Lava Kumar Singha: Sherman-Takeda theorem for the *-subalgebras of bounded operators on a quantized Hilbert domain

In this talk we explore a natural extension of the classical Sherman–Takeda theorem from the setting of normed C-algebras to the broader realm of locally C-algebras, which are inverse limits of C-algebras equipped with a locally convex topology. A key challenge in this context is that many of the familiar tools from normed operator algebras may fail without a norm, so new structural methods are needed. We begin by establishing core density results and identifying a crucial structural condition—termed the Kaplansky density property (KDP)—for an arbitrary locally C-algebra A. Using this, we construct a faithful continuous -representation of the bidual A∗∗ (with its unique Arens algebra product) into the space of locally bounded operators on a locally Hilbert space. This representation fits naturally inside the weak operator closure of the universal representation of A. The main theorem shows that if A is a Fréchet locally C-algebra with KDP, then its second strong dual is both algebraically and topologically-isomorphic to the weak operator closure of A under its universal representation. This result provides a direct analogue of the classical Sherman–Takeda theorem—originally stating that the bidual of a C-algebra is isomorphic to its enveloping von Neumann algebra—in the more general locally C-framework. We also discuss continuity properties of bilinear maps that naturally arise in this duality theory.