Banach *-algebra

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A Banach *-algebra A is a Banach algebra over the field of complex numbers, together with a map * : AA, called involution, that has the following properties:

  1. (x + y)* = x* + y* for all x, y in A.
  2. (\lambda x)^* = \bar{\lambda}x^* for every λ in C and every x in A; here, \bar{\lambda} denotes the complex conjugate of λ.
  3. (xy)* = y* x* for all x, y in A.
  4. (x*)* = x for all x in A.

In most natural examples, one also has that the involution is isometric, i.e.

  • ||x*|| = ||x||,

See also[edit]