Sherman-Takeda type theorems for locally C*-algebras

TL;DR

This paper extends Sherman-Takeda theorems to locally C*-algebras by establishing density results, defining a Kaplansky density property, and demonstrating an isomorphism between the second dual and a WOT-closure in a locally Hilbert space setting.

Contribution

It introduces a Kaplansky density property for locally C*-algebras and proves an analogue of Sherman-Takeda theorems in this broader context.

Findings

Established density results for locally C*-algebras.

Defined and utilized the Kaplansky density property.

Proved the second dual is isomorphic to a WOT-closure in the locally Hilbert space setting.

Abstract

In this article, we will first establish some density results for a locally $C^*$-algebra $\mathcal A$ and then identify a property, called Kaplansky density property (KDP). We then give a induced faithful continuous $*$-representation $\varphi$ of $\mathcal A^{**}$ (equipped with unique Arens product) on the space $B_{loc}(\mathcal H)$ such that $\varphi(\mathcal A^{**})\subset \overline{\pi(\mathcal A)}^{WOT}$, where $\pi:\mathcal A\to B_{loc}(\mathcal H)$ is the associated universal $*$-representation and $\mathcal H$ is the associated locally Hilbert space. Finally we show that for a Fr\'echet locally $C^*$-algebra $\mathcal A$ possessing KDP, the second strong dual is algebraically and topologically $*$-isomorphic to $ \overline{\pi(\mathcal A)}^{WOT}$, which is a direct analogue of the classical Sherman-Takeda theorem for $C^*$-algebras. We shall also observe the joint continuity of some associated bilinear maps in the running.