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Representation of Algebras with Involution

Published online by Cambridge University Press:  20 November 2018

George Maxwell
George Maxwell*
Affiliation:
University of British Columbia, Vancouver, British Columbia
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Let K be a field with an involution J. A *-algebra over K is an associative algebra A with an involution * satisfying (α.a)* = αJ.a*. A large class of examples may be obtained as follows. Let (V, φ) be an hermitian space over K consisting of a vector space V and a left hermitian (w.r.t. J) form φ on V which is nondegenerate in the sense that φ(V,v) = 0 implies v = 0. An endomorphism f of V may have an adjoint f* w.r.t. φ, defined by φ(f(u),v) = φ(u,f*(v)); due to the nondegeneracy of φ, f* is unique if it exists. The set B(V, φ) of all endomorphisms of V which do have an adjoint is easily verified to be a *-algebra.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 4 , August 1972 , pp. 592 - 597
Copyright
Copyright © Canadian Mathematical Society 1972

References

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