algebra without order

An algebraPlanetmathPlanetmath ( A is said to be \PMlinkescapephraseorder without order if it is commutativePlanetmathPlanetmathPlanetmath and
for each aA, there exists bA such that ab0.

The phrase algebra without order seems first in the book “Multipliers of Banach algebras” by Ronald Larsen. In noncommutative case, the concept is divied into two parts – without left/right order. However, in the noncommutative case, it is defined in terms of the injectivity of the left (right) regular representation given by xALxL(A).

Note that for an algebra A and an element xA, Lx:AA is the map defined by Lx(y)=xy. Then Lx is a linear operator on A. It is easy to see that A is without left order if and only if the map xALxL(A) is one-one; equivalently, the left idealMathworldPlanetmathPlanetmath {xA:xA}={0}. This ideal is is called the left annihilator of A.

Every commutative algebra with identityPlanetmathPlanetmathPlanetmathPlanetmath is without order.

Example: 2 with multiplication defined by (x1,x2)*(y1,y2)=(x1y1,0), ((x1,x2),(y1,y2)2) is not an algebra without order as multiplication of (0,1) with any other element gives (0,0).

Title algebra without order
Canonical name AlgebraWithoutOrder
Date of creation 2013-03-22 14:46:01
Last modified on 2013-03-22 14:46:01
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 11
Author mathcam (2727)
Entry type Definition
Classification msc 13A99