*-ALGEBRA

(Redirected from Star-algebra)
In mathematics, a '
★ -ring' is an associative ring with an antilinear, antiautomorphism
★ : ''A'' → ''A'' which is an involution. More precisely,
★ is required to satisfy the following properties:

★ $(x\; +\; y)^$
★ = x^
★ + y^
★ quad

★ $(x\; y)^$
★ = y^
★ x^
★ quad

★ $(x^$
★ )^
★ = x quad
for all ''x'',''y'' in ''A''.
A '
★ -algebra' is a
★ -ring that is an associative algebra over another
★ -ring, usually the
★ -ring of complex numbers (with
★ acting as complex conjugation).
The most obvious example of a
★ -algebra is the field of complex numbers 'C' where
★ is just complex conjugation. Another example is the algebra of ''n''×''n'' matrices over 'C' with
★ given by the conjugate transpose. Its generalization, the Hermitian adjoint of a linear operator on a Hilbert space is also a star-algebra.
An algebra homomorphism ''f'' : ''A'' → ''B'' is a '
★ -homomorphism' if it is compatible with the involutions of ''A'' and ''B'', i.e.

★ $f(a^$
★ ) = f(a)^
★ for all ''a'' in ''A''.
An element ''a'' in ''A'' is called self-adjoint if ''a''
★ = ''a''.

Contents

See also

See also

★ B
★ -algebra

★ C
★ -algebra

★ von Neumann algebra

★ Baer ring

★ operator algebra