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Sherman-Takeda type theorems for locally C*-algebras

Lav Kumar Singh, Aljoša Peperko

TL;DR

This work extends Sherman–Takeda-type identifications from $C^*$-algebras to locally $C^*$-algebras by analyzing their biduals under Arens products and introducing the Kaplansky density property (KDP). It constructs a faithful continuous $*$-representation $\varphi$ of the bidual $\mathcal{A}^{**}$ into $B_{loc}(\mathcal{H})$ with $\varphi(\mathcal{A}^{**})\subset \overline{\pi(\mathcal{A})}^{WOT}$ and shows that for Fréchet locally $C^*$-algebras with KDP, $\mathcal{A}^{**}$ is topologically and algebraically $*$-isomorphic to $\overline{\pi(\mathcal{A})}^{WOT}$. The paper also develops the continuity properties of bilinear maps arising from the universal representation and provides a framework to construct representations on locally bounded operator spaces. Together, these results yield a Sherman–Takeda-type theorem in the locally $C^*$-algebra setting and offer tools for analyzing second duals of non-normed operator algebras.

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 $\varphi(\mathcal A^{**})\subset \overline{π(\mathcal A)}^{WOT}$, where $π:\mathcal A\to B_{loc}(\mathcal H)$ is the associated universal $*$-representation and $\mathcal H$ is the associated locally Hilbert space. Finally we show that for a Fréchet locally $C^*$-algebra $\mathcal A$ possessing KDP, the second strong dual is algebraically and topologically $*$-isomorphic to $ \overline{π(\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.

Sherman-Takeda type theorems for locally C*-algebras

TL;DR

This work extends Sherman–Takeda-type identifications from -algebras to locally -algebras by analyzing their biduals under Arens products and introducing the Kaplansky density property (KDP). It constructs a faithful continuous -representation of the bidual into with and shows that for Fréchet locally -algebras with KDP, is topologically and algebraically -isomorphic to . The paper also develops the continuity properties of bilinear maps arising from the universal representation and provides a framework to construct representations on locally bounded operator spaces. Together, these results yield a Sherman–Takeda-type theorem in the locally -algebra setting and offer tools for analyzing second duals of non-normed operator algebras.

Abstract

In this article, we will first establish some density results for a locally -algebra and then identify a property, called Kaplansky density property (KDP). We then give a induced faithful continuous -representation of (equipped with unique Arens product) on the space such that , where is the associated universal -representation and is the associated locally Hilbert space. Finally we show that for a Fréchet locally -algebra possessing KDP, the second strong dual is algebraically and topologically -isomorphic to , which is a direct analogue of the classical Sherman-Takeda theorem for -algebras. We shall also observe the joint continuity of some associated bilinear maps in the running.
Paper Structure (7 sections, 22 theorems, 50 equations)

This paper contains 7 sections, 22 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Locally multiplicatively convex algebras
  4. Arens product on bidual of topological algebras.
  5. Kaplanski density property in locally $C^*$-algebras
  6. Bidual of locally $C^*$-algebras
  7. Continuity of some bilinear maps associated to a representation of locally $C^*$-algebras.

Key Result

Theorem 1.1

If $\mathcal{A}$ is a $C^*$-algebra and $\pi:\mathcal{A}\to B(\mathcal{H})$ is its universal representation on a Hilbert space $\mathcal{H}$, then the second dual $\mathcal{A}^{**}$ is a von Neumann algebra which is isometrically $*$-isomorphic to $\pi(\mathcal{A})^{cc}=\overline{\pi(\mathcal{A})}^{

Theorems & Definitions (46)

  • Theorem 1.1: Sherman-Takeda
  • Theorem 1.2
  • Definition 2.1: Inverse/Projective limit
  • Definition 2.2: Direct/Inductive limit
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1
  • Definition 2.3
  • Lemma 2.3
  • proof
  • ...and 36 more