Abstract

The notion of the completely dissipative maps on σ‐C*‐algebras is introduced. We show that a completely dissipative map induces a representation of a σ‐C*‐algebra. The classical Stinespring′s dilation‐type theorem is extended to a more general setting.

Highlights

  • There has been increased interest 1–12 in topological ∗-algebras that are inverse limits of C∗algebras, called P ro-C∗-algebras

  • A metrizable P ro-C∗-algebra is called a σ-C∗-algebra with its topology determined by a countable subfamily pn of S A, n ∈ N

  • If X and Y are vector spaces, we denote by X ⊗ Y their algebraic tensor product

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Summary

Introduction

There has been increased interest 1–12 in topological ∗-algebras that are inverse limits of C∗algebras, called P ro-C∗-algebras. These algebras were introduced in 5 as a generalization of C∗-algebras and were called locally C∗-algebras. We shall show that a completely positive definite map induces a representation of a σ-C∗-algebra. We get a classification of completely positive definite maps on σ-C∗-algebras. Each pα, α ∈ I, is a C∗-seminorm, that is, pα x∗x pα x 2 for any α ∈ I, x ∈ A, A is called to be an lmc-C∗-algebra. Given an lmc-algebra A denote by Aα the inverse system of Banach algebras corresponding to A, that is, Aα is the completion of the normed algebra. A metrizable P ro-C∗-algebra is called a σ-C∗-algebra with its topology determined by a countable subfamily pn of S A , n ∈ N

Preliminaries
The Stinespring’s Dilation-Type Theorem
Full Text
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