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- (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 other words, a Banach *-algebra is a Banach algebra over which is also a *-algebra.
In most natural examples, one also has that the involution is isometric, i.e.
- ||x*|| = ||x||,
Some authors include this isometric property in the definition of a Banach *-algebra.