Definition 2

The restricted dual

of the K-algebra A, denoted by Θ(A), is the union in $A^*$ of the orthogonals of the two-sided ideals of A of finite codimension.

By Proposition 1, we can give the following equivalent descriptions of Θ(A):

– the union of the orthogonals of the left ideals of A of finite codimension

– the union of the orthogonals of the right ideals of A of finite codimension – the union of the orthogonals of the two-sided ideals of A of finite codimension – the union of the left A-submodules of $A^*$ of finite dimension over K

– the union of the right A-submodules of $A^*$ of finite dimension over K

– the union of the $(A$, A)-sub-bimodules of $A^*$ of finite dimension over K.

We have $\Theta (A) = \Theta (A^o)$, and $\Theta (A) = A^*$ if A has finite degree over K. Let $f\in A^*$; then $f$ belongs to Θ(A) if and only if $Af$ (resp. $fA, AfA$) is a linear subspace of $A^*$ of finite dimension over K.

The sum of two $(A$, A)-sub-bimodules of $A^*$ of finite dimension over K has finite dimension over K. It follows that Θ(A) is an $(A$, A)-sub-bimodule of $A^*$.