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Adjoints

In §, we used the notation tex2html_wrap_inline4446 in conjunction with problems involving an algebra which possessed an involution, tex2html_wrap_inline7842 . We now make this notion precise.

We start by assuming that there is an involution on the coefficients, say tex2html_wrap_inline7844 .

By relabelling the knowns and unknowns if necessary, let us suppose that

(1)The polynomial q depends upon the knowns tex2html_wrap_inline7848 and upon the unknowns tex2html_wrap_inline7850 where tex2html_wrap_inline7852 and tex2html_wrap_inline7854 .
(2) The polynomial q does not depend on the knowns tex2html_wrap_inline7858 and does not depend on the unknowns tex2html_wrap_inline7860 .

Let us also suppose that there is an injective map tex2html_wrap_inline7862 from tex2html_wrap_inline7864 to tex2html_wrap_inline7866 and an injective map tex2html_wrap_inline7868 from tex2html_wrap_inline7870 to tex2html_wrap_inline7872 where tex2html_wrap_inline7852 and tex2html_wrap_inline7854 .

We now make the assumption which corresponds to the problem involving the algebra with involution:

If the mathematical problem we are trying to investigate involves elements of an algebra tex2html_wrap_inline3966 with involution, say tex2html_wrap_inline7842 , and we substitute tex2html_wrap_inline7882 for tex2html_wrap_inline7884 and tex2html_wrap_inline7886 for tex2html_wrap_inline4178 for tex2html_wrap_inline7890 and for tex2html_wrap_inline7892 , then

displaymath7894

There is a unique operation tex2html_wrap_inline7896 from tex2html_wrap_inline7898 to tex2html_wrap_inline4206 such that the following four conditions hold:
(1) tex2html_wrap_inline7896 is additive (i.e., if tex2html_wrap_inline7904 , then tex2html_wrap_inline7906 ).
(2) if tex2html_wrap_inline7904 , then tex2html_wrap_inline7910 .
(3) tex2html_wrap_inline7912 for tex2html_wrap_inline7914 .
(4) tex2html_wrap_inline7916 for tex2html_wrap_inline7918 .



Helton
Wed Jul 3 10:27:42 PDT 1996