# Banach-Alaoglu theorem for Hilbert $H^*$-module

## TL;DR

This paper extends the Banach-Alaoglu theorem to Hilbert H*-modules by constructing a suitable weak* topology, demonstrating the compactness of the unit ball in this new setting.

## Contribution

It introduces a novel Banach-Alaoglu type theorem for Hilbert H*-modules, establishing a weak* topology under which the unit ball is compact.

## Key findings

- Established a $\Lambda$-weak$^*$ topology on Hilbert H*-modules
- Proved the unit ball is compact in this topology
- Extended classical Banach-Alaoglu theorem to a new algebraic setting

## Abstract

We provided an analogue Banach-Alaoglu theorem for Hilbert $H^*$-module. We construct a $\Lambda$-weak$^*$ topology on a Hilbert $H^*$-module over a proper $H^*$-algebra $\Lambda$, such that the unit ball is compact with respect to $\Lambda$-weak$^*$ topology.