Publication

Abstract : In this article, we will first establish some density results for a locally C ∗ -algebra A and then identify a property, called Kaplansky density property (KDP). We then give a induced faithful continuous ∗ -representation φ of A ∗ ∗ (equipped with unique Arens product) on the space B l o c ( H ) such that φ ( A ∗ ∗ ) ⊂ π ( A ) ¯¯¯¯¯¯¯¯¯¯¯ W O T , where π : A → B l o c ( H ) is the associated universal ∗ -representation and H is the associated locally Hilbert space. Finally we show that for a Fréchet locally C ∗ -algebra A possessing KDP, the second strong dual is algebraically and topologically ∗ -isomorphic to π ( A ) ¯¯¯¯¯¯¯¯¯¯¯ W O T , 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.