Sherman-Takeda type theorems for locally C*-algebras

Abstract

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