* -algebra (an algebra with an involution , an algebra with a conjugation operation ) is an associative algebra with an involution that has properties similar to complex conjugation .
Content
* ring
* -ring - ring with unary operation *, which is
- anti-automorphism that is
- and involution that is
Such a ring is also called a ring with involution .
* algebra
* -algebra A is a * -ring, which is an associative algebra over another * -ring R , with the agreement of the operation * in
The base * -ring is usually complex numbers (where * is a complex conjugate).
Then * conjugate-linear, i.e.
- .
* -homomorphism Is a homomorphism of algebras that maps an involution in A to an involution in B , that is:
- Items for which are called self-conjugate , symmetric or hermitian .
- Items for which are called skew-conjugate , anti-symmetric or anti-hermitian .
- You can define a Hermitian form using the operation * in the form .
C * Algebra
C * -algebra - Banach * -algebra over the field of complex numbers, for which the C * -property is satisfied:
Both conditions are equivalent.
They are also equivalent to the * property
Examples
- The most famous example is complex numbers. with pairing operation.
- Square matrices with complex elements with the operation of Hermitian conjugation .
- Hermitian conjugations of a linear operator in a Hilbert space .
Properties
Many conjugation properties for complex numbers are stored in * -algebras:
- If the element 2 in the ring is reversible , then and is orthogonal idempotents . As a vector space , algebra decomposes into a direct sum of subspaces of symmetric and anti-symmetric (Hermitian and anti-Hermitian) elements.
- Hermitian elements * -algebras form Jordan algebra .
- The anti-Hermitian elements of an * -algebra form a Lie algebra .
Legend
The involution operation is usually written as an asterisk ( asterisk ), indicated after the operand, which is at the level of the midline or slightly raised above it:
- x ↦ x *
or
- x ↦ x ∗ ( Τ Ε Χ :
x^*
),
but not “ x ∗ ” because the star symbol for binary operations is below the centerline. Sometimes the superscript feature x is also used, as in complex conjugation, or x † (raised typographical cross ).
See also
- Operator algebras
- Cayley – Dixon procedure sometimes builds * algebra
Bibliography
- H. G. Dales, Banach algebras and automatic continuity , Clarence Press, Oxford, 2000, p. 142-150.