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A note on positive bilinear maps

Aaron C. H. Davey, Cristian Ivanescu and Adi Tcaciuc

Vol. 16 (2023), No. 4, 579–590
Abstract

This paper concerns positive maps between C-algebras, particularly when those positive maps are multilinear. We construct examples of positive bilinear maps that are not 2-positive, and therefore are not completely positive bilinear maps. Paulsen and Smith showed that completely bounded bilinear maps are in one-to-one correspondence with completely bounded linear maps. We show that a similar correspondence does not hold for positive bilinear and linear maps. In particular, we observe that a similar correspondence does not hold if we replace completely bounded with completely positive.

Keywords
positive maps, multilinear maps, completely bounded, C$^*$-algebras, positive bilinear maps, positive elements
Mathematical Subject Classification
Primary: 15A69, 47B65, 47C15, 46L05, 46L07
Milestones
Received: 2 October 2021
Revised: 12 November 2021
Accepted: 18 August 2022
Published: 31 October 2023

Communicated by Kenneth S. Berenhaut
Authors
Aaron C. H. Davey
Department of Mathematics and Statistics
MacEwan University
Edmonton, AB
Canada
Cristian Ivanescu
Department of Mathematics and Statistics
MacEwan University
Edmonton, AB
Canada
Adi Tcaciuc
Department of Mathematics and Statistics
MacEwan University
Edmonton, AB
Canada