Positivity 11 (2007), 369–374
2007 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020369-6, published online April 6, 2007
Corrigendum to the paper “Adjoining an Order
Unit to a Matrix Ordered Space”
Anil Kumar Karn
Abstract. An error has been detected (and also corrected) in Theorem 2.8
of the paper entitled “Adjoining an Order Unit to a Matrix Ordered Space”
(Positivity, (2005)9: 207–223; DOI 10.1007/s11117-003-2778-5). Accordingly,
some of the results of the paper have been modiﬁed. Also, a notion of C
matricially, Riesz normed spaces has been introduced.
Mathematics Subject Classiﬁcation (2000). Primary 46L07.
-matricially Riesz normed space.
In , Example 5.8, D. P. Blecher and M. Neal showed that the operator space
dual of a C
-algebra need not be order embedded in a C
-algebra. This fact, given
that the operator space dual of an L
-matricially Riesz normed space is again an
-matricially Riesz normed space (c.f. ), contradicts Theorem 2.8 of  (the
paper entitled “Adjoining an Order Unit to a Matrix Ordered Space” (Positivity
(2005)9: 207-223; DOI 10.1007/s11117-003-2778-5)). D. P. Blecher pointed out to
the author that the inequalities in page 211, lines 13, 20 and 27 are not true in
general. Thus Theorem 2.8 is not true in its present form. Since this theorem is
central in the paper, it becomes desirable to revise the entire paper.
In this short note we shall try to retrieve the results of this paper in their
best possible forms. For this purpose we need to incorporate certain changes which
we list below.
First, let us recall, from  (see also ), that an L
normed space is an approximate matrix order unit space, if for any natural number
n and u
< 1,i =1, 2 there exists u ∈ M
< 1 such that u
≤ u, i =1, 2.
Now we note that Theorem 2.8 holds in the following form:
Theorem 0.1. Let
be an L
-matricially Riesz normed space
Then there exists a matrix order unit space
and a completely contractive
complete order isomorphism φ : V −→
V such that the codimension of φ (V ) in