# Mixed limits of some functional spaces

## TL;DR

This paper introduces the concept of mixed limits of functional spaces, exploring conditions under which upper and lower limits coincide, and examines their properties, especially in relation to non-commutative $L_p$-spaces and Banach algebras.

## Contribution

It defines mixed limits of functional spaces and analyzes their properties, including their emergence as limits of non-commutative $L_p$-spaces and Banach algebras.

## Key findings

- Mixed limits generalize inductive and projective limits of functional spaces.
- Limit spaces of Banach algebras are (LF)-spaces when convergent.
- Conditions identified for upper and lower limits to coincide.

## Abstract

In this article we propose a conception of mixed limits of functional spaces as the case, when the upper limit (projective limit of inductive limits) and the lower limit (inductive limit of projective limits) coincide as topological spaces, which are generalization of inductive and projective limits of functional spaces. We show a cases where these mixed limits are naturally obtained as the limit spaces of non-commutative $L_p$-type spaces associated with the sequence of operators. Also, we obtain results on the properties of limit spaces, we show that limit spaces of Banach algebras are (LF)-spaces, if they converge.