Quark–lepton complementarity: Difference between revisions
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A '''Banach *-algebra''' ''A'' is a [[Banach algebra]] over the field of [[complex number]]s, together with a map * : ''A'' → ''A'' called ''[[Involution (mathematics)|involution]]'' which has the following properties: | |||
# (''x'' + ''y'')* = ''x''* + ''y''* for all ''x'', ''y'' in ''A''. | |||
# <math>(\lambda x)^* = \bar{\lambda}x^*</math> for every λ in '''C''' and every ''x'' in ''A''; here, <math>\bar{\lambda}</math> denotes the complex conjugate of λ. | |||
# (''xy'')* = ''y''* ''x''* for all ''x'', ''y'' in ''A''. | |||
# (''x''*)* = ''x'' for all ''x'' in ''A''. | |||
In most natural examples, one also has that the involution is [[isometry|isometric]], i.e. | |||
* ||''x''*|| = ||''x''||, | |||
==See also== | |||
*[[Algebra over a field]] | |||
*[[Associative algebra]] | |||
*[[*-algebra]] | |||
*[[C*-algebra]]. | |||
[[Category:Banach algebras]] |
Revision as of 01:48, 19 October 2013
A Banach *-algebra A is a Banach algebra over the field of complex numbers, together with a map * : A → A called involution which has the following properties:
- (x + y)* = x* + y* for all x, y in A.
- for every λ in C and every x in A; here, denotes the complex conjugate of λ.
- (xy)* = y* x* for all x, y in A.
- (x*)* = x for all x in A.
In most natural examples, one also has that the involution is isometric, i.e.
- ||x*|| = ||x||,