# Convergent Star Products for Projective Limits of Hilbert Spaces

## TL;DR

This paper develops a topology on the symmetric algebra of a locally convex space to ensure the continuity of a star product, comparing it with other approaches in finite and infinite-dimensional settings.

## Contribution

It introduces a new topology on symmetric algebras for locally convex spaces that guarantees the continuity of exponential star products, extending previous finite-dimensional results.

## Key findings

- The star product becomes continuous under the new topology.
- The approach applies to Hilbert and nuclear spaces.
- Comparison with existing methods highlights advantages of the new topology.

## Abstract

Given a locally convex vector space with a topology induced by Hilbert seminorms and a continuous bilinear form on it we construct a topology on its symmetric algebra such that the usual star product of exponential type becomes continuous. Many properties of the resulting locally convex algebra are explained. We compare this approach to various other discussions of convergent star products in finite and infinite dimensions. We pay special attention to the case of a Hilbert space and to nuclear spaces.