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The R-Shilov boundary for a local operator space

Maria Joiţa, Gheorghe-Ionuţ Şimon

TL;DR

This work extends Arveson–Hamana boundary theory to the setting of locally convex operator spaces by formulating and establishing the existence and uniqueness of the injective $\mathcal{R}$-envelope and the $\mathcal{R}$-$C^{*}$-envelope for unital local operator spaces. It shows the injective $\mathcal{R}$-envelope for local operator spaces coincides with Dosi's construction and develops the corresponding local $\mathcal{R}$-$C^{*}$-envelope, including a local Hamana–Ruan extension analogue. The authors introduce and analyze the local $\mathcal{R}$-Shilov boundary ideal, proving its existence and maximality and establishing universal properties for the corresponding local envelopes. The results connect the local (unbounded) framework to the bounded-part theory, providing a robust foundation for noncommutative boundary concepts in locally $C^{*}$-algebra settings and enabling further applications in locally convex noncommutative geometry.

Abstract

To extend the notion of the injective envelope of a unital operator space to the locally convex case, Dosi (2014) first introduced the notion of the injective R-envelope for a unital operator space and then defined the injective R-envelope for a unital local operator space as the closure of the injective R-envelope for its bounded part. In this paper, we investigate the existence of the Shilov boundary ideal in this context, as defined by Arveson (1969). To do this, by following the conceptual frameworks underlying Hamana's constructions of the injective envelope and the C*-envelope, respectively, we define the notions of the injective R-envelope and the R-C*-envelope for a unital local operator space. Furthermore, we show that the injective R-envelope construction given by us coincides with the one given by Dosi (2014).

The R-Shilov boundary for a local operator space

TL;DR

This work extends Arveson–Hamana boundary theory to the setting of locally convex operator spaces by formulating and establishing the existence and uniqueness of the injective -envelope and the --envelope for unital local operator spaces. It shows the injective -envelope for local operator spaces coincides with Dosi's construction and develops the corresponding local --envelope, including a local Hamana–Ruan extension analogue. The authors introduce and analyze the local -Shilov boundary ideal, proving its existence and maximality and establishing universal properties for the corresponding local envelopes. The results connect the local (unbounded) framework to the bounded-part theory, providing a robust foundation for noncommutative boundary concepts in locally -algebra settings and enabling further applications in locally convex noncommutative geometry.

Abstract

To extend the notion of the injective envelope of a unital operator space to the locally convex case, Dosi (2014) first introduced the notion of the injective R-envelope for a unital operator space and then defined the injective R-envelope for a unital local operator space as the closure of the injective R-envelope for its bounded part. In this paper, we investigate the existence of the Shilov boundary ideal in this context, as defined by Arveson (1969). To do this, by following the conceptual frameworks underlying Hamana's constructions of the injective envelope and the C*-envelope, respectively, we define the notions of the injective R-envelope and the R-C*-envelope for a unital local operator space. Furthermore, we show that the injective R-envelope construction given by us coincides with the one given by Dosi (2014).
Paper Structure (13 sections, 17 theorems, 28 equations)

This paper contains 13 sections, 17 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Locally $C^*$-algebras
  4. Quantized domains
  5. Local positive maps
  6. Injective local operator spaces
  7. The $\mathcal{R}-C^{\ast }$ -envelope of an operator space
  8. The injective $\mathcal{R}$-envelope of an operator space DD
  9. The $\mathcal{R}-C^{\ast }$-envelope of an operator space
  10. The $\mathcal{R}-C^{\ast }$ -envelope of a local operator space
  11. The injective $\mathcal{R}$-envelope of a local operator space
  12. The $\mathcal{R}-C^{\ast }$-envelope of a local operator space
  13. The $\mathcal{R}$-Shilov boundary ideal for a local operator space

Key Result

Theorem 2.4

For any unital locally $C^{\ast }$-algebra $\mathcal{A}$, there exist a commutative domain $\{\mathcal{H};\mathcal{E}; \mathcal{D}_{\mathcal{E}}\}$ and a local isometric $\ast$-morphism $\pi : \mathcal{A}\rightarrow C^{\ast }(\mathcal{D}_{\mathcal{E}})$.

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: D1, Proposition 3.1
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Theorem 2.7: D, Corollary 5.5
  • Proposition 3.1
  • Definition 3.2
  • ...and 36 more