Banach Algebra

A Banach algebra over a complex or real field is an associative algebra , which is a Banach space . In this case, the multiplication in it must be consistent with the norm:

\forall x,y\in A,\|x\,y\|\ \leq \|x\|\,\|y\|

{\ displaystyle \ forall x, y \ in A, \ | x \, y \ | \ \ leq \ | x \ | \, \ | y \ |}

\ forall x, y \ in A, \ | x \, y \ | \ \ leq \ | x \ | \, \ | y \ |

.

This property is required for the continuity of the operation of multiplication relative to the norm.

A Banach algebra is called a unitary or Banach algebra with a unit if it has a unit (i.e. such an element $\mathbf {1}$ ${\ displaystyle \ mathbf {1}}$ $\ mathbf {1}$ that for everyone $x\in A$ ${\ displaystyle x \ in A}$ $x \ in A$ rightly $x\mathbf {1} =\mathbf {1} x=x$ ${\ displaystyle x \ mathbf {1} = \ mathbf {1} x = x}$ $x \ mathbf {1} = \ mathbf {1} x = x$ ) It usually requires that the norm of the unit be equal to 1. If the unit exists, then it is unique. Every Banach algebra $A$ ${\ displaystyle A}$ $A$ can be isometrically embedded in the corresponding unital Banach algebra $A_{e}$ ${\ displaystyle A_ {e}}$ $A_e$ as a closed bilateral ideal .

A Banach algebra is called commutative if the multiplication operation in it is commutative .

Examples

Fields of complex numbers or real numbers - $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ and $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ relative to standard addition and multiplication operations. These are unitary commutative algebras.
Algebras of complex or real matrices with respect to matrix multiplication and submultiplicative matrix norms .
The quaternion algebra is a real algebra with a norm - a module.
$C(\Omega )$ ${\ displaystyle C (\ Omega)}$ - the algebra of continuous functions on a compact with respect to pointwise multiplication with respect to the sup- norm . A more general example is $C_{0}(\Omega )$ ${\ displaystyle C_ {0} (\ Omega)}$ Is the space of complex-valued functions disappearing at infinity, where $\Omega$ ${\ displaystyle \ Omega}$ - locally compact space .
Algebra of bounded operators acting in a Banach space, with respect to the operator norm and composition as multiplication. The set of compact operators with respect to the same operations is a closed ideal in this algebra.
If a $G$ ${\ displaystyle G}$ Is a locally compact Hausdorff topological group with Haar measure $\mu$ ${\ displaystyle \ mu}$ then the Banach space $L_{1}(G)$ ${\ displaystyle L_ {1} (G)}$ integrable with respect to the measure $\mu$ ${\ displaystyle \ mu}$ complex-valued functions on $G$ ${\ displaystyle G}$ is a Banach algebra with respect to multiplication-convolution, defined by the formula

(xy)(g)=\int _{G}x(h)y(h^{-1}g)\,\mathrm {d} \mu (h),\;g\in G

{\ displaystyle (xy) (g) = \ int _ {G} x (h) y (h ^ {- 1} g) \, \ mathrm {d} \ mu (h), \; g \ in G}

.

$L_{1}(\mathbb {R} )$ ${\ displaystyle L_ {1} (\ mathbb {R})}$ - algebra of functions summable on a line with convolution as multiplication. This is a special case of the previous example.
C * -algebra is an algebra with * - involution consistent with the norm: $\forall a\ ||a^{*}a||=||a||^{2}$ ${\ displaystyle \ forall a \ || a ^ {*} a || = || a || ^ {2}}$

Properties

Some elementary functions can be determined using power series for elements of a Banach algebra. In particular, it is possible to determine the exponent of an element of a Banach algebra, trigonometric functions, and, in the general case, any entire function . For elements of a Banach algebra, the formula for the sum of an infinitely decreasing geometric progression ( Neumann series ) remains valid.

Many reversible elements $\mathrm {Inv} (A)$ ${\ displaystyle \ mathrm {Inv} (A)}$ algebras $A$ ${\ displaystyle A}$ is an open set. Moreover, the mapping $\mathrm {Inv}$ ${\ displaystyle \ mathrm {Inv}}$ associating an inverse to each invertible element is a homeomorphism . In this way, $\mathrm {Inv} (A)$ ${\ displaystyle \ mathrm {Inv} (A)}$ - topological group.

In unitary algebra, a unit cannot be a commutator: $xy-yx\neq \mathbf {1}$ ${\ displaystyle xy-yx \ neq \ mathbf {1}}$ for any x , y ∈ A. It follows that $\lambda \mathbf {1} ,\ \lambda \neq 0$ ${\ displaystyle \ lambda \ mathbf {1}, \ \ lambda \ neq 0}$ also is not a switch.

The Gelfand – Mazur theorem holds : every unital complex Banach algebra in which all nonzero elements are invertible is isomorphic $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ .

Spectral Theory

In unitary Banach algebras, the concept of spectrum is introduced, which extends the concept of the spectrum of an operator to a more general class of objects.

Element $a\in A$ ${\ displaystyle a \ in A}$ algebras $A$ ${\ displaystyle A}$ called reversible if there is such an element $a^{-1}\in A$ ${\ displaystyle a ^ {- 1} \ in A}$ , what $aa^{-1}=a^{-1}a=\mathbf {1}$ ${\ displaystyle aa ^ {- 1} = a ^ {- 1} a = \ mathbf {1}}$ . Spectrum $\sigma (a)$ ${\ displaystyle \ sigma (a)}$ element $a$ ${\ displaystyle a}$ called a lot of such $\lambda \in \mathbb {C} ,$ ${\ displaystyle \ lambda \ in \ mathbb {C},}$ what element $a-\lambda \mathbf {1}$ ${\ displaystyle a- \ lambda \ mathbf {1}}$ irreversible. The spectrum of every element of a unital complex Banach algebra is a nonempty compact. On the other hand, for any compact $K\subset \mathbb {C}$ ${\ displaystyle K \ subset \ mathbb {C}}$ element spectrum $w$ ${\ displaystyle w}$ from algebra $C(K)$ ${\ displaystyle C (K)}$ defined by the formula $w(z)=z$ ${\ displaystyle w (z) = z}$ coincides with $K$ ${\ displaystyle K}$ , therefore, there are no other restrictions on the spectrum of an element in an arbitrary Banach algebra.

Spectral radius $\mathrm {r} (x)$ ${\ displaystyle \ mathrm {r} (x)}$ element $x\in A$ ${\ displaystyle x \ in A}$ called quantity

\mathrm {r} (x)=\sup\{|\lambda |:\lambda \in \sigma (x)\}

{\ displaystyle \ mathrm {r} (x) = \ sup \ {| \ lambda |: \ lambda \ in \ sigma (x) \}}

.

The Burling-Gelfand formula for the spectral radius is valid :

\mathrm {r} (x)=\lim _{n\to \infty }\|x^{n}\|^{1/n}.

{\ displaystyle \ mathrm {r} (x) = \ lim _ {n \ to \ infty} \ | x ^ {n} \ | ^ {1 / n}.}

Resolvent set element $a\in A$ ${\ displaystyle a \ in A}$ called set $\rho (a)=\mathbb {C} \setminus \sigma (a)$ ${\ displaystyle \ rho (a) = \ mathbb {C} \ setminus \ sigma (a)}$ . The resolvent set of an element of a Banach algebra is always open. Resolvent element $a\in A$ ${\ displaystyle a \ in A}$ called a function of a complex variable $R_{a}\colon \rho (a)\to A$ ${\ displaystyle R_ {a} \ colon \ rho (a) \ to A}$ defined by the formula $R_{a}(\lambda )=(\lambda \mathbf {1} -a)^{-1}$ ${\ displaystyle R_ {a} (\ lambda) = (\ lambda \ mathbf {1} -a) ^ {- 1}}$ . The resolvent of an element of a Banach algebra is a holomorphic function .

If a $f$ ${\ displaystyle f}$ - holomorphic in the neighborhood $D\subset \mathbb {C}$ ${\ displaystyle D \ subset \ mathbb {C}}$ spectrum $\sigma (a)$ ${\ displaystyle \ sigma (a)}$ function, you can define $f(a)\in A$ ${\ displaystyle f (a) \ in A}$ according to the formula

f(a)={\frac {1}{2\pi i}}\int _{\gamma }f(\lambda )R_{a}(\lambda )\,\mathrm {d} \lambda

{\ displaystyle f (a) = {\ frac {1} {2 \ pi i}} \ int _ {\ gamma} f (\ lambda) R_ {a} (\ lambda) \, \ mathrm {d} \ lambda }

,

Where $\gamma$ ${\ displaystyle \ gamma}$ - rectifiable Jordan contour lying in $D$ ${\ displaystyle D}$ containing the spectrum of the element $x$ ${\ displaystyle x}$ and oriented positively as well $R_{a}$ ${\ displaystyle R_ {a}}$ - element resolvent $a$ ${\ displaystyle a}$ . In particular, using this formula, we can determine the exponential of an element from a Banach algebra.

Ideals and characters

Let A be a unital commutative Banach algebra over a field of complex numbers. A character χ of an algebra A is a nonzero linear functional possessing the multiplicativity property: for any a , b ∈ A , χ ( ab ) = χ ( a ) χ ( b ) and χ ( 1 ) = 1 holds. That is, the character is a nonzero homomorphism of algebras A and $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ . One can verify that every character in a Banach algebra is continuous and its norm is 1.

The core of the character is the maximum ideal in A. If a ${\mathfrak {m}}$ ${\ displaystyle {\ mathfrak {m}}}$ Is the maximum ideal, then the quotient algebra $A/{\mathfrak {m}}$ ${\ displaystyle A / {\ mathfrak {m}}}$ is a field and a Banach algebra, then, by the Gelfand-Mazur theorem, it is isomorphic $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ . Therefore, each maximum ideal ${\mathfrak {m}}$ ${\ displaystyle {\ mathfrak {m}}}$ one can associate a unique character χ such that ker χ = ${\mathfrak {m}}$ ${\ displaystyle {\ mathfrak {m}}}$ . This character is defined as a composition of factor mapping and isomorphism. $A/{\mathfrak {m}}$ ${\ displaystyle A / {\ mathfrak {m}}}$ at $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ . Thus, between the set of characters and the set of maximum ideals, a bijection is established.

The set of all characters is called the space of maximal ideals or the spectrum of the algebra A and is denoted by Spec A. This set can be endowed with a topology inherited from the weak * topology (the topology of pointwise convergence) in the adjoint space A ^* . It follows from the Banach-Alaoglu theorem and the closeness of Spec A that Spec A is a compact Hausdorff topological space .

Gelfand Element Transformation $a$ ${\ displaystyle a}$ algebra A is called a continuous function ${\hat {a}}\colon \mathrm {Spec} \,A\to \mathbb {C}$ ${\ displaystyle {\ hat {a}} \ colon \ mathrm {Spec} \, A \ to \ mathbb {C}}$ determined by the formula ${\hat {a}}(\chi )=\chi (a)$ ${\ displaystyle {\ hat {a}} (\ chi) = \ chi (a)}$ for all characters χ. The Gelfand transform carries out a contracting homomorphism of the algebra A into the algebra C (Spec A) of continuous functions on a compact set.

The radical of A is the intersection of all its maximal ideals. If the radical consists only of zero, the algebra A is called semisimple . The kernel of the Gelfand transform coincides with the radical of the algebra; therefore, the Gelfand transform is injective if and only if the algebra A is semisimple. Thus, every semisimple commutative Banach algebra with unity coincides, up to isomorphism, with some algebra of functions continuous on a compact - with the image of the Gelfand transform.

Literature

Naimark M. A. Normalized rings. - M .: Nauka, 1968 .-- 664 p.
Khelemsky A. Ya. Lectures on functional analysis. - M .: MCCNMO, 2004 .-- ISBN 5-94057-065-8 .
Khelemsky A. Ya. Banakhovs and polynormalized algebras: general theory, representations, homology. - M .: Nauka, 1989 .-- ISBN 5-02-014192-5 .