NationMaster - Encyclopedia: Star algebra

In mathematics, a *-algebra is an associative algebra over the field of complex numbers with an antilinear antiautomorphism * : A->A which is an involution. More precisely, * is required to satisfy the following properties: for all a,b in A, and z a complex number,

$(x + y)^* = x^* + y^* \quad$
$(z x)^* = \overline{z} x^*$
$(x y)^* = y^* x^* \quad$
$(x^*)^* = x \quad$

The field of complex numbers C is a *-algebra with * being complex conjugation.

An algebra homomorphism f : A->B is a *-homomorphism if, in addition, is compatible with the involutions of A and B. What this means is that

$f (a *) = f (a) *$ for all a in A.

If a*=a, then a is called self-adjoint.

Categories: Abstract algebra

Results from FactBites:

Deformation Quantization Homepage: Selected (1772 words)
	Soon later Fedosov gave his geometric construction of star products on symplectic manifolds in 1985 which was unfortunately unnoticed till the early nineties, see his book for a detailed description [fedosov:1996a].
	Roughly, a strict deformation quantization of a (Poisson) C* algebra A is the data of a continuous field of C* algebras over an interval containing zero.
	This algebraic GNS construction is also the starting point for a general representation theory of the deformed algebras.

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