Talk:Gelfand–Naimark theorem
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Hm, seems to me the article's got some problems. In particular, the statement that "...Gelfand–Naimark representation depends only on the GNS construction and...In general it will not be a faithful representation." is confusing and misleading, since the very content of the theorem is that every C*-algebra is isomorphic to a norm closed *-algebra of operator. The GNS construction for a single state will not yield a faithful representation, and I assume that's what the article meant. Also, the "C*-seminorm" as defined is a norm, in fact the very C*-norm that is on A to begin with. This is again the content of the theorem. Mct mht (talk) 11:58, 30 December 2010 (UTC)
Just the opposite
edit"This result was proven by Israel Gelfand and Mark Naimark in 1943 and was a significant point in the development of the theory of C*-algebras since it established the possibility of considering a C*-algebra as an abstract algebraic entity without reference to particular realizations as an operator algebra."
Just the opposite ! This statement is fungey !!
You can "consider" abstract C*-algebra abstractly, without needing the Gelfand-Naimark to justify your work.
How about:
"This result was proven by Israel Gelfand and Mark Naimark in 1943 and was a significant point in the development of the theory of C*-algebras, since it shows that to obtain a result for abstract C*-algebra it suffices to prove it for concrete C*-algebras (i.e. operator algebras on a Hilbert space)."