Positivity (2017) 21:817–823
Ordered Banach algebras and multi-norms: some open
A. W. Wickstead
Received: 18 March 2017 / Accepted: 23 March 2017 / Published online: 30 March 2017
© The Author(s) 2017. This article is an open access publication
Abstract This is a list of open problems posed at the workshop on Ordered Banach
Algebras held at the Lorentz Center, Leiden, during the week 21–25 July, 2014.
Keywords Banach lattice algebras · Ordered Banach algebras · Multi-norms
Mathematics Subject Classiﬁcation 46B40 · 46B42 · 46H05 · 46H15 · 47A10 ·
47B65 · 47L10
We call A a Banach lattice algebra if it is both a Banach lattice and a Banach algebra
and furthermore the product of two positive elements is positive.
Problem 1 Characterize those Banach lattice algebras A for which the left regular
representation a → L
, where L
(x) = ax, into the space of regular operators L
preserves ﬁnite suprema and inﬁma, as well as being an algebra homomorphism. Even
an answer for Dedekind complete Banach lattice algebras would be of interest, when
we are asking about it being a lattice homomorphism. In addition, consider when such
representations are either faithful or isometric. The natural norm to take on L
would be the regular norm, but it is conceivable that the question might be of interest
for the operator norm as well.
This list of problems is based on three preliminary reports produced by Gerard Buskes, Garth Dales and
Sonja Mouton. The author would like to extend thanks to them for their work in preparing the original list
and to them, Marcel de Jeu and Vladimir Troitsky for further input into this ﬁnal list.
A. W. Wickstead
Mathematical Sciences Research Centre, Queen’s University Belfast, BT7 1NN, Belfast,