On Partially Ordering Operator Algebras

Abstract

In this paper, we consider linear spaces and algebras with real scalars. It is well known that if X is a Banach space and is the set of all bounded linear operators which map X into itself, then is a Banach algebra. In this paper we shall show that can be partially ordered so that it becomes a partially ordered algebra in which norm convergence is equivalent to order convergence. This motivates a study of Banach algebras of operators in which one uses the order structure to obtain various results. In addition, it encourages a study of partially ordered algebras in general, since our result shows that among such algebras one finds all real Banach algebras of operators. Of course, there are many other real algebras which are naturally partially ordered and which have been studied from that point of view.

On Elements with Negative Squares

“…It is appropriate, therefore, to construct a proof within the more general framework of partially ordered linear algebras. The main theorem will then be valid not only for finite matrices but also for (row-finite) infinite matrices as well as operators on a Banach space [1]. We also generalize a result due to Birkhoff and Pierce [2] which states that the complex number field regarded as a linear algebra over the reals admits no lattice order.…”

Fully ordered groups in partially ordered linear algebras

“…In fact, DeMarr regards general groups as groups in a dsc-pola and studies these groups using linear algebra techniques. A similar approach to investigate linear operators (both bounded and unbounded) using dscpola methods appeared in [4], [5], [6]. This idea of studying algebraic structures (and other important mathematical subjects) in a linear algebra setting can also be found in a rather inspiring paper by Eckmann [7].…”

Partially Ordered Linear Algebras With Multiplicative Diagonal Map

“…This paper is motivated by the work of Kadison and Singer [6] on triangular operator algebras, but instead of dealing with a normed linear algebra of operators, we consider an abstract partially ordered linear algebra (pola). It should be pointed out that the algebra of all norm bounded operators on a real Banach space can always be regarded as a pola [3]. Although the basic concept is the same as in [6], our approach is somewhat different and centers on the fact that in an algebra of upper triangular matrices the diagonal of the product of two matrices is equal to the product of the diagonals (see §4).…”