On the Norm of Jordan Elementary Operators in Tensor Product of 𝐶∗-Algebras

Winnie Kerubo Kegwaro; Denis Njue King’ang’i; Collins Kimutai Meli

doi:https://doi.org/10.14445/22315373/IJMTT-V72I6P106

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 72 | Issue 6 | Year 2026 | Article Id. IJMTT-V72I6P106 | DOI : https://doi.org/10.14445/22315373/IJMTT-V72I6P106

On the Norm of Jordan Elementary Operators in Tensor Product of 𝐶∗-Algebras

Winnie Kerubo Kegwaro, Denis Njue King’ang’i, Collins Kimutai Meli

Received	Revised	Accepted	Published
24 Apr 2026	30 May 2026	17 Jun 2026	30 Jun 2026

Citation :

Winnie Kerubo Kegwaro, Denis Njue King’ang’i, Collins Kimutai Meli, "On the Norm of Jordan Elementary Operators in Tensor Product of 𝐶∗-Algebras," International Journal of Mathematics Trends and Technology (IJMTT), vol. 72, no. 6, pp. 40-44, 2026. Crossref, https://doi.org/10.14445/22315373/IJMTT-V72I6P106

Abstract

Elementary operators have been studied over years, with their norms being of significant interest in operator theory. The study includes the derivation of formulas that describe norms in terms of their coefficient operators, which is traced back to Stampfli`s Theorem that used the property of Numerical range as a foundation for the study of norms. Other properties of Elementary operators have been studied ever since, but little is known on Jordan Elementary operators in tensor products of 𝐶∗- algebras. This paper aims to extend the determination of norms of Jordan Elementary operators in the tensor product of 𝐶∗- algebra by determining the lower bound of the norm using the maximal numerical range by employing the technique of tensor product, finite rank operator and inner product. A lower bound of Jordan elementary operator in tensor product of 𝐶∗algebras is obtained.

Keywords

Maximal numerical range, Elementary operator, Rank-one-operator, Tensor Product, 𝐶∗- algebras.

References

[1] International Organization for Standardization, ISO 4190-1: Lifts (Elevators) and Service Lifts-Part 1: Planning and Selection, Geneva, Switzerland, 2010.
[Publisher Link]

[2] Verein Deutscher Ingenieure, VDI 4707 Part 1: Energy Efficiency of Lifts, Düsseldorf, Germany, 2009.
[Publisher Link]

[3] Gina C. Barney, and Lutfi Al-Sharif, Elevator Traffic Handbook: Theory and Practice, 2^nd Ed., London, U.K.: Routledge, 2015.
[Google Scholar] [Publisher Link]

[4] George R. Strakosch, The Vertical Transportation Handbook, 4^th Ed., Hoboken, NJ, USA: Wiley, 2010.
[Google Scholar] [Publisher Link]

[5] Robert H. Crites, and Andrew G. Barto, “Elevator Group Control using Multiple Reinforcement Learning Agents,” Machine Learning, vol. 33, no. 2-3, pp. 235-262, 1998.
[CrossRef] [Google Scholar] [Publisher Link]

[6] Pablo Cortés, J. Larrañeta, and Luis Onieva, “Genetic Algorithm for Controllers in Elevator Groups: Analysis and Simulation during Lunchpeak Traffic,” Applied Soft Computing, vol. 4, no. 2, pp. 159-174, 2004.
[CrossRef] [Google Scholar] [Publisher Link]

[7] Qinglai Wei et al., “Optimal Elevator Group Control via Deep Asynchronous Actor-Critic Learning,” IEEE Transactions on Neural Networks and Learning Systems, vol. 31, no. 12, pp. 5245-5256, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[8] David Pepyne, and Christos G. Cassandras, “Optimal Dispatching Control for Elevator Systems during Uppeak Traffic,” IEEE Transactions on Control Systems Technology, vol. 5, no. 6, pp. 629-643, 1997.
[CrossRef] [Google Scholar] [Publisher Link]

[9] David Pepyne, and Christos G. Cassandras, “Design and Implementation of an Adaptive Dispatching Controller for Elevator Systems during Uppeak Traffic,” IEEE Transactions on Control Systems Technology, vol. 6, no. 5, pp. 635-650, 1998.
[CrossRef] [Google Scholar] [Publisher Link]

[10] Daniel Nikovski, and Matt Brand, “Exact Calculation of Expected Waiting Times for Group Elevator Control,” IEEE Transactions on Automatic Control, vol. 49, no. 10, pp. 1820-1823, 2004.
[CrossRef] [Google Scholar] [Publisher Link]

[11] Mirko Ruokokoski et al., “Assignment Formulation for the Elevator Dispatching Problem with Destination Control and Its Performance Analysis,” European Journal of Operational Research, vol. 252, no. 2, pp. 397-406, 2016.
[CrossRef] [Google Scholar] [Publisher Link]

[12] Mirko Ruokokoski, Harri Ehtamo, and Panos M. Pardalos, “Elevator Dispatching Problem: A Mixed Integer Linear Programming Formulation And Polyhedral Results,” Journal of Combinatorial Optimization, vol. 29, no. 4, pp. 750-780, 2015.
[CrossRef] [Google Scholar] [Publisher Link]

[13] ChangBum Kim et al., “Design and Implementation of a Fuzzy Elevator Group Control System,” IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, vol. 28, no. 3, pp. 277-287, 1998.
[CrossRef] [Google Scholar] [Publisher Link]

[14] Joaquin R. Fernandez et al., “Dynamic Fuzzy Logic Elevator Group Control System With Relative Waiting Time Consideration,” IEEE Transactions on Industrial Electronics, vol. 61, no. 9, pp. 4912-4919, 2014.
[CrossRef] [Google Scholar] [Publisher Link]

[15] Berna Bolat et al., “Optimal Car Dispatching for Elevator Groups Using Genetic Algorithms,” Intelligent Automation & Soft Computing, vol. 16, no. 1, pp. 89-99, 2010.
[CrossRef] [Google Scholar] [Publisher Link]

[16] Marja-Liisa Siikonen, “Planning and Control Models for Elevators in High-Rise Buildings,” Ph.D. Dissertation, Espoo, Finland, 1997.
[Google Scholar] [Publisher Link]

[17] Eduardo F. Camacho, and Carlos Bordons, Model Predictive Control, 2^nd Ed., 2007.
[CrossRef] [Google Scholar] [Publisher Link]

[18] Bao Ding et al., “A Hybrid Approach for the Analysis and Prediction of Elevator Passenger Flow in an Office Building,” Automation in Construction, vol. 35, pp. 69-78, 2013.
[CrossRef] [Google Scholar] [Publisher Link]

[19] Richard S. Sutton, and Andrew G. Barto, Reinforcement Learning: An Introduction, 2^nd Ed., Cambridge, MA, USA: MIT Press, 2018.
[Google Scholar] [Publisher Link]

[20] Lutfi Al-Sharif et al., “Establishing the Upper Performance Limit of Destination Elevator Group Control using Idealised Optimal Benchmarks,” Building Services Engineering Research & Technology, vol. 36, no. 5, pp. 546-566, 2015.
[CrossRef] [Google Scholar] [Publisher Link]

[21] Jörg Lauener, Traffic Performance of Elevators with Destination Control, Elevator World, vol. 55, no. 9, 2007.
[Google Scholar]

[22] Jinglong Zhang, and Qun Zong, “Energy-Saving-Oriented Group-Elevator Dispatching Strategy for Multi-Traffic Patterns,” Building Services Engineering Research & Technology, vol. 35, no. 5, pp. 543-568, 2015.
[CrossRef] [Google Scholar] [Publisher Link]

[23] Mo Shi et al., “Prediction and Analysis of Elevator Traffic Flow under the LSTM Neural Network,” Intelligent Control and Automation, vol. 15, no. 2, pp. 63-82, 2024.
[CrossRef] [Google Scholar] [Publisher Link]

[24] Nathan Vaartjes, and Vincent Francois-Lavet, “Novel RL Approach for Efficient Elevator Group Control Systems,” arXiv preprint arXiv Preprint, pp. 1-15, 2025.
[CrossRef] [Google Scholar] [Publisher Link]

[25] Shu Wang et al., “Smart Dispatching and Optimal Elevator Group Control through Real-Time Occupancy-Aware Deep Learning of Usage Patterns,” Advanced Engineering Informatics, vol. 48, 2021.
[CrossRef] [Google Scholar] [Publisher Link]

[26] Matthew J Page et al., “The PRISMA 2020 Statement: An Updated Guideline for Reporting Systematic Reviews,” BMJ, vol. 372, pp. 1-9, 2021.
[CrossRef] [Google Scholar] [Publisher Link]

[27] Leonard Kleinrock, Queueing Systems, Volume 1: Theory. New York, NY, USA: Wiley-Interscience, 1975.
[Google Scholar] [Publisher Link]

[28] Gordon Frank Newell, Applications of Queueing Theory, 2^nd Ed., London, U.K.: Chapman and Hall, 1982.
[Google Scholar] [Publisher Link]

[29] Lutfi Al-Sharif, Ahmad M Abu Alqumsan, and Rasha Khaleel, “Derivation of a Universal Elevator Round Trip Time Formula under Incoming Traffic,” Building Services Engineering Research & Technology, vol. 35, no. 2, pp. 198-213, 2015.
[CrossRef] [Google Scholar] [Publisher Link]