Talk:Gelfand–Naimark theorem

Latest comment: 4 years ago by 144.200.0.161 in topic Just the opposite

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)Reply

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)."

144.200.0.161 (talk) 20:53, 17 April 2020 (UTC)Reply