From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: Involution
Date: 20 Jun 1999 01:19:58 -0400
Newsgroups: sci.math
Keywords: What are C*-algebras?
In article ,
JOHN STANTON wrote:
:I was looking at an introduction to C*-algebra which began with a
:definition of involution, which looked like complex conjugation to me.
:What are examples of involution? What motivates the definition? I have
:looked around in the library but just can't get my fingers wrapped around
:the general concept. Thanks for any help with this.
You got half of the explanation: an involution is an abstraction of taking
the complex conjugate transpose of a matrix.
The basic examples of C*-algebras are B(H), the algebras of all continuous
linear operators on a Hilbert space H, for various choices of H.
Later on, with more theory in your toolbox, you will see that every
C*-algebra is a C*-subalgebra of some B(H) anyway.
And the involution in B(H) takes the operator adjoint. If you pick an
orthonormal basis of H and represent an operator T by its matrix, then the
adjoint of T is represented by the complex conjugate transpose of the
matrix representing T.
Finally, knowing the adjoint is useful, among others, because it relates
orthogonal complements, nullspaces and (closures of) ranges, and in
general, because we get better insight into the performance of an operator
by knowing how it composes with linear functionals (members of the dual
space).
In a condensed form, you can find these and much more advanced facts in
Kenneth R. Davidson: C*-algebras by Example,
AMS 1991, Providence, R.I., ISBN 0-8218-0599-1.
Hope it helps, ZVK(Slavek).