algebra without order
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 .
Note that for an algebra and an element , is the map defined by . Then is a linear operator on . It is easy to see that is without left order if and only if the map is one-one; equivalently, the left ideal . This ideal is is called the left annihilator of .
Example: with multiplication defined by , () is not an algebra without order as multiplication of (0,1) with any other element gives (0,0).
|Title||algebra without order|
|Date of creation||2013-03-22 14:46:01|
|Last modified on||2013-03-22 14:46:01|
|Last modified by||mathcam (2727)|