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A HELLY-TYPE THEOREM FOR DUAL p-OPERATOR SPACES

Part of:Selfadjoint operator algebras

Published online by Cambridge University Press:  08 June 2026

YAFEI ZHAOOpen the ORCID record for YAFEI ZHAO[Opens in a new window] and
ZHE DONGOpen the ORCID record for ZHE DONG[Opens in a new window]
YAFEI ZHAO*
Affiliation:
Department of Mathematics, Zhejiang International Studies University , Hangzhou 310023, PR China
ZHE DONG
Affiliation:
School of Mathematical Sciences, Zhejiang University , Hangzhou 310058, PR China e-mail: dongzhe@zju.edu.cn
*
e-mail: yfzhao@zisu.edu.cn
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Abstract

We prove a Helly-type theorem for dual p-operator spaces. As a consequence, any dual p-operator space is weakly p-locally reflexive.

Keywords

Helly-type theoremdual p-operator spaceweakly p-local reflexivity

MSC classification

Primary:46L07: Operator spaces and completely bounded maps

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 4
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Copyright
© The Author(s), 2026. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

1 Introduction

The local theory of operator spaces has been studied extensively for several decades, and is closely tied to local reflexivity, exactness and nuclearity (see [Reference Effros, Junge and Ruan4–Reference Effros and Ruan6, Reference Pisier9]). Among the basic notions in this theory, local reflexivity plays a particularly important role. It has been used to study the structure of operator spaces and $C^*$ upper C Superscript asterisk -algebras, and is also closely connected with the behaviour of dual operator spaces. In particular, Effros et al. [Reference Effros, Junge and Ruan4] proved that the dual of every $C^*$ upper C Superscript asterisk -algebra is locally reflexive.

Classical local reflexivity for operator spaces admits a finite-dimensional reformulation based on Helly’s lemma (see [Reference Effros and Ruan6]). Recently, Dong et al. [Reference Dong, Jiang and Zhao3] introduced the notion of weakly local reflexivity and proved that every dual operator space is weakly locally reflexive. This result shows that, even when full local reflexivity is not available, one may still recover a weaker form of dual approximation. For $1<p<\infty $ 1 less than p less than infinity , the corresponding theory of p-operator spaces has also been developed in considerable detail. Le Merdy [Reference Le Merdy7] studied factorisation of p-completely bounded multilinear maps. Daws [Reference Daws2] established the basic framework of p-operator spaces and their tensor products. The p-approximation property was investigated by An et al. [Reference An, Lee and Ruan1]. Lee [Reference Lee8] studied the local structure of p-operator spaces through conditions $C_p$ upper C Subscript p , $C^{\prime }_p$ upper C prime Subscript p and $C^{\prime \prime }_p$ upper C double prime Subscript p . More recently, Zhao and Dong [Reference Zhao and Dong10] introduced p-mapping spaces, and applied them to the study of p-local reflexivity.

If V is a p-operator space, then its dual space $V^*=\mathit {CB}_p(V,\mathbb C)$ upper V Superscript asterisk Baseline equals italic upper C upper B Subscript p Baseline left parenthesis upper V comma double struck upper C right parenthesis is naturally a p-operator space and agrees isometrically with the Banach dual (see [Reference Daws2, Reference Lee8]). Motivated by [Reference Dong, Jiang and Zhao3], we prove a Helly-type theorem for dual p-operator spaces. As a consequence, the dual of every p-operator space is weakly p-locally reflexive.

2 A Helly-type theorem

We begin by recalling the operator-space definition of local reflexivity and the Banach-space form of Helly’s lemma (see [Reference Effros and Ruan6]).

Definition 2.1. An operator space V is said to be locally reflexive if for every finite-dimensional operator space L and every complete contraction $\varphi :L\to V^{**}$ phi colon upper L right arrow upper V Superscript asterisk asterisk , there exists a net of complete contractions $\varphi _\alpha :L\to V$ phi Subscript alpha Baseline colon upper L right arrow upper V such that

$$ \begin{align*} \langle \varphi_\alpha(x),f\rangle\longrightarrow \langle \varphi(x),f\rangle \quad \text{for all } x\in L,\ f\in V^*. \end{align*} $$ StartLayout 1st Row left angle bracket phi Subscript alpha Baseline left parenthesis x right parenthesis comma f right angle bracket long right arrow left angle bracket phi left parenthesis x right parenthesis comma f right angle bracket for all x element of upper L comma f element of upper V Superscript asterisk Baseline period EndLayout

Lemma 2.2 (Helly’s lemma).

Let E be a Banach space, let $L\subseteq E^*$ upper L subset of or equal to upper E Superscript asterisk be finite-dimensional and let $\widetilde {x}_0\in E^{**}$ x overtilde Subscript 0 Baseline element of upper E Superscript asterisk asterisk satisfy $\|\widetilde {x}_0\|\le 1$ StartMetric x overtilde Subscript 0 Baseline EndMetric less than or equals 1 . Then, for every $\varepsilon>0$ epsilon greater than 0 , there exists $x_0\in E$ x 0 element of upper E such that

$$ \begin{align*} \|x_0\|<1+\varepsilon \end{align*} $$ StartMetric x 0 EndMetric less than 1 plus epsilon

and

$$ \begin{align*} \langle f,x_0\rangle=\langle \widetilde{x}_0,f\rangle \quad \text{for all } f\in L. \end{align*} $$ left angle bracket f comma x 0 right angle bracket equals left angle bracket x overtilde Subscript 0 Baseline comma f right angle bracket for all f element of upper L period

By Helly’s lemma, [Reference Effros and Ruan6, Lemma 14.3.3] gives an equivalent characterisation of local reflexivity for operator spaces. Motivated by this result, we prove the following result in the setting of p-operator spaces.

Theorem 2.3. Let V be a p-operator space. Then, for any p-operator space E and every p-complete contraction $\varphi :E\to V^{***}$ phi colon upper E right arrow upper V Superscript asterisk asterisk asterisk , there exists a p-complete contraction

$$ \begin{align*}\psi:E\to V^*\end{align*} $$ psi colon upper E right arrow upper V Superscript asterisk

such that

$$ \begin{align*}\langle \psi(x),v\rangle =\langle \varphi(x),\kappa_V(v)\rangle \quad \text{for all } x\in E,\ v\in V.\end{align*} $$ left angle bracket psi left parenthesis x right parenthesis comma v right angle bracket equals left angle bracket phi left parenthesis x right parenthesis comma kappa Subscript upper V Baseline left parenthesis v right parenthesis right angle bracket for all x element of upper E comma v element of upper V period

Proof. Let E be a p-operator space and let $\varphi :E\to V^{***}$ phi colon upper E right arrow upper V Superscript asterisk asterisk asterisk be a p-complete contraction. By the standard duality for the p-projective tensor product (see [Reference Daws2, Proposition 4.9]), we have the p-complete isometry

$$ \begin{align*} (E\stackrel{\wedge_{p}}{\otimes} V^{**})^* \cong \mathit{CB}_p(E,V^{***}). \end{align*} $$ left parenthesis upper E circled times Overscript and Subscript p Baseline Endscripts upper V Superscript asterisk asterisk Baseline right parenthesis Superscript asterisk Baseline approximately equals italic upper C upper B Subscript p Baseline left parenthesis upper E comma upper V Superscript asterisk asterisk asterisk Baseline right parenthesis period

Hence, $\varphi $ phi determines a functional

$$ \begin{align*} F_\varphi\in (E\stackrel{\wedge_{p}}{\otimes} V^{**})^* \end{align*} $$ upper F Subscript phi Baseline element of left parenthesis upper E circled times Overscript and Subscript p Baseline Endscripts upper V Superscript asterisk asterisk Baseline right parenthesis Superscript asterisk

with

$$ \begin{align*} \|F_\varphi\|=\|\varphi\|_{pcb}\le 1 \end{align*} $$ StartMetric upper F Subscript phi Baseline EndMetric equals StartMetric phi EndMetric Subscript p c b Baseline less than or equals 1

such that

$$ \begin{align*} F_\varphi(x\otimes \eta)=\langle \varphi(x),\eta\rangle \quad \text{for all } x\in E, \eta\in V^{**}. \end{align*} $$ StartLayout 1st Row upper F Subscript phi Baseline left parenthesis x circled times eta right parenthesis equals left angle bracket phi left parenthesis x right parenthesis comma eta right angle bracket for all x element of upper E comma eta element of upper V Superscript asterisk asterisk Baseline period EndLayout

Since V is a p-operator space, the canonical embedding $\kappa _V:V\to V^{**}$ kappa Subscript upper V Baseline colon upper V right arrow upper V Superscript asterisk asterisk is a p-complete contraction by [Reference Daws2, Proposition 4.4]. Therefore, the tensor map

$$ \begin{align*} \mathrm{id}_E\otimes \kappa_V: E\stackrel{\wedge_{p}}{\otimes} V\to E\stackrel{\wedge_{p}}{\otimes} V^{**} \end{align*} $$ id Subscript upper E Baseline circled times kappa Subscript upper V Baseline colon upper E circled times Overscript and Subscript p Baseline Endscripts upper V right arrow upper E circled times Overscript and Subscript p Baseline Endscripts upper V Superscript asterisk asterisk

is p-completely contractive (see the discussion following [Reference Daws2, Proposition 4.9]). In particular, it is contractive as a Banach space map. Now, define

$$ \begin{align*} G:=F_\varphi\circ(\mathrm{id}_E\otimes \kappa_V) \in (E\stackrel{\wedge_{p}}{\otimes} V)^*. \end{align*} $$ upper G colon equals upper F Subscript phi Baseline ring left parenthesis id Subscript upper E Baseline circled times kappa Subscript upper V Baseline right parenthesis element of left parenthesis upper E circled times Overscript and Subscript p Baseline Endscripts upper V right parenthesis Superscript asterisk Baseline period

Then,

$$ \begin{align*} \|G\| \le \|F_\varphi\|\,\|\mathrm{id}_E\otimes \kappa_V\| \le 1. \end{align*} $$ StartMetric upper G EndMetric less than or equals StartMetric upper F Subscript phi Baseline EndMetric StartMetric id Subscript upper E Baseline circled times kappa Subscript upper V Baseline EndMetric less than or equals 1 period

Again by the duality for the p-projective tensor product, we have the p-complete isometry

$$ \begin{align*} (E\stackrel{\wedge_{p}}{\otimes} V)^* \cong \mathit{CB}_p(E,V^*). \end{align*} $$ left parenthesis upper E circled times Overscript and Subscript p Baseline Endscripts upper V right parenthesis Superscript asterisk Baseline approximately equals italic upper C upper B Subscript p Baseline left parenthesis upper E comma upper V Superscript asterisk Baseline right parenthesis period

Hence, G corresponds to a unique map $\psi \in \mathit {CB}_p(E,V^*)$ psi element of italic upper C upper B Subscript p Baseline left parenthesis upper E comma upper V Superscript asterisk Baseline right parenthesis such that

$$ \begin{align*} \|\psi\|_{pcb}=\|G\|\le 1 \end{align*} $$ StartMetric psi EndMetric Subscript p c b Baseline equals StartMetric upper G EndMetric less than or equals 1

and

$$ \begin{align*} G(x\otimes v)=\langle \psi(x),v\rangle \quad \text{for all } x\in E,\ v\in V. \end{align*} $$ upper G left parenthesis x circled times v right parenthesis equals left angle bracket psi left parenthesis x right parenthesis comma v right angle bracket for all x element of upper E comma v element of upper V period

Thus, $\psi :E\to V^*$ psi colon upper E right arrow upper V Superscript asterisk is a p-complete contraction. Finally, for every $x\in E$ x element of upper E and $v\in V$ v element of upper V ,

$$ \begin{align*} \langle \psi(x),v\rangle = G(x\otimes v) = F_\varphi(x\otimes \kappa_V(v)) = \langle \varphi(x),\kappa_V(v)\rangle. \end{align*} $$ StartLayout 1st Row left angle bracket psi left parenthesis x right parenthesis comma v right angle bracket equals upper G left parenthesis x circled times v right parenthesis equals upper F Subscript phi Baseline left parenthesis x circled times kappa Subscript upper V Baseline left parenthesis v right parenthesis right parenthesis equals left angle bracket phi left parenthesis x right parenthesis comma kappa Subscript upper V Baseline left parenthesis v right parenthesis right angle bracket period EndLayout

This completes the proof.

Remark 2.4. Theorem 2.3 can be viewed as a Helly-type theorem for dual p-operator spaces. Our main result differs from the classical Banach-space form of Helly’s lemma in several ways. It preserves p-complete contractivity without any $1+\varepsilon $ 1 plus epsilon loss and it applies to arbitrary p-operator spaces rather than only to finite-dimensional ones. However, it only preserves the pairing on the canonical copy $\kappa _V(V)\subseteq V^{**}$ kappa Subscript upper V Baseline left parenthesis upper V right parenthesis subset of or equal to upper V Superscript asterisk asterisk and so does not recover the full flexibility of Helly’s lemma.

We now introduce the definition of weakly p-local reflexivity, which generalises the corresponding notion for dual operator spaces in [Reference Dong, Jiang and Zhao3].

Definition 2.5. Let V be a p-operator space. We say that the dual p-operator space $V^{*}$ upper V Superscript asterisk is weakly p-locally reflexive if for every finite-dimensional p-operator space L and every p-complete contraction $\varphi :L\to V^{***}$ phi colon upper L right arrow upper V Superscript asterisk asterisk asterisk , there exists a net of p-complete contractions $\varphi _{\alpha }:L\to V^{*}$ phi Subscript alpha Baseline colon upper L right arrow upper V Superscript asterisk such that

$$ \begin{align*}\langle \varphi_{\alpha}(x),v\rangle \longrightarrow \langle \varphi(x),\kappa_V(v)\rangle \quad \text{for all } x\in L,\ v\in V,\end{align*} $$ StartLayout 1st Row left angle bracket phi Subscript alpha Baseline left parenthesis x right parenthesis comma v right angle bracket long right arrow left angle bracket phi left parenthesis x right parenthesis comma kappa Subscript upper V Baseline left parenthesis v right parenthesis right angle bracket for all x element of upper L comma v element of upper V comma EndLayout

where $\kappa _V:V\to V^{**}$ kappa Subscript upper V Baseline colon upper V right arrow upper V Superscript asterisk asterisk denotes the canonical embedding.

As a corollary of our Helly-type theorem, any dual p-operator space is weakly p-locally reflexive.

Corollary 2.6. Let V be a p-operator space. Then, $V^*$ upper V Superscript asterisk is weakly p-locally reflexive.

Proof. Let L be a finite-dimensional p-operator space and let $\varphi :L\to V^{***}$ phi colon upper L right arrow upper V Superscript asterisk asterisk asterisk be a p-complete contraction. By Theorem 2.3, there exists a p-complete contraction

$$ \begin{align*} \psi:L\to V^* \end{align*} $$ psi colon upper L right arrow upper V Superscript asterisk

such that

$$ \begin{align*} \langle \psi(x),v\rangle = \langle \varphi(x),\kappa_V(v)\rangle \quad \text{for all } x\in L,\ v\in V. \end{align*} $$ left angle bracket psi left parenthesis x right parenthesis comma v right angle bracket equals left angle bracket phi left parenthesis x right parenthesis comma kappa Subscript upper V Baseline left parenthesis v right parenthesis right angle bracket for all x element of upper L comma v element of upper V period

Now, let ${\varphi _\alpha }$ phi Subscript alpha be the constant net with value $\psi $ psi , that is, $\varphi _\alpha =\psi $ phi Subscript alpha Baseline equals psi for all $\alpha $ alpha . Then, for every $x\in L$ x element of upper L and $v\in V$ v element of upper V ,

$$ \begin{align*} \langle \varphi_\alpha(x),v\rangle = \langle \psi(x),v\rangle = \langle \varphi(x),\kappa_V(v)\rangle. \end{align*} $$ StartLayout 1st Row left angle bracket phi Subscript alpha Baseline left parenthesis x right parenthesis comma v right angle bracket equals left angle bracket psi left parenthesis x right parenthesis comma v right angle bracket equals left angle bracket phi left parenthesis x right parenthesis comma kappa Subscript upper V Baseline left parenthesis v right parenthesis right angle bracket period EndLayout

Hence, ${\varphi _\alpha }$ phi Subscript alpha converges to $\varphi $ phi in the required sense and $V^*$ upper V Superscript asterisk is weakly p-locally reflexive.

Footnotes

The project was partially supported by Scientific Research Fund of Zhejiang Provincial Education Department under Grant No. Y202456228.

References

An, G., Lee, J.-J. and Ruan, Z.-J., ‘On $p$ p -approximation properties for $p$ p -operator spaces’, J. Funct. Anal. 259 (2010), 933–974.10.1016/j.jfa.2010.04.007CrossRefGoogle Scholar
Daws, M., ‘ $p$ p -operator spaces and Figà–Talamanca–Herz algebras’, J. Operator Theory 63 (2010), 47–83.Google Scholar
Dong, Z., Jiang, J.-Z. and Zhao, Y.-F., ‘Any dual operator space is weakly locally reflexive’, Bull. Aust. Math. Soc. 110(2024), 158–160.10.1017/S0004972723001120CrossRefGoogle Scholar
Effros, E. G., Junge, M. and Ruan, Z.-J., ‘Integral mappings and the principle of local reflexivity for non-commutative ${L}^1$ upper L Superscript 1 -spaces’, Ann. of Math. 151 (2000), 59–92.10.2307/121112CrossRefGoogle Scholar
Effros, E. G., Ozawa, N. and Ruan, Z.-J., ‘On injectivity and nuclearity for operator spaces’, Duke Math. J. 110 (2001), 489–521.10.1215/S0012-7094-01-11032-6CrossRefGoogle Scholar
Effros, E. G. and Ruan, Z.-J., Operator Spaces, London Mathematical Society Monographs (New Series), 23 (Clarendon Press, Oxford, 2000).Google Scholar
Le Merdy, C., ‘Factorization of $p$ p -completely bounded multilinear maps’, Pacific J. Math. 172 (1996), 187–213.10.2140/pjm.1996.172.187CrossRefGoogle Scholar
Lee, J.-J., ‘Conditions ${C}_p$ upper C Subscript p , ${C}_p^{\prime }$ upper C prime Subscript p , and ${C}_p^{{\prime\prime} }$ upper C double prime Subscript p for $p$ p -operator spaces’, Oper. Matrices 8 (2014), 1079–1093.10.7153/oam-08-60CrossRefGoogle Scholar
Pisier, G., Introduction to Operator Space Theory, London Mathematical Society Lecture Note Series, 294 (Cambridge University Press, Cambridge, 2003).10.1017/CBO9781107360235CrossRefGoogle Scholar
Zhao, Y.-F., and Dong, Z., ‘ $p$ p -mapping spaces for $p$ p -operator spaces on ${L}_p$ upper L Subscript p spaces’, Illinois J. Math. 58 (2014), 1071–1091.10.1215/ijm/1446819302CrossRefGoogle Scholar