EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6527 ISSN 1307-5543 – ejpam.com Published by New York Business Global Equivalence and Stability of Compactness in Operator Spaces over the Non-Commutative Torus Mortada S. Ali1,∗, Abd Elmotaleb A. M. A.2, Ibtisam M. O. Mohammed1 1 Department of Mathematics, College of Science, Al-Baha University, P.O. Box 1988, KSA. 2 Department of Mathematics, College of Science and Humanity, Prince Sattam bin Abdulaziz University, Sulail, Al-Kharj 11942, KSA. Abstract. We investigate compactness in operator spaces over the non-commutative torus Aθ, applying the structure of non-commutative C∗-algebras as well as compact operators acting on Hilbert Aθ-modules, and provide a characterization of compactness in the framework of operator spaces. Key results include the equivalence between classical and complete compactness, and the stability of compactness under tensor products. Applications and examples of compact operators in operator spaces over the non-commutative torus Aθ are presented. We also discuss limitations and propose future research directions to extend these results to more general settings. 2020 Mathematics Subject Classifications: 46L08, 47L25, 46L87 Key Words and Phrases: Compactness, Operator Spaces, Non-Commutative Torus, Hilbert Modules, Quantum Metric Spaces, Tensor Products 1. Introduction Compactness plays a foundational role in the theory of operators on Banach and Hilbert spaces, as highlighted in [1, 2]. Such an operator is called compact if it sends bounded subsets to relatively compact subsets, and it has significant properties in operator theory, as well as connections to compactness in topological spaces, spectral theory, and even functional calculus. While the classical theory of compact operators focuses on spaces of functions and matrices, the study of non-commutative operator algebras presents an intriguing new context for this theory. The non-commutative torus represents one of the simplest and most studied examples of a non-commutative C∗-algebra[3, 4]. It plays a central role in various branches of mathematics and theoretical physics, ranging from quantum mechanics and topological ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6527 Email addresses: mortada@bu.edu.sa (M. S. Ali), aa.alameen@psau.edu.sa (A. E. A. M. A. Elamin), a.mahjoub@bu.edu.sa (I. M. O. Mohammed) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6527 2 of 9 dynamics to non-commutative geometric spaces. The non-commutative torus Aθ is defined as the algebra generated by two elements, U and V , with the relation UV = e2πiθV U, where θ is a positive angle argument. For instance, this algebra possesses a rich represen- tation theory and serves as a non-commutative generalization of the classical two-torus. In other words, the non-commutative torus has a rich history, related to studies in spectral theory and representation theory of C∗-algebras[5]. In this paper, we study the compactness properties of operators on operator spaces related to Aθ. Operator spaces are subspaces of operator algebras equipped with a matrix norm structure that satisfies the Ruan axioms[6–8]. Operator spaces provide a rigor- ous framework for analyzing operator behavior, particularly in the context of completely bounded maps. These spaces have demonstrated their fundamental role as a powerful tool across various disciplines, including quantum computing, harmonic analysis, and operator algebras[9, 10]. Compactness for operator spaces is of special importance in the non-commutative realm, as it combines classical notions of compactness with the well-developed theory of completely bounded maps. This provides a synthesis of functional analysis, operator alge- bras, and the structure of the non-commutative torus for compact operators on operator spaces over Aθ[11–13] . The notion of compactness in operator spaces, although distinct from its classical analogue in Banach spaces, serves as a conceptual bridge between these two domains. Earlier studies and research works have introduced notions of compactness in Hilbert C∗-modules, the behavior of compact operators, their uses in geometry, as well as in quan- tum and non-commutative contexts[14–17]. However, previous studies have not systemat- ically explored the relationship between classical compactness and complete compactness within the context of operator spaces over the non-commutative torus. This study bridges these two notions and demonstrates the stability of compactness under multi-fold tensor products, which is an important feature describing the structure of non-commutative operator spaces [18–20]. In addition, we also describe theoretical applications that illustrate the effect of compact operators in quantum metric spaces, thus broadening the applied scope of our results in comparison to previous work. Hence, this study makes a significant contribution to the deeper understanding of non-commutative operator space theory and the rising opportunities for applications in modern mathematics and physics. Our motivation arises from fundamental questions in non-commutative geometry and quantum theory [16, 21], which are further explored in the applications section. We further establish connections to quantum theory, in which compactness plays a key role in determining the behavior of quantum channels [22, 23]. Our results build on what other researchers have done before and give us new ways to think about compact operators in non-commutative spaces. M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6527 3 of 9 2. Comparison with Related Work The concept of compactness in operator algebras and in Hilbert C∗-modules has been investigated in both classical and non-commutative settings. Earlier work (e.g., Lance[15]) developed a theory for compact operators on Hilbert C∗-modules. Ruan[8] and Pisier[10] generalized these concepts to operator spaces with completely boundedness and matrix norm structures. Moreover, recent developments have introduced refined and innovative tools that build upon these classical foundations. Junge and Sherman [7] provide an overview of operator spaces in noncommutative ℓp spaces, which builds on this more analytic foundation. Hiai and Ueda[12] have explored novel aspects of non-commutative operator theory, particularly concerning bounded and compact maps, bringing new insights into the structure and behavior of such mappings. In particular, a modern approach to compact quantum metric spaces was proposed by Latrémolière[23], unifying compactness and geometry via quantum Gromov–Hausdorff convergence. We contribute to this framework by relating operator-theoretic compactness specifically through adjointable maps and complete compactness, while also establishing the stability of these properties under the tensor product operation. Similarly, Caspers and Skalski[22] studied the behavior of compactness in quantum information channels, and how this concept can be leveraged to maximize the transfer of information. Our paper builds on this foundation by providing a functional-analytic perspective on compact operators in operator spaces over the non-commutative torus explicitly highlighting how compact operators act as information-preserving maps. Besnard and Latrémolière[3] focused on convergence issues in quantum metric geome- try, whereas our results concentrate on functional structures and equivalence theorems for classical and non-classical compactness in operator spaces. Therefore, this paper serves both as a complement and an extension to the modern literature, offering a unified treat- ment of compactness for non-commutative tori with broad theoretical and applied impli- cations. 3. Preliminaries In this section, we review key concepts related to compact operators, Hilbert Aθ- Modules, Compactness, and Complete Compactness. Definition 1. Compact Operators [19] In Hilbert C∗-modules, an operator is said to be compact if it lies in the norm closure of operators of the form ξ 7→ η⟨ζ, ξ⟩, where η, ζ are fixed elements of the module, and ⟨·, ·⟩ denotes the C∗-valued inner product. Definition 2. Hilbert Aθ-Modules [3, 13] M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6527 4 of 9 A Hilbert Aθ-module is a right Aθ-module equipped with an Aθ-valued inner product ⟨·, ·⟩ : E × E → Aθ, which is positive-definite and complete with respect to the norm induced by this inner product. Definition 3. Compactness [15] An operator T : E → E on a Hilbert Aθ-module E is said to be compact if it can be represented as the norm limit of finite-rank operators: Tk(ξ) = nk∑ i=1 η (k) i · ⟨ζ(k)i , ξ⟩, where η (k) i , ζ (k) i ∈ E, with order k ∈ N and lim k→∞ ∥T − Tk∥ = 0. Definition 4. Complete Compactness [18, 20, 22] Let T : E → E be an adjointable operator on a Hilbert Aθ-module E. The operator T is said to be completely compact if its matrix amplifications Tn := In ⊗ T : Mn(E) → Mn(E) are compact for all n ∈ N. Definition 5. Stability under Tensor Products[19, 20] Let T, S be compact adjointable operators on Hilbert Aθ-modules E,F respectively. Then the tensor product operator T ⊗ S : E ⊗ F → E ⊗ F is also compact. Definition 6. completely bounded [10] Let T : E → F be a compact linear-transformation between operator spaces E and F . The operator T is said to be completely bounded if the sequence of amplifications Tn : Mn(E) → Mn(F ) is uniformly bounded, that is, ∥T∥cb := sup n∈N ∥Tn∥ < ∞, where Tn denotes the matrix amplification of T acting on Mn(E). M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6527 5 of 9 4. Main Results and Applications 4.1. Main Results In this subsection, we will prove our main theorems on compactness in operator spaces over Aθ. Theorem 1. An operator T is compact if and only if it is completely compact in the operator space sense, provided that T is an adjointable operator. Proof. Let T ∈ L(E) be adjointable. If T is compact, then by definition, there exists a sequence of finite-rank operators {Tk} such that lim k→∞ ∥T − Tk∥ = 0, where each Tk is of the form Tk(ξ) = nk∑ i=1 η (k) i · ⟨ζ(k)i , ξ⟩, with η (k) i , ζ (k) i ∈ E. For each n ∈ N, consider the matrix amplification Tn := In ⊗ T : Mn(E) → Mn(E). Since matrix norms in operator spaces are defined via Ruan’s axioms, the compactness of T implies that the sequence Tn,k := In ⊗ Tk converges uniformly to Tn. Therefore, Tn is compact for all n, and thus T is completely compact. Conversely, if T is completely compact, then for every n, Tn is compact. In particular, T1 = T is compact. Hence, the two notions coincide under adjointability. Corollary 1. Let T : E → E be an adjointable operator on a Hilbert Aθ-module E. If T ∗ is compact, then T is compact. Proof. In a Hilbert Aθ-module, the polar decomposition T = U |T | exists, where U is a partial isometry and adjointable.Theorem 1 establishes that compactness and complete compactness coincide for adjointable operators. Since |T | = (T ∗T )1/2 is obtained via continuous functional calculus applied to the compact operator T ∗, it follows that |T | is compact. Then T = U |T | is the product of an adjointable operator and a compact operator, and hence is compact. Proposition 1. A compact operator T is completely bounded if ∥T∥cb = ∥T∥. The con- verse does not necessarily hold. Proof. Suppose that T is compact and that ∥T∥cb = ∥T∥. Since T is a bounded linear operator between operator spaces, and ∥T∥cb < ∞, it follows directly from the definition that T is completely bounded. Therefore, the condition ∥T∥cb = ∥T∥ implies ∥T∥cb < ∞, which confirms that T is completely bounded. However, the converse does not necessarily hold. That is, there exist compact operators T that are completely bounded but satisfy ∥T∥cb > ∥T∥. M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6527 6 of 9 Theorem 2. Let T : E → E and S : F → F be compact adjointable operators on Hilbert Aθ-modules E and F , respectively. Then the tensor product operator T ⊗ S : E ⊗Aθ F → E ⊗Aθ F is also compact. Moreover, the compactness property is preserved under finite internal direct sums of compact adjointable operators. That is, if T1, T2 are compact on E1, E2, then T1 ⊕ T2 is compact on E1 ⊕ E2. Proof. Since T and S are compact, there exist sequences of finite-rank operators {T (k)} and {S(ℓ)} such that lim k→∞ ∥T − T (k)∥ = 0, lim ℓ→∞ ∥S − S(ℓ)∥ = 0. Then the tensor product T ⊗ S = lim k,ℓ→∞ T (k) ⊗ S(ℓ) is a norm-limit of finite-rank operators (since the tensor of two finite-rank operators is again finite-rank), and hence is compact. For the direct sum part: let T1 and T2 be compact on E1 and E2, respectively. Then their direct sum T1 ⊕ T2 : E1 ⊕ E2 → E1 ⊕ E2 is defined by (T1⊕T2)(x1, x2) = (T1x1, T2x2), and since both components are compact, so is the operator. This follows from the fact that the operator norm and compactness are stable under finite direct sums. Corollary 2. Let T1, . . . , Tm be compact adjointable operators acting on Hilbert Aθ- modules E1, . . . , Em, respectively. Then the (internal) m-fold tensor product T1 ⊗̂ · · · ⊗̂Tm is compact on the tensor-product module E1 ⊗̂ · · · ⊗̂Em. Proof. Theorem 2 gives the result for m = 2. We proceed by induction. Suppose the result holds for m = k. Then for m = k + 1, consider the operator (T1 ⊗̂ · · · ⊗̂Tk) ⊗̂Tk+1. By the induction hypothesis, the k-fold tensor product is compact, and since Tk+1 is compact,Theorem 2 implies that their tensor product is compact. Thus, by induction, the m-fold tensor product is compact. M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6527 7 of 9 4.2. Applications and Examples In this section, we illustrate applications and examples of compact operators in oper- ator spaces over the non-commutative torus Aθ. Example 1 (Finite-Rank Operator). Let T : An θ → An θ be defined by T (ξ) = η⟨ζ, ξ⟩, for fixed η, ζ ∈ An θ . Then T is a finite-rank operator, since its range is contained in the span of η. Hence, T is compact. Example 2 (Toeplitz-Type Operators). Suppose {en}∞n=1 is an orthonormal basis of a Hilbert Aθ-module, and define an operator T by T (en) = λnen, where λn → 0 as n → ∞. Then T is the norm limit of finite-rank diagonal operators and is therefore compact. This structure is similar to classical **Toeplitz** and **Hankel** operators in Hilbert spaces, where sequences {λn} represent symbol decay. A detailed comparison with Hankel operators (see [6, 9]) may yield further insights into non-self-adjoint analogues in Aθ- modules. Example 3 (Numerical Example – Truncated Matrix Representation). Let Aθ be ap- proximated by finite matrices (via rational θ ≈ p/q). Consider Aθ ∼= Mq(C) for q = 3. Let T = 1 0 0 0 1 2 0 0 0 1 3  , which has eigenvalues tending to zero. Then T is compact as a limit of finite-rank diagonal matrices. This illustrates compactness numerically in quantum tori approximated by finite- dimensional matrix algebras. Example 4 (Quantum Information Channels). Let Φ : Aθ → Aθ be a quantum channel defined by Φ(x) = k∑ i=1 VixV ∗ i , where ∑ i V ∗ i Vi = I and Vi ∈ Aθ. If Φ = Ψ+noise with Ψ completely compact, then Φ has effectively finite-dimensional range, preserving key properties in quantum information. Such structure is useful for error correction, compression, and modeling decoherence. Compactness ensures containment of quantum evolution within finite metric spaces, as noted in [22]. M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6527 8 of 9 Application 1 (Quantum Metric Spaces). In Rieffel’s framework of compact quantum metric spaces, define the Lipschitz semi- norm: L(x) = sup {∥[D,x]∥ : D = D∗, D unbounded} , and consider the unit ball {x ∈ Aθ : L(x) ≤ 1, ∥x∥ ≤ 1}. Compact operators help define the metric on the state space via approximations of the identity operator. This concept is critical for analyzing convergence in non-commutative Gromov–Hausdorff spaces (see [23]). Application 2 (Non-Commutative Harmonic Analysis). Let T (f) = u ∗ f ∗ v∗, where u, v are unitaries in Aθ. If u has rapidly decaying Fourier coefficients (i.e., in the smooth subalgebra of Aθ), then T acts as a **low-pass filter**, and is compact. Such compact operators localize frequency energy in non-commutative spaces, enabling signal representation with geometric and spectral coherence. This parallels classical har- monic filters and contributes to a developing theory of non-commutative signal analysis. 5. Conclusion We have characterized compact operators in operator spaces over the non-commutative torus, including studying their action on tensor products and their relation to complete boundedness. This structure has many important applications in quantum geometry and harmonic analysis, and these results are anticipated to inspire further developments in the realm of quantum groups. A key direction for future work involves the generalization of these results to more general operator spaces, in particular those of non-adjointable or unbounded operators. Furthermore, the theoretical framework could be further strength- ened by incorporating numerical simulations or case studies from quantum computation and non-commutative signal analysis. Acknowledgements We would like to extend our sincere appreciation to our colleagues in the Department of Mathematics for their invaluable guidance, constructive feedback, and continuous support throughout the development of this work. 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