States and\( ^ *\)-representations. I
type: Item
instance of: scholarly article
title: States and\( ^ *\)-representations. I
zbMATH Open document ID: 0624.46035
DOI: 10.1007/BF01848645
published in: Periodica Mathematica Hungarica
publication date: 1986-01-01T00:00:00Z
review text: The author introduces and studies a concept of an \(S^*\)-algebra. It is a complex *-algebra A posessing a \(C^*\)-seminorm \(p_ A\) such that every state f on A is continuous with respect to \(p_ A\) (a state is a positive functional satisfying \(| f(a)|^ 2\leq M_ ff(a^*a)\) for all a in A). Among other results the author shows that A is an \(S^*\)-algebra if and only if \[ \sup \{f(b^*a^*ab):b\in A,\text{ f is a state and }f(b^*b)\leq 1\}<\infty \] for all a in A, and that all \(C^*\)-seminorms on an \(S^*\)-algebra A are majorized by \(p_ A\). There is also given a general form of completely positive maps from an \(S^*\)-algebra into B(H) and it is shown that every Schwartz map on a commutative \(S^*\)-algebra is automatically completely positive.
reviewed by: Wiesław Tadeusz Żelazko
Mathematics Subject Classification ID: 46K10
Mathematics Subject Classification ID: 46K05
Mathematics Subject Classification ID: 46L30
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46H10
zbMATH DE Number: 4013312
zbMATH Keywords: semigroup
zbMATH Keywords: \(S^ *\)-algebra
zbMATH Keywords: \(C^ *\)-seminorm
zbMATH Keywords: state
zbMATH Keywords: completely positive maps
zbMATH Keywords: Schwartz map
author: Zoltán Sebestyén
MaRDI profile type: Publication
cites work: Q3215824
cites work: The reducing ideal is a radical
cites work: Completely Positive Maps on U ∗ -Algebras
cites work: Q3857319
cites work: Every \(C^*\)-seminorm is automatically submultiplicative
cites work: On representability of linear functionals on *-algebras
cites work: Positive Functions on C ∗ -Algebras
full work available at URL: https://doi.org/10.1007/bf01848645
OpenAlex ID: W4237739747
description: scientific article; zbMATH DE number 4013312