Local operator systems are locally convex version of operator systems. Dosiev (2008) defined a local operator system as a unital self-adjoint subspace of the multinormed $C^*$-algebra $C^*_\mathcal{E} (\mathcal{D})$, where $H$ is a fixed Hilbert space with an upward filtered family of closed subspaces $\mathcal{E} = \{H_{\alpha}\}_{\alpha \in \Lambda}$ such that their union $\mathcal{D}$ is a dense subspace in $H$ with $p = \{P_{\alpha}\}_{\alpha \in \Lambda}$ family of projections in $B(H)$ onto the subspaces $H_{\alpha},\alpha \in \Lambda$. In fact, local operator systems are projective limits of operator systems, which are simply unital self-adjoint subspaces of the space of bounded linear operator on a Hilbert space.

Choi and Effros (1977) proved that an operator system can be defined independently of Hilbert space, purely in terms of $*$-vector space and matrix ordering. Motivated by this, we propose an abstract definition of local operator systems that turns out to be equivalent to the definition given by Dosiev. Based upon this definition and the work of Kavruk et al. (2011), we construct a minimal local operator system LOMIN and a maximal local operator system LOMAX, analogous to minimal operator system OMIN and maximal operator system OMAX, respectively. We also introduce and explore the theory of tensor products in the category of local operator systems.

This talk is based on joint work with Surbhi Beniwal and Ajay Kumar.