# Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online July 8, 2022

## Inductive limit in the category of $C^{\ast}$-ternary rings

Arpit Kansal, Ajay Kumar, and Vandana Rajpal

University of Delhi, Shivaji College, University of Delhi

### Abstract

We show the existence of inductive limit in the category of $C^{\ast}$-ternary rings. It is proved that the inductive limit of $C^{\ast}$-ternary rings commutes with the functor $\mathcal{A}$ in the sense that if $(M_n, \phi_n)$ is an inductive system of $C^{\ast}$-ternary rings then $\varinjlim \mathcal{A}(M_n)=\mathcal{A}(\varinjlim M_n)$. Some local properties(such as nuclearity, exactness and simplicity) of inductive limit of $C^{\ast}$-ternary rings have been investigated. Finally we obtain $\varinjlim M_n^{\ast\ast}=(\varinjlim M_n)^{\ast\ast}$.

Keywords: Inductive limit, C$^*$-ternary rings, TRO, C$^*$-algebras

MSC numbers: 46L06; 46L07; 46M40