*-algebra

* -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

\ (x+y)^{*}=x^{*}+y^{*}

{\ displaystyle \ (x + y) ^ {*} = x ^ {*} + y ^ {*}}

\ (xy)^{*}=y^{*}x^{*}

{\ displaystyle \ (xy) ^ {*} = y ^ {*} x ^ {*}}

\ 1^{*}=1

{\ displaystyle \ 1 ^ {*} = 1}

and involution that is

\ (x^{*})^{*}=x.

{\ displaystyle \ (x ^ {*}) ^ {*} = x.}

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 $R\subset A.$ ${\ displaystyle R \ subset A.}$

The base * -ring is usually complex numbers (where * is a complex conjugate).

Then * conjugate-linear, i.e.

(\lambda x+\mu y)^{*}=\lambda ^{*}x^{*}+\mu ^{*}y^{*}\quad \lambda ,\mu \in R;\;\;x,y\in A

{\ displaystyle (\ lambda x + \ mu y) ^ {*} = \ lambda ^ {*} x ^ {*} + \ mu ^ {*} y ^ {*} \ quad \ lambda, \ mu \ in R; \; \; x, y \ in A}

.

* -homomorphism $\ f:A\to B$ ${\ displaystyle \ f: A \ to B}$ Is a homomorphism of algebras that maps an involution in A to an involution in B , that is:

f(x^{*})=f(x)^{*}\quad \forall x\in A.

{\ displaystyle f (x ^ {*}) = f (x) ^ {*} \ quad \ forall x \ in A.}

Items for which $\ x^{*}=x$ ${\ displaystyle \ x ^ {*} = x}$ are called self-conjugate , symmetric or hermitian .
Items for which $\ x^{*}=-x$ ${\ displaystyle \ x ^ {*} = - x}$ are called skew-conjugate , anti-symmetric or anti-hermitian .
You can define a Hermitian form using the operation * in the form $\phi (x,y)=x^{*}\cdot y$ ${\ displaystyle \ phi (x, y) = x ^ {*} \ cdot y}$ .

C * Algebra

C * -algebra - Banach * -algebra over the field of complex numbers, for which the C * -property is satisfied:

\|x^{*}x\|=\|x\|\|x^{*}\|,

{\ displaystyle \ | x ^ {*} x \ | = \ | x \ | \ | x ^ {*} \ |}

\|xx^{*}\|=\|x\|\|x^{*}\|.

{\ displaystyle \ | xx ^ {*} \ | = \ | x \ | \ | x ^ {*} \ |.}

Both conditions are equivalent.

They are also equivalent to the * property

\|xx^{*}\|=\|x\|^{2}.

{\ displaystyle \ | xx ^ {*} \ | = \ | x \ | ^ {2}.}

Examples

The most famous example is complex numbers. $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ 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 ${\frac {1}{2}}(1-*)$ ${\ displaystyle {\ frac {1} {2}} (1- *)}$ and ${\frac {1}{2}}(1+*)$ ${\ displaystyle {\ frac {1} {2}} (1 + *)}$ 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 ).

Bibliography

H. G. Dales, Banach algebras and automatic continuity , Clarence Press, Oxford, 2000, p. 142-150.