}$$

$$K(x,t;X_0,X_1)=\inf \left\{{\Vert x_0\Vert }_{X_0}+t{\Vert x_1\Vert }_{X_1}|x=x_0+x_1,x_0\in X_0,x_1\in X_1\right\}.$$

$${\left({\int }_0^{\infty }{\left(t^{-\theta }K(x,t;X_0,X_1)\right)}^q\frac{dt}{t}\right)}^{1/q}<\infty .$$

$$a^{\frac{1}{2}}\phi a^{\frac{1}{2}}(x)=\underset{\lambda \to +\infty }{\lim }\phi (a_{\lambda }^{\frac{1}{2}}x{a}_{\lambda }^{\frac{1}{2}})\text{ with }x\in \mathcal{M}$$

$$\inf \{\lambda |-\lambda \widehat{a^{\frac{1}{2}}1a^{\frac{1}{2}}}\le \widehat{a^{\frac{1}{2}}x{a}^{\frac{1}{2}}}\le \lambda \widehat{a^{\frac{1}{2}}1a^{\frac{1}{2}}}\}=\Vert x\Vert \text{ for any }x\in {\mathcal{M}}^{\mathrm{sa}}$$

$$u:x\in \mathcal{M}\mapsto \widehat{a^{\frac{1}{2}}x{a}^{\frac{1}{2}}}\in L_{\infty }(a)$$

$$v:\phi \in {\mathfrak{D}}_a(\subset L_1(a))\mapsto a^{\frac{1}{2}}\phi a^{\frac{1}{2}}\in {\mathcal{M}}_{\ast }$$

$$\mathrm{id}:x\in {\mathcal{B}}_1\mapsto x\in {\mathcal{B}}_2$$

$$\begin{array}{ccccc}\begin{array}{ccc}{\mathcal{B}}^{k_1,n_1}& \subset & {\mathcal{B}}^{k_1,n_2}\\ \cup & & \cup \\ {\mathcal{B}}^{k_2,n_1}& \subset & {\mathcal{B}}^{k_2,n_2}\end{array}& & \text{ and }& & \begin{array}{ccc}\Vert \cdot {\Vert }_{k_1,n_1}& \succ & \Vert \cdot {\Vert }_{k_1,n_2}\\ ⋏& & ⋏\\ \Vert \cdot {\Vert }_{k_2,n_1}& \succ & \Vert \cdot {\Vert }_{k_2,n_2}\end{array}\end{array}.$$

$$\begin{array}{ccc}{\tau }^{k_1,n_1}{|}_{k_2}& \supset & {\tau }^{k_1,n_2}{|}_{k_2,n_1}\\ \cap & & \cap \\ {\tau }^{k_2,n_1}& \supset & {\tau }^{k_2,n_2}{|}_{n_1}\end{array},$$

$${\mathfrak{B}}^k=\bigcup _{n>0}{\mathcal{B}}^{k,n}\text{ и }{\mathfrak{B}}_n=\bigcap _{k>0}{\mathcal{B}}^{k,n}$$

$${\phi }_n^k:x\in ({\mathcal{B}}^{k,n},\Vert \cdot {\Vert }_{k,n})\mapsto x\in {\mathfrak{B}}^k$$

$${\psi }_k^n:x\in {\mathfrak{B}}_n\mapsto x\in ({\mathcal{B}}^{k,n},\Vert \cdot {\Vert }_{k,n})$$

$${\mathfrak{B}}^n=\bigcup _{\gamma >0}{\mathcal{B}}^{n,\gamma },{\mathfrak{B}}^k=\bigcup _{\gamma >0}{\mathcal{B}}^{k,\gamma }\text{ и }{\mathcal{B}}^{n,\gamma }\subset {\mathcal{B}}^{k,\gamma }.$$

$$x\in \bigcap _{\gamma >0}{\mathcal{B}}^{\gamma ,n}={\mathfrak{B}}^n.$$

$$\overline{\mathfrak{L}}=\overline{\lim }{\mathcal{B}}^{k,k}\equiv \bigcap _{k>0}\bigcup _{n\ge k}{\mathcal{B}}^{n,n}$$

$$\underline{\mathfrak{L}}=\underline{\lim }{\mathcal{B}}^{k,k}\equiv \bigcup _{k>0}\bigcap _{n\ge k}{\mathcal{B}}^{n,n}$$

$$\overline{\mathfrak{L}}\supset \underline{\mathfrak{L}}.$$

$$\overline{\mathfrak{L}}=\bigcap _{k>0}\bigcup _{n>0}{\mathcal{B}}^{k,n}=\bigcap _{k>0}\bigcup _{n\ge k}{\mathcal{B}}^{k,n}\supset \bigcap _{k>0}\bigcup _{n\ge k}{\mathcal{B}}^{n,n}.$$

$$\bigcup _{k>0}\bigcap _{n\ge k}{\mathcal{B}}^{n,n}\supset \bigcup _{k>0}\bigcap _{n\ge k}{\mathcal{B}}^{n,k}=\bigcup _{k>0}\bigcap _{n>0}{\mathcal{B}}^{n,k}=\underline{\mathfrak{L}}.$$

$${\tau }^k{|}_n:=\{X\cap {\mathfrak{B}}^n|X\in {\tau }^k\}$$

Full text

$*$ e-mail: a.hobukob@gmail.com, Kazan Federal University, Kremlievskaia ul. 18, Kazan, Tatarstan, 420008, Russia

$\circ$ e-mail: zohreh.eskandarian@gmail.com,Kazan Federal University, Kremlievskaia ul. 18, Kazan, Tatarstan, 420008, Russia

$\dagger$ e-mail: zamira.kholmatova@gmail.com, 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 $L_{\infty}(a)$ associated with positive operator affiliated with the von Neumann algebra. Further in [20, 14] we have considered the commutative constructions of the limits spaces $L_{\infty}(f^{\alpha})$, $L_{1}(f^{\alpha})$ and $L_{\infty}^{*}(f^{\alpha})$, and found that they are total for each other in the dualities $\langle\lim L_{1}(f^{\alpha}),\lim L_{\infty}(f^{\alpha})\rangle$ and $\langle\lim L_{\infty}(f^{\alpha}),\lim L_{\infty}^{*}(f^{\alpha})\rangle.$ In this work we start to apply the same methodology for the noncommutative $L_{\infty}(a)$ 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 $\mathcal{B}_{1},\mathcal{B}_{2}$ be Banach spaces with the norms $\|\cdot\|_{1}$ and $\|\cdot\|_{2}$ such that $\mathcal{B}_{1}\subset\mathcal{B}_{2}$. We will write $\|\cdot\|_{1}\succ\|\cdot\|_{2}$ if

[TABLE]

Throughout this paper we adhere to the following notation. By $\mathcal{M}$ we denote a von Neumann algebra that acts on a Hilbert space $H$ with the scalar product $\langle\cdot,\cdot\rangle$. We denote its selfadjoint part by $\mathcal{M}^{\mathrm{sa}}$, and the set of all projections in $\mathcal{M}$ by $\mathcal{M}^{\mathrm{pr}}$. Let $p\in\mathcal{M}^{\mathrm{pr}}$ and $x\in\mathcal{M}$, then by $x_{p}$ we denote the restriction of $pxp$ to $pH$ (i.e. $x_{p}:=pxp|_{pH}$), also we denote the reduction of $\mathcal{M}$ to $pH$ by $\mathcal{M}_{p}$. By $\mathfrak{C}(\mathcal{M})$ we denote the center of $\mathcal{M}$. By $\mathcal{M}_{*}$ and $\mathcal{M}_{*}^{h}$ we denote the predual of $\mathcal{M}$ and its Hermitian part, respectively. If an operator $x$ is affiliated with $\mathcal{M}$ then we write $x\eta\mathcal{M}$. We denote the domain of an operator $x$ by $D(x)$. The adjoint operator is denoted by $x^{*}$. We denote the identity operator, the zero operator and the zero vector by $\mathbf{1}$,$\mathbf{0}$ and $\mathit{0}$, respectively. We use standard notation for multiplication of a functional $\varphi\in\mathcal{M}^{*}$ by an operator $x\in\mathcal{M}$, namely, $x\varphi$, $\varphi x$ and $x\varphi x$ denote the linear functionals $y\mapsto\varphi(xy)$, $y\mapsto\varphi(yx)$ and $y\mapsto\varphi(xyx)$, respectively.

We also consider partial order for positive selfadjoint operators affiliated with $\mathcal{M}$. If $x$ is affiliated with $\mathcal{M}$ we denote it as $x\eta\mathcal{M}$. For positive selfadjoint $x$, $y$ $\eta$ $\mathcal{M}$ we write $x\leq y$ if and only if

[TABLE]

If for an increasing net $(x_{j})_{j\in J}$ of operators affiliated with $\mathcal{M}$ there exists $x=\sup\limits_{j\in J}x_{j}$, then we write $x_{j}\nearrow x.$ For a positive selfadjoint operator $x\eta\mathcal{M}$ we use $x_{\lambda}$ to denote $\lambda x(\lambda+x)^{-1}$ with $\lambda\in\mathbb{R}^{+}\setminus\{0\}$. From the Spectral theorem it follows that the mapping $\lambda\mapsto x_{\lambda}\in\mathcal{M}^{+}$ is monotone operator-valued function and $\lim\limits_{\lambda\to+\infty}x^{\frac{1}{2}}_{\lambda}f=x^{\frac{1}{2}}f$ for all $f\in D(x^{\frac{1}{2}})$, therefore $x_{\lambda}\nearrow x$. For an unbounded $x$ and $\varphi\in\mathcal{M}^{+}_{*}$ we define $\varphi(x)$ as $\varphi(x):=\lim\limits_{\lambda\to+\infty}\varphi(x_{\lambda})$.

From now on $a$ stands for a positive selfadjoint operator affiliated with $\mathcal{M}$. We consider

[TABLE]

[TABLE]

Note that if operator $a$ is bounded, then $\mathfrak{D}_{a}^{+}=\mathcal{M}_{*}^{+},\mathfrak{D}_{a}^{h}=\mathcal{M}^{h}_{*}$ and $\mathfrak{D}_{a}=\mathcal{M}_{*}.$ We define a seminorm $\|\cdot\|_{a}$ on $\mathfrak{D}^{h}_{a}$ as

[TABLE]

Also, Theorem 2 from [7] states, that if operator a is bounded, then

[TABLE]

If $\|\cdot\|_{a}$ is a norm, then we call it the $a$-norm. Note that the $\mathbf{1}$-norm coincides with the restriction of the standard norm in $\mathcal{M}_{*}$ onto $\mathcal{M}_{*}^{h}$.

definition 1** ([16]).**

By $L_{1}^{h}(a)$ we denote the completion of the real normed space $\mathfrak{D}_{a}^{h}=\mathrm{lin}_{\mathbb{R}}\mathfrak{D}_{a}^{+}$ with the norm

[TABLE]

where $\mathfrak{D}_{a}^{+}=\{\varphi\in\mathcal{M}^{+}_{*}\ |\ \varphi(a)<+\infty\}$.

The dual of $L^{h}_{1}(a)$ is ($L^{\mathrm{sa}}_{\infty}(a),\|\cdot\|_{a}$), where

[TABLE]

and $\|x\|_{a}\equiv\inf\{\lambda\in\mathbb{R}\ |\ -\lambda a\leq x\leq\lambda a\}$. We identify the elements of $\mathfrak{D}^{h}_{a}$ with the corresponding elements in $L^{h}_{1}(a)$. Further for an injective operator a we always assume that $L^{\mathrm{sa}}_{\infty}(a)$ is equiped with the a-norm.

For $x\in\mathcal{M}$ we define the sesquilinear form a $a^{\frac{1}{2}}xa^{\frac{1}{2}}$ on $D(a^{\frac{1}{2}})\times D(a^{\frac{1}{2}})$ by the equality $\widehat{a^{\frac{1}{2}}xa^{\frac{1}{2}}}(f,g):=\langle xa^{\frac{1}{2}}f,a^{\frac{1}{2}}g\rangle.$ The set of all such sesquilinear forms is denoted by

[TABLE]

We consider partial order on $\mathcal{S}_{a}(\mathcal{M}^{\mathrm{sa}})$, such that

[TABLE]

if and only if $\widehat{a^{\frac{1}{2}}xa^{\frac{1}{2}}}(f,f)\leq\widehat{a^{\frac{1}{2}}ya^{\frac{1}{2}}}(f,f)$ for all $f\in D(a^{\frac{1}{2}})$. By $\mathcal{S}_{a}(\mathcal{M}^{\mathrm{sa}})$ we denote the seminormed space of sesquilinear forms $\{a^{\frac{1}{2}}xa^{\frac{1}{2}}\ |\ x\in\mathcal{M}^{\mathrm{sa}}\}$ equiped with the seminorm $p_{a}(\widehat{a^{\frac{1}{2}}xa^{\frac{1}{2}}}):=\inf\{\lambda\in\mathbb{R}^{+}\ |\ -\lambda\widehat{a^{\frac{1}{2}}\mathbf{1}a^{\frac{1}{2}}}\leq\widehat{a^{\frac{1}{2}}xa^{\frac{1}{2}}}\leq\lambda\widehat{a^{\frac{1}{2}}\mathbf{1}a^{\frac{1}{2}}}\}.$

definition 2** ([1]).**

Let $(X_{0},X_{1})$ be the pair of Banach spaces. For $t>0$ and $x\in X_{0}+X_{1}$ let

[TABLE]

By $K$-method of interpolation we call the construction of the space $K_{\theta,q}(X_{0},X_{1})$ as the linear subspace of the sum $X_{0}+X_{1}$ such that

[TABLE]

definition 3**.**

By $L_{p,q}^{h}(a)$ we denote the noncommutative Lorentz space with $p,q\in(1,+\infty)$ which is the interpolation space $K_{(p-1)/p,q}(L_{1}^{h}(a),L_{\infty}^{h}(a))$

definition 4**.**

By $L_{p}^{h}(a)$ we denote the noncommutative Lebesgue space which is interpolation space $L_{p,p}^{h}(a)$.

2 Preliminaries

For $\varphi\in\mathfrak{D}_{a}$ the equality

[TABLE]

defines the normal functional $a^{\frac{1}{2}}\varphi a^{\frac{1}{2}}\in\mathcal{M}_{*}.$

If an operator $a$ is injective then

[TABLE]

and the latter implies that the mapping $u_{1}:x\mapsto\widehat{a^{\frac{1}{2}}xa^{\frac{1}{2}}}$ is an isometrical isomorphism of $\mathcal{M}$ onto $\mathcal{S}_{a}(\mathcal{M}).$

For an injective operator $a$ the mapping

[TABLE]

is an isometrical isomporphism of $\mathcal{M}$ onto $L_{\infty}(a)$. Thus, $L_{\infty}(a)$ is isometricaly isomorphic to $\mathcal{S}_{a}(\mathcal{M}).$ Further we call the isomorphism $u_{1}^{-1}u$ the canonical isomorphism of $\mathcal{S}_{a}(\mathcal{)}\mapsto\mathcal{M}$ onto $L_{\infty}(a)$ and identify the corresponding elements. Moreover, the adjoint mapping $u^{t}$ is an isometrical isomorphism of $L_{\infty}(a)^{*}$ onto $\mathcal{M}^{*}.$

For an injective operator $a$ the continue of the mapping

[TABLE]

is an isometrical isomorphism of $L_{1}(a)$ onto $\mathcal{M}_{*}$

Let $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ be Banach spaces and $\|\cdot\|_{1}\ \succ\ \|\cdot\|_{2}$, then the embedding

[TABLE]

is continuous.

3 Mixed Limits of Banach Spaces

By $\mathcal{B}^{k,n}$ we denote two-indexed family of Banach spaces. Let $\tau^{k,n}$ be the topology of the norm $\|\cdot\|_{k,n}$ which is natural norm of the Banach space $\mathcal{B}^{k,n}.$

Let $k_{1}<k_{2}$ and $n_{1}<n_{2}$, $k_{i},n_{i}\in\mathbb{N},$ and consider that

[TABLE]

Note that

[TABLE]

with $\tau|_{k_{2}}\equiv\tau|_{n_{1}}\equiv\tau|_{k_{2},n_{1}}:=\{X\cap\mathcal{B}^{k_{2},n_{1}}\ |\ X\in\tau\}.$

Consider the limits

[TABLE]

with the topology $\tau^{k}$ and the topology $\tau_{n}$, respectively. The topology $\tau^{k}$ is the strongest topology on $\mathfrak{B}^{k}$ such that the mappings

[TABLE]

are continuous and the topology $\tau_{n}$ is the weakest on $\mathfrak{B}_{n}$ such that the mappings

[TABLE]

are coninuous.

Lemma 1**.**

Let $k<n,k,n\in\mathbb{N}$, then

1. (i)

$\mathfrak{B}^{k}\supset\mathfrak{B}^{n}$; 2. (ii)

$\mathfrak{B}_{k}\subset\mathfrak{B}_{n}.$**

Proof.

$(i)$ We have

[TABLE]

Evidently, if $x\in\mathfrak{B}^{n}$, then there exists $\gamma_{0}$ such that $x\in\mathcal{B}^{n,\gamma_{0}}\subset\mathcal{B}^{k,\gamma_{0}}.$ Therefore, $x\in\mathfrak{B}^{k}.$

$(ii)$ If $x\in\mathfrak{B}_{k}$, then $\forall\gamma>0\ x\in\mathcal{B}^{\gamma,k}\subset\mathcal{B}^{\gamma,n}.$ Thus,

[TABLE]

∎

definition 5**.**

We call

[TABLE]

an upper limit.

definition 6**.**

And we call

[TABLE]

a lower limit.

Proposition 1**.**

[TABLE]

Proof.

If $k_{1}<k_{2}$, then $\mathcal{B}^{k_{1},n}\supset\mathcal{B}^{k_{2},n}.$ Therefore,

[TABLE]

If $n_{1}<n_{2}$, then $\mathcal{B}^{k,n_{1}}\subset\mathcal{B}^{k,n_{2}}.$ Thus,

[TABLE]

∎

definition 7**.**

The family $\mathcal{B}^{k,n}$ is called converging if $\underline{\mathfrak{L}}=\overline{\mathfrak{L}}$.

Lemma 2**.**

Let $k<n,k,n\in\mathbb{N}$ and

[TABLE]

(i.e. the topology induced by the $\tau^{k}$ on $\mathfrak{B}_{n}$), then

[TABLE]

Proof.

Let $X_{0}\in\tau^{k}|_{n}$, i.e. $X_{0}=X\cap\mathfrak{B}^{k}$, where $X\in\tau^{k}$. Then $(\varphi^{k}_{\gamma})^{-1}(X)$ is open in $(\mathcal{B}^{k,\gamma},\|\cdot\|_{k,\gamma})$. The embedding

[TABLE]

is continuous, therefore

[TABLE]

is open in $(\mathcal{B}^{n,\gamma},\|\cdot\|_{n,\gamma})$ for any $\gamma>0.$

The topology $\tau^{n}$ is the strongest topology, such that any embedding $\varphi_{\gamma}^{n}$ is continuous. If $X_{0}\notin\tau^{n}$, then there exists topology

[TABLE]

stronger, then the topology $\tau^{n}$, $X_{0}\in\tau$. We show that for any $A\in\tau$ the preimage $(\varphi^{n}_{\gamma})^{-1}(A)$ is open.

Consider three cases $A\in\tau^{n}$, $A=X_{0}\cap Y$, $A=X_{0}\cup Y$ ($Y\in\tau^{n}$).

If $A\in\tau^{n}$, then $(\varphi_{\gamma}^{\beta})^{-1}(A)$ is open. 2. 2.

If $A=X_{0}\cap Y$, then

[TABLE]

[TABLE]

is open, since $\left(\varphi^{k}_{\gamma}m_{\gamma}^{n,k}\right)^{-1}(X)$ and $\left(\varphi_{\gamma}^{n}\right)^{-1}(Y)$ are open. 3. 3.

If $A=X_{0}\cup Y$, then

[TABLE]

[TABLE]

is open, since$\left(\varphi^{k}_{\gamma}m_{\gamma}^{n,k}\right)^{-1}(X)$ and $\left(\varphi_{\gamma}^{n}\right)^{-1}(Y)$ are open.

Thus, we get the contradiction with the maximality of the topology $\tau^{n}$, therefore $X_{0}\in\tau^{n}.$ ∎

Lemma 3**.**

Let $k<n,k,n\in\mathbb{N}$ and

[TABLE]

be the topology induced by the topology $\tau_{n}$ on $\mathfrak{B}_{k},$ then

[TABLE]

Proof.

The topology $\tau_{k}$ is determinded by the family of semi-norms $\{\|\cdot\|_{\gamma,k}\}_{\gamma=1}^{\infty}$, and the topology $\tau_{n}|_{k}$ is determined by the family $\{\|\cdot\|_{\gamma,n}\}_{\gamma=1}^{\infty}.$ Evidently, that $\|\cdot\|_{\gamma,n}\prec\|\cdot\|_{\gamma,k}.$ ∎

For $\underline{\mathfrak{L}}$ define the topology $\overline{\tau}$ the strongest topology, such that any embedding

[TABLE]

is continuous.

Also, determine the topology $\underline{\tau},$ such that it will be the weakest topology on $\overline{\mathfrak{L}}$ 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 $n_{0}>0$ we have the inclusion

[TABLE]

Thus

[TABLE]

и

[TABLE]

Note, that by reduction we obtain $\mathfrak{B}_{n_{0}}$ space with the topology:

[TABLE]

But then $\tau^{k}|_{n_{0}}=\bigcap_{n\geq n_{0}}\tau^{k,n}|_{n_{0}},$ thus

[TABLE]

∎

4 Limits of noncommutative $L_{\infty}$ spaces

We split this section into four parts. Firstly, we consider general properties for $L_{\infty}(a)$ spaces and its norms. Secondly, we consider case of bounded $a$, in the third case we consider unbounded $a$ such that $a^{-1}$ is bounded, and at last we consider the general case of unbounded $a$.

4.1 Case of bounded operator

Lemma 4**.**

Let $a\eta\mathcal{M}$ (a is affiliated with $\mathcal{M}$, $a\geq\mathbf{0}$), then $\mathcal{S}_{a}(\mathcal{M})=\mathcal{S}_{\lambda a}(\mathcal{M})$ for any $\lambda>0.$

Proof.

Let $\widehat{a^{\frac{1}{2}}xa^{\frac{1}{2}}}\in\mathcal{S}_{a}(\mathcal{M})$. Evidently, $D(a^{\frac{1}{2}})=D((\lambda a)^{\frac{1}{2}})=D(\lambda^{\frac{1}{2}}a^{\frac{1}{2}})$, thus

[TABLE]

[TABLE]

Hence, $\lambda\widehat{a^{\frac{1}{2}}xa^{\frac{1}{2}}}=\widehat{(\lambda a)^{\frac{1}{2}}x(\lambda a)^{\frac{1}{2}}}$, and since $\mathcal{S}_{a}(\mathcal{M})$, $\mathcal{S}_{\lambda a}(\mathcal{M})$ are linear spaces, we have $\mathcal{S}_{a}(\mathcal{M})=\mathcal{S}_{\lambda a}(\mathcal{M})$.∎

Particularly, if $a$ is bounded, then $\mathcal{S}_{a}(\mathcal{M})=\mathcal{S}_{a_{0}}(\mathcal{M}),$ where $a_{0}=a/\|a\|$, if $a$ is such that $a^{-1}$ is bounded, then $\mathcal{S}_{a}(\mathcal{M})=\mathcal{S}_{a_{\infty}}(\mathcal{M})$, where $a_{\infty}=\|a^{-1}\|a$.

Proposition 2**.**

Let $a\eta\mathcal{M}$, $a\geq\mathbf{0}$, then $\|\cdot\|_{a}$ in $L_{\infty}(a)$ is equivalent to $\|\cdot\|_{\lambda a}$, where $\lambda>0$.

Proof.

Since, $L_{\infty}(a)$ is isometrically isomorphic to $\mathcal{S}_{a}(\mathcal{M})$ for $x\in L_{\infty}(a)$ we may consider $x=\widehat{a^{\frac{1}{2}}ya^{\frac{1}{2}}},$ where $y\in\mathcal{M}$, then $\|x\|_{a}=\|y\|$. On the other hand,

[TABLE]

[TABLE]

∎

Particularly, for the bounded $a\in\mathcal{M}^{+}$ we have $\|\cdot\|_{a}$ is equivalent to $\|\cdot\|_{a_{0}}$, where $a_{0}=a/\|a\|.$ If $a^{-1}$ is bounded and we consider $a_{\infty}=\|a^{-1}\|a$, then $\|\cdot\|_{a_{\infty}}$ is equivalent to $\|\cdot\|_{a}$.

Now, consider $L_{\infty}(a^{\alpha}).$

Lemma 5**.**

For a bounded $a\in\mathcal{M}^{+}$ and $\alpha,\beta\in\mathbb{R}^{+}$, if $\alpha<\beta$, then

[TABLE]

Proof.

Let $y\in L_{\infty}(a^{\beta})$, then we consider $y=\widehat{a^{\beta/2}xa^{\beta/2}},$ where $x\in\mathcal{M}$. Thus,

[TABLE]

[TABLE]

Note, that $x^{\prime}:=a^{(\beta-\alpha)/2}xa^{(\beta-\alpha)/2}\in\mathcal{M}$, thus

[TABLE]

∎

The following definition is standard.

definition 8**.**

Let $X\supset Y$ be normed spaces with the norms $\|\leavevmode\nobreak\ \cdot\leavevmode\nobreak\ \|_{X},\|\leavevmode\nobreak\ \cdot\leavevmode\nobreak\ \|_{Y}$, respectively. We write $\|\cdot\|_{X}\prec\|\cdot\|_{Y}$ if and only if there exists $C\in\mathbb{R}^{+}$, such that for any $x\in Y$ the inequality $\|x\|_{X}\leq C\|x\|_{Y}$ holds.

Lemma 6**.**

For a bounded $a\in\mathcal{M}^{+}$ and $\alpha,\beta\in\mathbb{R}^{+}$. If $\alpha<\beta$, then

[TABLE]

Proof.

Let $y\in L_{\infty}(a^{\beta}),$ then $y=\widehat{a^{\beta/2}xa^{\beta/2}}$, where $x\in\mathcal{M}$, then

[TABLE]

[TABLE]

∎

Now, let us consider the limit space.

definition 9**.**

For the bounded $a\in\mathcal{M}^{+}$ we define the topological space $(\mathcal{L}_{\infty}(a),\tau(a))$ as the limit space for $L_{\infty}(a^{\alpha})$, where $\mathcal{L}_{\infty}(a):=\bigcap\limits_{\alpha>1}L_{\infty}(a^{\alpha})$ and $\tau(a):=\bigcup\limits_{\alpha>1}\tau_{\infty}(a^{\alpha})$, $\tau_{\infty}(a^{\alpha})=\{X\cap\mathcal{L}_{\infty}(a)\ |\ X\in\tau(a^{\alpha})\}$, $\tau(a^{\alpha})$ is the topology on $L_{\infty}(a^{\alpha})$ of the norm $\|\cdot\|_{a^{\alpha}}$.

The latter definition essentially means, that $\tau(a)$ is the initial topology on the $\mathcal{L}_{\infty}(a)$ for the family of mapping

[TABLE]

Also, we can describe this topology as the topology on $\mathcal{L}_{\infty}(a)$ defined by the family of the seminorms $\{\|\cdot\|_{a^{\alpha}}\}_{\alpha>1}$. The familiy of the seminorms $\{\|\cdot\|_{a^{n}}\}_{n\in\mathbb{N}}$ describes the same topology, thus we get the following theorem.

Theorem 2**.**

For the bounded $a\in\mathcal{M}^{+}$ the space $(\mathcal{L}_{\infty}(a),\tau(a))$ 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 $a\eta\mathcal{M},a\geq\mathbf{0}$, a is not bounded, but $a^{-1}$ is bounded.

Lemma 7**.**

For a $a\eta\mathcal{M}$, $a\geq\mathbf{0}$ and $\alpha,\beta\in\mathbb{R}^{+}$. If $\alpha<\beta$, then

[TABLE]

Proof.

Let $y\in L_{\infty}(a^{\alpha})$, then we consider $y=\widehat{a^{\alpha/2}xa^{\alpha/2}},$ where $x\in\mathcal{M}$. Thus,

[TABLE]

[TABLE]

Note, that $x^{\prime}:=(a^{-1})^{(\beta-\alpha)/2}x(a^{-1})^{(\beta-\alpha)/2}\in\mathcal{M}$, therefore we have that $y=\widehat{a^{\beta/2}x^{\prime}a^{\beta/2}}\in L_{\infty}(a^{\beta})$. ∎

Lemma 8**.**

For $a\eta\mathcal{M}$, $a\geq\mathbf{0}$ such that $a^{-1}$ is bounded and $\alpha,\beta\in\mathbb{R}^{+}$, such that $\alpha<\beta$ we have

[TABLE]

Proof.

Let $y\in L_{\infty}(a^{\alpha}),$ then $y=\widehat{a^{\alpha/2}xa^{\alpha/2}}$, where $x\in\mathcal{M}$, then

[TABLE]

[TABLE]

[TABLE]

∎

definition 10**.**

For the unbounded $a\in\mathcal{M}^{+}$ with bounded inverse $a^{-1}$ we define the topological space $(\mathcal{L}_{\infty}(a),\tau_{\infty}(a))$ as the limit space for $L_{\infty}(a^{\alpha})$, where $\mathcal{L}_{\infty}(a):=\bigcup\limits_{\alpha>1}L_{\infty}(a^{\alpha})$ and $\tau(a)$ is the strongest topology such that for all $\alpha>0$ the mappings $m_{\alpha}:x\in L_{\infty}(a^{\alpha})\mapsto x\in(\mathcal{L}_{\infty}(a),\tau_{\infty}(a))$ are continuous.

Evidently, $(\mathcal{L}_{\infty}(a),\tau_{\infty}(a))$ is an (LB)-space.

4.3 Case of unbounded operator with unbounded inverse

Now, consider case, when $a\eta\mathcal{M}$, $a\geq\mathbf{0}$, $a$ and $a^{-1}$ are unbounded, simultaneously. Then we take spectra gecomposition

[TABLE]

and determine

[TABLE]

Note, that

[TABLE]

[TABLE]

[TABLE]

Evidently, $H=p_{0}H\oplus p_{\infty}H$, thus we represent $\mathcal{M}$ as a subalgebra of the algebra of matrices $\mathbb{M}_{2}(\mathcal{M})$:

[TABLE]

meaning that if $x$ acts on $h\in H$, then

[TABLE]

[TABLE]

[TABLE]

Evidently, $a^{\frac{1}{2}}\mathcal{M}a^{\frac{1}{2}}$ in this notation is represented as

[TABLE]

where $\mathcal{S}_{a_{0}^{1/2},a_{\infty}^{1/2}}(\mathcal{M})$ and $\mathcal{S}_{a_{\infty}^{1/2},a_{0}^{1/2}}(\mathcal{M})$ are defined by the following definitions

definition 11**.**

Let $a,b\eta\mathcal{M}$, $x\in\mathcal{M}$, then the sesquilinear form

[TABLE]

is defined by the equality

[TABLE]

definition 12**.**

Let $a,b\eta\mathcal{M}$. We define the linear space of sesqulinear forms

[TABLE]

endowed with the norm $\|\widehat{axb}\|_{a,b}:=\|x\|.$

Let $a$ be injective, then $a_{0}|_{p_{0}H}$ is injective bounded operator in $\mathcal{M}_{p_{0}}$ acting on $p_{0}H$ and $a_{\infty}|_{p_{\infty}H}$ is injective operator affiliated with $\mathcal{M}_{p_{\infty}}$ (acting on $p_{\infty}H$) with bounded inverse. Moreover, $L_{\infty}(a_{0})$ is isometrically isomorphic to $L_{\infty}(a_{0}|_{p_{0}H})$ as well as $L_{\infty}(a_{\infty})$ is isometrically isomorphic to $L_{\infty}(a_{\infty}|_{p_{\infty}H})$ by construction. Also, it is evident, that there exists the isomorphism $\widehat{axb}\mapsto\overline{\widehat{bxa}}$ between the spaces $\mathcal{S}_{a,b}(\mathcal{M})$ and $\mathcal{S}_{b,a}(\mathcal{M})$. Thus, we only need to consider the limits for $\mathcal{S}_{a_{0}^{\alpha},a_{\infty}^{\beta}}(\mathcal{M})$ for $\alpha,\beta\to+\infty.$

Further we also use the notation

[TABLE]

and

[TABLE]

Let $\alpha_{1}<\alpha_{2}$ and $\beta_{1}<\beta_{2}$, $\alpha_{i},\beta_{i}\in\mathbb{R}$, then note, that the following schemes

[TABLE]

As for the norms,

[TABLE]

By $\tau_{0,\infty}^{\alpha,\beta}$ we denote the topology of the norm $\|\cdot\|_{a_{0}^{\alpha},a_{0}\infty^{\beta}}$ on the space $\mathcal{S}_{0,\infty}^{\alpha,\beta}.$ Note, that

[TABLE]

where

[TABLE]

Sine there exists the isomorphism between $\mathcal{S}_{0,\infty}^{\alpha,\beta}$ and $\mathcal{S}_{\infty,0}^{\alpha,\beta}$, we will only consider one case. Consider the limits

[TABLE]

with the topologies $\tau_{0}^{\alpha}$ and $\tau_{\infty}^{\beta}$, respectively. Topology $\tau_{0}^{\alpha}$ is the strongest topology on $\mathfrak{S}_{0}^{\alpha}$, such that all the mappings

[TABLE]

are continuous and the topology $\tau_{\infty}^{\beta}$ is the weakest topology on $\mathfrak{S}_{\infty}^{\beta}$ that all the mappings

[TABLE]

are continuous.

Remark Note, that any $(\mathcal{S}_{0,\infty}^{\alpha,\beta},\|\cdot\|_{a_{0}^{\alpha},a_{\infty}^{\beta}})$ is Banach space, thus $(\mathfrak{S}_{0}^{\alpha},\tau_{0}^{\alpha})$ is an $(LB)$-space.

Remark Note, that $(\mathfrak{S}_{\infty}^{\alpha},\tau_{\infty}^{\alpha})$ is a Frechet space, since its topology is determined by the countable set of the seminorms.

Lemma 9**.**

Let $\alpha<\beta$, $\alpha,\beta\in\mathbb{R}^{+}$, then

1. (i)

$\mathfrak{S}_{0}^{\alpha}\supset\mathfrak{S}_{0}^{\beta}$; 2. (ii)

$\mathfrak{S}_{\infty}^{\alpha}\subset\mathfrak{S}_{\infty}^{\beta}.$**

Proof.

$(i)$ Since

[TABLE]

it follows that if $x\in\mathfrak{S}_{0}^{\beta}$, then there exists $\gamma_{0}$ such that $x\in\mathcal{S}_{0,\infty}^{\beta,\gamma_{0}}\subset\mathcal{S}_{0,\infty}^{\alpha,\gamma_{0}}$, thus $x\in\mathfrak{S}_{0}^{\alpha}.$

$(ii)$ If $x\in\mathfrak{S}_{\infty}^{\alpha}$, then for all $\gamma>0$ $x\in\mathcal{S}_{0,\infty}^{\gamma,\alpha}\subset\mathcal{S}_{0,\infty}^{\gamma,\beta},$ thus

[TABLE]

∎

We consider the following constructions:

[TABLE]

Note that

[TABLE]

Proposition 3**.**

[TABLE]

Proof.

Note, that if $\alpha_{1}<\alpha_{2}$, then $\mathcal{S}_{0,\infty}^{\alpha_{1},\beta}\supset\mathcal{S}_{0,\infty}^{\alpha_{2},\beta},$ thus

[TABLE]

Note, that if $\beta_{1}<\beta_{2}$, then $\mathcal{S}_{0,\infty}^{\alpha,\beta_{1}}\subset\mathcal{S}_{0,\infty}^{\alpha,\beta_{2}},$ thus

[TABLE]

∎

definition 13**.**

We call the family $\mathcal{S}_{0,\infty}^{\alpha,\beta}$ converging if $\underline{\mathfrak{L}_{0,\infty}}=\overline{\mathfrak{L}_{0,\infty}}$ and denote it as $\mathcal{S}_{0,\infty}^{\alpha,\beta}\xrightarrow{\alpha,\beta}\mathfrak{L}_{0,\infty}.$

Corollary 1**.**

If the family $\mathcal{S}_{0,\infty}^{\alpha,\beta}$ is converging, then

[TABLE]

Lemma 10**.**

Let $\alpha<\beta,$ $\alpha,\beta\in\mathbb{R}^{+}$ and

[TABLE]

i.e. the topology induced by $\tau_{0}^{\alpha}$ on $\mathfrak{S}_{0}^{\beta}$, then

[TABLE]

Proof.

Let $X_{0}\in\tau_{0}^{\alpha}|_{\beta}$, i.e. $X_{0}=X\cap\mathfrak{S}_{0}^{\beta}$ with $X\in\tau_{0}^{\alpha}$. Then $(\varphi^{\alpha}_{\gamma})^{-1}(X)$ is open in $(\mathcal{S}_{0,\infty}^{\alpha,\gamma},\|\cdot\|_{a_{0}^{\alpha},a_{\infty}^{\gamma}})$. Note, that the embedding

[TABLE]

is also continuous, thus

[TABLE]

is open in $(\mathcal{S}_{0,\infty}^{\beta,\gamma},\|\cdot\|_{a_{0}^{\beta},a_{\infty}^{\gamma}})$ for any $\gamma>0.$

The topology $\tau_{0}^{\beta}$ is the strongest topology, such that all of the embeddings $\varphi_{\gamma}^{\beta}$ are continuous. If $X_{0}\notin\tau_{0}^{\beta}$, then there exists the toplogy

[TABLE]

such that it is stronger, then $\tau_{0}^{\beta}$, $X_{0}\in\tau$ and for any $A\in\tau$ the preimages $(\varphi^{\beta}_{\gamma})^{-1}(A)$ are open. Further we explain why it is open.

Consider three cases $A\in\tau_{0}^{\beta}$, $A=X_{0}\cap Y$, $A=X_{0}\cup Y$ ($Y\in\tau_{0}^{\beta}$).

1. Let $A\in\tau_{0}^{\beta}$, then evidently $(\varphi_{\gamma}^{\beta})^{-1}(A)$ is open.

2. Let $A=X_{0}\cap Y$, then

[TABLE]

[TABLE]

is open, since $\left(\varphi^{\alpha}_{\gamma}m_{\gamma}^{\beta,\alpha}\right)^{-1}(X)$ is open and $\left(\varphi_{\gamma}^{\beta}\right)^{-1}(Y)$ is open.

3. Let $A=X_{0}\cup Y$, then

[TABLE]

[TABLE]

is open, since $\left(\varphi^{\alpha}_{\gamma}m_{\gamma}^{\beta,\alpha}\right)^{-1}(X)$ is open and $\left(\varphi_{\gamma}^{\beta}\right)^{-1}(Y)$ is open.

Thus, we get a contradiction with the maximality of $\tau_{0}^{\beta}$, therefore $X_{0}\in\tau_{0}^{\beta}.$∎

Lemma 11**.**

Let $\alpha<\beta,\alpha,\beta\in\mathbb{R}^{+}$ and

[TABLE]

i.e. the topology induced by $\tau_{\infty}^{\beta}$ on $\mathfrak{S}_{\infty}^{\alpha},$ then

[TABLE]

Proof.

The topology $\tau_{\infty}^{\alpha}$ is determined with the set of the seminorms $\{\|\cdot\|_{a_{0}^{\gamma},a_{\infty}^{\alpha}}\}_{\gamma=1}^{\infty}$ and the topology $\tau_{\infty}^{\beta}|_{\alpha}$ is determined by the system $\{\|\cdot\|_{a_{0}^{\gamma},a_{\infty}^{\beta}}\}_{\gamma=1}^{\infty}.$ It is sufficient to note, that $\|\cdot\|_{a_{0}^{\gamma},a_{\infty}^{\beta}}\prec\|\cdot\|_{a_{0}^{\gamma},a_{\infty}^{\alpha}}.$ ∎

For $\underline{\mathfrak{L}_{0,\infty}}$ it is natural to define the topology $\overline{\tau_{0,\infty}}$ 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 $\underline{\tau_{0,\infty}}$ which would be the weakest topology on $\overline{\mathfrak{L}_{0,\infty}}$ 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 $\beta_{0}>0$ we have the inclusion

[TABLE]

Thus,

[TABLE]

and

[TABLE]

Note, that the reductions always lead to the space $\mathfrak{S}_{\infty}^{\beta_{0}}$ and may be rewrited as

[TABLE]

But then $\tau_{0}^{\alpha}|_{\beta_{0}}=\bigcap_{\beta\geq\beta_{0}}\tau_{0,\infty}^{\alpha,\beta}|_{\beta_{0}},$ therefore

[TABLE]

∎

Remark Note, that $\overline{\tau_{0,\infty}}$ is a topology of $(LF)$-space and $\underline{\tau_{0,\infty}}$ is a topology of the projective limit of $(LB)$-spaces.

Remark All the constructions of this section may be applied to the system $\mathcal{S}_{\infty,0}^{\alpha,\beta}$. We will distinguish such constructions by the notation $\overline{\mathfrak{L}_{\infty,0}}$, $\underline{\mathfrak{L}_{\infty,0}}$, $\mathfrak{L}_{\infty,0}$, $\tau_{\infty,0}^{\alpha,\beta}$ and so on.

4.4 Bring it all together

Evidently we may consider different limits of $L_{\infty}(a^{\alpha})$ spaces, using the constructions above, particularly, we may define

definition 14**.**

Let $a$ be a positive operator affiliated with von Neumann algebra $\mathcal{M}$ acting on the Hilbert space $H$, such that not $a$ nor $a^{-1}$ are necessarily bounded. Then we define the lower limit of the spaces $\underline{\mathrm{lim}}L_{\infty}(a^{\alpha})$ as a vector space of sesquilinear forms formally wriiten as

[TABLE]

definition 15**.**

Let $a$ be a positive operator affiliated with von Neumann algebra $\mathcal{M}$ acting on the Hilbert space $H$, such that not $a$ nor $a^{-1}$ are necessarily bounded. Then we define the upper limit of the spaces $\overline{\mathrm{lim}}L_{\infty}(a^{\alpha})$ as a vector space of sesquilinear forms formally wriiten as

[TABLE]

Theorem 4**.**

Let $\underline{\mathfrak{L}_{0,\infty}}=\overline{\mathfrak{L}_{0,\infty}}$, $\underline{\tau_{0,\infty}}=\overline{\tau_{0,\infty}},$ then

[TABLE]

is $(LF)$-space.

Corollary 2**.**

If $a\geq 0$ is affiliated with the center $\mathfrak{C}(\mathcal{M})$ of the von Neumann algebra $\mathcal{M}$, then the limits space $\mathfrak{L}_{\infty}(a)=\lim\limits_{\alpha}L_{\infty}(a^{\alpha})$ is an (LF)-space.

5 $L_{p}$-spaces

Let $\mathcal{M}$ be a von Neumann algebra and $a\in\mathcal{M}$ be a positive bounded injective linear operator. We consider $L_{1}(a)$ and $L_{\infty}(a),$ descirbed previously (also see [16]).

It is easy to see that for any $\alpha<\beta,\alpha,\beta\in\mathbb{R}$

[TABLE]

We also obtain, that $L_{1}^{h}(a)$ is isometrically isomorphic to $L_{1}^{h}(a/\|a\|)$.

Note, that for any $a$ such that $\|a\|\leq 1$ and any pair $\alpha<\beta$ the inequalities

[TABLE]

[TABLE]

hold.

Thus, we for any bounded $a$ the inclusion

[TABLE]

[TABLE]

hold.

Theorem 5**.**

Let $a$ be positive injective bounded operator from the von Neumann algebra $\mathcal{M}$ and $\|a\|\leq 1$.

[TABLE]

[TABLE]

with $L_{p}(a^{\alpha})$ being the interpolation space $K_{(p-1)/p,p}(L_{1}^{h}(a^{\alpha}),L_{\infty}^{h}(a^{\alpha}))$.

Proof.

The inclusions are proved analogously. Consider as the example the embedding $K_{(p-1)/p,p}(L_{1}^{h}(a^{\alpha},a^{\beta}))$ в $L_{p}(a^{\alpha}).$ First of all note, that

[TABLE]

After that note, that

[TABLE]

[TABLE]

thus if $x\in K_{(p-1)/p,p}(L_{1}^{h}(a^{\alpha},a^{\beta}))$, then $x\in L_{p}(a^{\alpha}).$ 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 $p\in(0,1)$.

Let $\mathfrak{L}_{p}(a^{\alpha})$ be the inductive limit (by $\beta\to\infty$) of $K_{(p-1)/p,p}(L_{1}^{h}(a^{\beta}),L_{\infty}^{h}(a^{\alpha}))$ and $\mathfrak{R}_{p}(a^{\alpha})$ be the projective limit (by $\beta\to\infty$) of $K_{(p-1)/p,p}(L_{1}^{h}(a^{\alpha}),L_{\infty}^{h}(a^{\beta}))$. The projective limit of $\mathfrak{L}_{p}(a^{\alpha})$ will be the upper limit $\overline{\lim}L_{p}(a^{\alpha})$ and the inductive limit of $\mathfrak{R}_{p}(a^{\alpha})$ will be the lower limit $\underline{{\lim}}L_{p}(a^{\alpha})$. If they coincide tas the topological spaces, then $L_{p}(a^{\alpha})$ has the mixed limit for $\alpha$ converging to infinity.

Acknowledgments

This work was supported by Russian Foundation for Basic Research Grant 18-31-00218.