Editing C*-algebra (section)

== Some history: B*-algebras and C*-algebras ==
The term B*-algebra was introduced by [[Charles Earl Rickart|C. E. Rickart]] in 1946 to describe [[Banach *-algebra]]s that satisfy the condition:

* <math>\lVert x x^* \rVert = \lVert x \rVert ^2</math> for all ''x'' in the given B*-algebra. (B*-condition)

This condition automatically implies that the *-involution is isometric, that is, <math>\lVert x \rVert = \lVert x^* \rVert </math>. Hence, <math>\lVert xx^*\rVert = \lVert x \rVert \lVert x^*\rVert</math>, and therefore, a B*-algebra is also a C*-algebra. Conversely, the C*-condition implies the B*-condition. This is nontrivial, and can be proved without using the condition <math>\lVert x \rVert = \lVert x^* \rVert</math>.<ref>{{harvnb|Doran|Belfi|1986|pp=5–6}}, [https://books.google.com/books?id=6jNbsnJVjMoC&pg=PA5 Google Books].</ref> For these reasons, the term B*-algebra is rarely used in current terminology, and has been replaced by the term 'C*-algebra'.

The term C*-algebra was introduced by [[Irving Segal|I. E. Segal]] in 1947 to describe norm-closed subalgebras of ''B''(''H''), namely, the space of bounded operators on some Hilbert space ''H''. 'C' stood for 'closed'.<ref>{{harvnb|Doran|Belfi|1986|p=6}}, [https://books.google.com/books?id=6jNbsnJVjMoC&pg=PA6 Google Books].</ref><ref>{{harvnb|Segal|1947}}</ref> In his paper Segal defines a C*-algebra as a "uniformly closed, self-adjoint algebra of bounded operators on a Hilbert space".<ref>{{harvnb|Segal|1947|p=75}}</ref>