Banach algebra

<h2 id="examples">Examples</h2>
The prototypical example of a Banach algebra is 
 
 
 
 
 C
 
 0
 
 
 (
 X
 )
 
 
 {\displaystyle C_{0}(X)}
 
, the space of (complex-valued) continuous functions, defined on a <a href="/facts/Locally_compact_Hausdorff_space/hesalT7q">locally compact Hausdorff space</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
, that <a href="/facts/Vanish_at_infinity/nCAl6pYS">vanish at infinity</a>. 
 
 
 
 
 C
 
 0
 
 
 (
 X
 )
 
 
 {\displaystyle C_{0}(X)}
 
 is unital if and only if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is <a href="/facts/Compactness/d0cgXJH7">compact</a>. The <a href="/facts/Complex_conjugation/7TEaoQDe">complex conjugation</a> being an <a href="/facts/Involution_(mathematics)/kKxYrUVa">involution</a>, 
 
 
 
 
 C
 
 0
 
 
 (
 X
 )
 
 
 {\displaystyle C_{0}(X)}
 
 is in fact a <a href="/facts/C*-algebra/B3tnGHIo">C*-algebra</a>. More generally, every C*-algebra is a Banach algebra by definition.

<ul><li>The set of real (or complex) numbers is a Banach algebra with norm given by the <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a>.</li>
<li>The set of all real or complex 
 
 
 
 n
 
 
 {\displaystyle n}
 
-by-
 
 
 
 n
 
 
 {\displaystyle n}
 
 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> becomes a <a href="/facts/Unital_algebra/8T8ulWJb">unital</a> Banach algebra if we equip it with a sub-multiplicative <a href="/facts/Matrix_norm/UJvsb9fF">matrix norm</a>.</li>
<li>Take the Banach space 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 (or 
 
 
 
 
 
 C
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {C} ^{n}}
 
) with norm 
 
 
 
 ‖
 x
 ‖
 =
 
 max

|
 
 
 x
 
 i
 
 
 
 |
 
 
 
 {\displaystyle \|x\|=\max _{}|x_{i}|}
 
 and define multiplication componentwise: 
 
 
 
 
 (
 
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 
 )
 
 
 (
 
 
 y
 
 1
 
 
 ,
 …
 ,
 
 y
 
 n
 
 
 
 )
 
 =
 
 (
 
 
 x
 
 1
 
 
 
 y
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 
 y
 
 n
 
 
 
 )
 
 .
 
 
 {\displaystyle \left(x_{1},\ldots ,x_{n}\right)\left(y_{1},\ldots ,y_{n}\right)=\left(x_{1}y_{1},\ldots ,x_{n}y_{n}\right).}
 
</li>
<li>The <a href="/facts/Quaternion/T3tnu7mX">quaternions</a> form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.</li>
<li>The algebra of all bounded real- or complex-valued functions defined on some set (with pointwise multiplication and the <a href="/facts/Supremum/oXCKVopz">supremum</a> norm) is a unital Banach algebra.</li>
<li>The algebra of all bounded <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous</a> real- or complex-valued functions on some <a href="/facts/Locally_compact_space/hesalT7q">locally compact space</a> (again with pointwise operations and supremum norm) is a Banach algebra.</li>
<li>The algebra of all <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous</a> <a href="/facts/Linear_transformation/Q1PBKNVV">linear</a> operators on a Banach space 
 
 
 
 E
 
 
 {\displaystyle E}
 
 (with functional composition as multiplication and the <a href="/facts/Operator_norm/a5l0TkFW">operator norm</a> as norm) is a unital Banach algebra. The set of all <a href="/facts/Compact_operator/KNkcK4mE">compact operators</a> on 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is a Banach algebra and closed ideal. It is without identity if 
 
 
 
 dim
 ⁡
 E
 =
 ∞
 .
 
 
 {\displaystyle \dim E=\infty .}
 
<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a></li>
<li>If 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is a <a href="/facts/Locally_compact/hesalT7q">locally compact</a> <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> <a href="/facts/Topological_group/cAgNMMRU">topological group</a> and 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is its <a href="/facts/Haar_measure/2IcmTcuU">Haar measure</a>, then the Banach space 
 
 
 
 
 L
 
 1
 
 
 (
 G
 )
 
 
 {\displaystyle L^{1}(G)}
 
 of all 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
-integrable functions on 
 
 
 
 G
 
 
 {\displaystyle G}
 
 becomes a Banach algebra under the <a href="/facts/Convolution/PVPrdz9J">convolution</a> 
 
 
 
 x
 y
 (
 g
 )
 =
 ∫
 x
 (
 h
 )
 y
 
 (
 
 
 h
 
 −
 1
 
 
 g
 
 )
 
 d
 μ
 (
 h
 )
 
 
 {\displaystyle xy(g)=\int x(h)y\left(h^{-1}g\right)d\mu (h)}
 
 for 
 
 
 
 x
 ,
 y
 ∈
 
 L
 
 1
 
 
 (
 G
 )
 .
 
 
 {\displaystyle x,y\in L^{1}(G).}
 
<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li>
<li><a href="/facts/Uniform_algebra/3Sdr8rwE">Uniform algebra</a>: A Banach algebra that is a subalgebra of the complex algebra 
 
 
 
 C
 (
 X
 )
 
 
 {\displaystyle C(X)}
 
 with the supremum norm and that contains the constants and separates the points of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (which must be a compact Hausdorff space).</li>
<li><a href="/facts/Uniform_algebra/3Sdr8rwE">Natural Banach function algebra</a>: A uniform algebra all of whose characters are evaluations at points of 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
</li>
<li><a href="/facts/C*-algebra/B3tnGHIo">C*-algebra</a>: A Banach algebra that is a closed *-subalgebra of the algebra of bounded operators on some <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a>.</li>
<li><a href="/facts/Measure_algebra/C1Qan3TE">Measure algebra</a>: A Banach algebra consisting of all <a href="/facts/Radon_measure/21adq3XN">Radon measures</a> on some <a href="/facts/Locally_compact_group/AGPC8vnY">locally compact group</a>, where the product of two measures is given by <a href="/facts/Convolution/PVPrdz9J">convolution of measures</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li>
<li>The algebra of the <a href="/facts/Quaternion/T3tnu7mX">quaternions</a> 
 
 
 
 
 H
 
 
 
 {\displaystyle \mathbb {H} }
 
 is a real Banach algebra, but it is not a complex algebra (and hence not a complex Banach algebra) for the simple reason that the center of the quaternions is the real numbers, which cannot contain a copy of the complex numbers.</li>
<li>An <a href="/facts/Affinoid_algebra/2p3fUBHc">affinoid algebra</a> is a certain kind of Banach algebra over a nonarchimedean field. Affinoid algebras are the basic building blocks in <a href="/facts/Rigid_analytic_space/2p3fUBHc">rigid analytic geometry</a>.</li></ul>
<h2 id="properties">Properties</h2>
Several <a href="/facts/List_of_functions/bua9ERtP">elementary functions</a> that are defined via <a href="/facts/Power_series/vYh6sTps">power series</a> may be defined in any unital Banach algebra; examples include the <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a> and the <a href="/facts/Trigonometric_functions/czej7ur0">trigonometric functions</a>, and more generally any <a href="/facts/Entire_function/XNluKzu7">entire function</a>. (In particular, the exponential map can be used to define <a href="/facts/Abstract_index_group/uAeK4g5m">abstract index groups</a>.) The formula for the <a href="/facts/Geometric_series/Ne5TBH57">geometric series</a> remains valid in general unital Banach algebras. The <a href="/facts/Binomial_theorem/hd4SmkUd">binomial theorem</a> also holds for two commuting elements of a Banach algebra.
The set of <a href="/facts/Invertible_element/GJSBu8Y7">invertible elements</a> in any unital Banach algebra is an <a href="/facts/Open_set/QfTCIqMu">open set</a>, and the inversion operation on this set is continuous (and hence is a homeomorphism), so that it forms a <a href="/facts/Topological_group/cAgNMMRU">topological group</a> under multiplication.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
If a Banach algebra has unit 
 
 
 
 
 1
 
 ,
 
 
 {\displaystyle \mathbf {1} ,}
 
 then 
 
 
 
 
 1
 
 
 
 {\displaystyle \mathbf {1} }
 
 cannot be a <a href="/facts/Commutator_(ring_theory)/eYxzFmFY">commutator</a>; that is, 
 
 
 
 x
 y
 −
 y
 x
 ≠
 
 1
 
 
 
 {\displaystyle xy-yx\neq \mathbf {1} }
 
  for any 
 
 
 
 x
 ,
 y
 ∈
 A
 .
 
 
 {\displaystyle x,y\in A.}
 
 This is because 
 
 
 
 x
 y
 
 
 {\displaystyle xy}
 
 and 
 
 
 
 y
 x
 
 
 {\displaystyle yx}
 
 have the same <a href="/facts/Spectrum_(functional_analysis)/cA1evTo6">spectrum</a> except possibly 
 
 
 
 0.
 
 
 {\displaystyle 0.}

The various algebras of functions given in the examples above have very different properties from standard examples of algebras such as the reals. For example:

<ul><li>Every real Banach algebra that is a <a href="/facts/Division_algebra/xp5Bepx3">division algebra</a> is isomorphic to the reals, the complexes, or the quaternions. Hence, the only complex Banach algebra that is a division algebra is the complexes. (This is known as the <a href="/facts/Gelfand%E2%80%93Mazur_theorem/9Je8DVak">Gelfand–Mazur theorem</a>.)</li>
<li>Every unital real Banach algebra with no <a href="/facts/Zero_divisor/5GraAU8o">zero divisors</a>, and in which every <a href="/facts/Principal_ideal/Cga66qgM">principal ideal</a> is <a href="/facts/Closed_set/upYnRTUj">closed</a>, is isomorphic to the reals, the complexes, or the quaternions.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li>
<li>Every commutative real unital <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian</a> Banach algebra with no zero divisors is isomorphic to the real or complex numbers.</li>
<li>Every commutative real unital Noetherian Banach algebra (possibly having zero divisors) is finite-dimensional.</li>
<li>Permanently singular elements in Banach algebras are <a href="/facts/Topological_divisor_of_zero/q44s7DH5">topological divisors of zero</a>, that is, considering extensions 
 
 
 
 B
 
 
 {\displaystyle B}
 
 of Banach algebras 
 
 
 
 A
 
 
 {\displaystyle A}
 
 some elements that are singular in the given algebra 
 
 
 
 A
 
 
 {\displaystyle A}
 
 have a multiplicative inverse element in a Banach algebra extension 
 
 
 
 B
 .
 
 
 {\displaystyle B.}
 
 Topological divisors of zero in 
 
 
 
 A
 
 
 {\displaystyle A}
 
 are permanently singular in any Banach extension 
 
 
 
 B
 
 
 {\displaystyle B}
 
 of 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
</li></ul>
<h2 id="spectral-theory">Spectral theory</h2>
Main article: <a href="/facts/Spectral_theory/8sAEEIMM">Spectral theory</a>
Unital Banach algebras over the complex field provide a general setting to develop spectral theory. The spectrum of an element 
 
 
 
 x
 ∈
 A
 ,
 
 
 {\displaystyle x\in A,}
 
 denoted by 
 
 
 
 σ
 (
 x
 )
 
 
 {\displaystyle \sigma (x)}
 
, consists of all those complex <a href="/facts/Scalar_(mathematics)/MkDbA4Eo">scalars</a> 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 such that 
 
 
 
 x
 −
 λ
 
 1
 
 
 
 {\displaystyle x-\lambda \mathbf {1} }
 
 is not invertible in 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
 The spectrum of any element 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is a closed subset of the closed disc in 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 with radius 
 
 
 
 ‖
 x
 ‖
 
 
 {\displaystyle \|x\|}
 
 and center 
 
 
 
 0
 ,
 
 
 {\displaystyle 0,}
 
 and thus is <a href="/facts/Compact_space/d0cgXJH7">compact</a>. Moreover, the spectrum 
 
 
 
 σ
 (
 x
 )
 
 
 {\displaystyle \sigma (x)}
 
 of an element 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is <a href="/facts/Non-empty/ffEk3eA6">non-empty</a> and satisfies the <a href="/facts/Spectral_radius/j8T2xrLu">spectral radius</a> formula:

sup
        {
        
          |
        
        λ
        
          |
        
        :
        λ
        ∈
        σ
        (
        x
        )
        }
        =
        
          lim
          
            n
            →
            ∞
          
        
        ‖
        
          x
          
            n
          
        
        
          ‖
          
            1
            
              /
            
            n
          
        
        .
      
    
    {\displaystyle \sup\{|\lambda |:\lambda \in \sigma (x)\}=\lim _{n\to \infty }\|x^{n}\|^{1/n}.}

Given 
 
 
 
 x
 ∈
 A
 ,
 
 
 {\displaystyle x\in A,}
 
 the <a href="/facts/Holomorphic_functional_calculus/xtWI9f4V">holomorphic functional calculus</a> allows to define 
 
 
 
 f
 (
 x
 )
 ∈
 A
 
 
 {\displaystyle f(x)\in A}
 
 for any function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a> in a neighborhood of 
 
 
 
 σ
 (
 x
 )
 .
 
 
 {\displaystyle \sigma (x).}
 
 Furthermore, the spectral mapping theorem holds:<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

σ
        (
        f
        (
        x
        )
        )
        =
        f
        (
        σ
        (
        x
        )
        )
        .
      
    
    {\displaystyle \sigma (f(x))=f(\sigma (x)).}

When the Banach algebra 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is the algebra 
 
 
 
 L
 (
 X
 )
 
 
 {\displaystyle L(X)}
 
 of bounded linear operators on a complex Banach space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (for example, the algebra of square matrices), the notion of the spectrum in 
 
 
 
 A
 
 
 {\displaystyle A}
 
 coincides with the usual one in <a href="/facts/Operator_theory/Ays1ofel">operator theory</a>. For 
 
 
 
 f
 ∈
 C
 (
 X
 )
 
 
 {\displaystyle f\in C(X)}
 
 (with a compact Hausdorff space 
 
 
 
 X
 
 
 {\displaystyle X}
 
), one sees that:

σ
        (
        f
        )
        =
        {
        f
        (
        t
        )
        :
        t
        ∈
        X
        }
        .
      
    
    {\displaystyle \sigma (f)=\{f(t):t\in X\}.}

The norm of a normal element 
 
 
 
 x
 
 
 {\displaystyle x}
 
 of a C*-algebra coincides with its spectral radius. This generalizes an analogous fact for normal operators.
Let 
 
 
 
 A
 
 
 {\displaystyle A}
 
 be a complex unital Banach algebra in which every non-zero element 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is invertible (a division algebra). For every 
 
 
 
 a
 ∈
 A
 ,
 
 
 {\displaystyle a\in A,}
 
 there is 
 
 
 
 λ
 ∈
 
 C
 
 
 
 {\displaystyle \lambda \in \mathbb {C} }
 
 such that

a
 −
 λ
 
 1
 
 
 
 {\displaystyle a-\lambda \mathbf {1} }
 
 is not invertible (because the spectrum of 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is not empty) hence 
 
 
 
 a
 =
 λ
 
 1
 
 :
 
 
 {\displaystyle a=\lambda \mathbf {1} :}
 
 this algebra 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is naturally isomorphic to 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 (the complex case of the Gelfand–Mazur theorem).

<h2 id="ideals-and-characters">Ideals and characters</h2>
Let 
 
 
 
 A
 
 
 {\displaystyle A}
 
 be a unital commutative Banach algebra over 
 
 
 
 
 C
 
 .
 
 
 {\displaystyle \mathbb {C} .}
 
 Since 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is then a commutative ring with unit, every non-invertible element of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 belongs to some <a href="/facts/Maximal_ideal/nf4DnKVS">maximal ideal</a> of 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
 Since a maximal ideal 
 
 
 
 
 
 m
 
 
 
 
 {\displaystyle {\mathfrak {m}}}
 
 in 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is closed, 
 
 
 
 A
 
 /
 
 
 
 m
 
 
 
 
 {\displaystyle A/{\mathfrak {m}}}
 
 is a Banach algebra that is a field, and it follows from the Gelfand–Mazur theorem that there is a bijection between the set of all maximal ideals of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 and the set 
 
 
 
 Δ
 (
 A
 )
 
 
 {\displaystyle \Delta (A)}
 
 of all nonzero homomorphisms from 
 
 
 
 A
 
 
 {\displaystyle A}
 
 to 
 
 
 
 
 C
 
 .
 
 
 {\displaystyle \mathbb {C} .}
 
 The set 
 
 
 
 Δ
 (
 A
 )
 
 
 {\displaystyle \Delta (A)}
 
 is called the "<a href="/facts/Structure_space/3BTHdpg2">structure space</a>" or "character space" of 
 
 
 
 A
 ,
 
 
 {\displaystyle A,}
 
 and its members "characters".
A character 
 
 
 
 χ
 
 
 {\displaystyle \chi }
 
 is a linear functional on 
 
 
 
 A
 
 
 {\displaystyle A}
 
 that is at the same time multiplicative, 
 
 
 
 χ
 (
 a
 b
 )
 =
 χ
 (
 a
 )
 χ
 (
 b
 )
 ,
 
 
 {\displaystyle \chi (ab)=\chi (a)\chi (b),}
 
 and satisfies 
 
 
 
 χ
 (
 
 1
 
 )
 =
 1.
 
 
 {\displaystyle \chi (\mathbf {1} )=1.}
 
 Every character is automatically continuous from 
 
 
 
 A
 
 
 {\displaystyle A}
 
 to 
 
 
 
 
 C
 
 ,
 
 
 {\displaystyle \mathbb {C} ,}
 
 since the kernel of a character is a maximal ideal, which is closed. Moreover, the norm (that is, operator norm) of a character is one. Equipped with the topology of pointwise convergence on 
 
 
 
 A
 
 
 {\displaystyle A}
 
 (that is, the topology induced by the weak-* topology of 
 
 
 
 
 A
 
 ∗
 
 
 
 
 {\displaystyle A^{*}}
 
), the character space, 
 
 
 
 Δ
 (
 A
 )
 ,
 
 
 {\displaystyle \Delta (A),}
 
 is a Hausdorff compact space.
For any 
 
 
 
 x
 ∈
 A
 ,
 
 
 {\displaystyle x\in A,}

σ
        (
        x
        )
        =
        σ
        (
        
          
            
              x
              ^
            
          
        
        )
      
    
    {\displaystyle \sigma (x)=\sigma ({\hat {x}})}

where 
 
 
 
 
 
 
 x
 ^
 
 
 
 
 
 {\displaystyle {\hat {x}}}
 
 is the <a href="/facts/Gelfand_representation/QV54f7Jm">Gelfand representation</a> of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 defined as follows: 
 
 
 
 
 
 
 x
 ^
 
 
 
 
 
 {\displaystyle {\hat {x}}}
 
 is the continuous function from 
 
 
 
 Δ
 (
 A
 )
 
 
 {\displaystyle \Delta (A)}
 
 to 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 given by 
 
 
 
 
 
 
 x
 ^
 
 
 
 (
 χ
 )
 =
 χ
 (
 x
 )
 .
 
 
 {\displaystyle {\hat {x}}(\chi )=\chi (x).}
 
 The spectrum of 
 
 
 
 
 
 
 x
 ^
 
 
 
 ,
 
 
 {\displaystyle {\hat {x}},}
 
 in the formula above, is the spectrum as element of the algebra 
 
 
 
 C
 (
 Δ
 (
 A
 )
 )
 
 
 {\displaystyle C(\Delta (A))}
 
 of complex continuous functions on the compact space 
 
 
 
 Δ
 (
 A
 )
 .
 
 
 {\displaystyle \Delta (A).}
 
 Explicitly,

σ
        (
        
          
            
              x
              ^
            
          
        
        )
        =
        {
        χ
        (
        x
        )
        :
        χ
        ∈
        Δ
        (
        A
        )
        }
        .
      
    
    {\displaystyle \sigma ({\hat {x}})=\{\chi (x):\chi \in \Delta (A)\}.}

As an algebra, a unital commutative Banach algebra is <a href="/facts/Semisimple_algebra/JIXNLzck">semisimple</a> (that is, its <a href="/facts/Jacobson_radical/tTTfKySl">Jacobson radical</a> is zero) if and only if its Gelfand representation has trivial kernel. An important example of such an algebra is a commutative C*-algebra. In fact, when 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism between 
 
 
 
 A
 
 
 {\displaystyle A}
 
 and 
 
 
 
 C
 (
 Δ
 (
 A
 )
 )
 .
 
 
 {\displaystyle C(\Delta (A)).}
 
<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="banach--algebras">Banach *-algebras</h2>
A Banach *-algebra 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a Banach algebra over the field of <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, together with a map

∗
 
 
 :
 A
 →
 A
 
 
 {\displaystyle {}^{*}:A\to A}
 
 that has the following properties:

<ol><li>
 
 
 
 
 
 (
 
 x
 
 ∗
 
 
 )
 
 
 ∗
 
 
 =
 x
 
 
 {\displaystyle \left(x^{*}\right)^{*}=x}
 
 for all 
 
 
 
 x
 ∈
 A
 
 
 {\displaystyle x\in A}
 
 (so the map is an <a href="/facts/Involution_(mathematics)/kKxYrUVa">involution</a>).</li>
<li>
 
 
 
 (
 x
 +
 y
 
 )
 
 ∗
 
 
 =
 
 x
 
 ∗
 
 
 +
 
 y
 
 ∗
 
 
 
 
 {\displaystyle (x+y)^{*}=x^{*}+y^{*}}
 
 for all 
 
 
 
 x
 ,
 y
 ∈
 A
 .
 
 
 {\displaystyle x,y\in A.}
 
</li>
<li>
 
 
 
 (
 λ
 x
 
 )
 
 ∗
 
 
 =
 
 
 
 λ
 ¯
 
 
 
 
 x
 
 ∗
 
 
 
 
 {\displaystyle (\lambda x)^{*}={\bar {\lambda }}x^{*}}
 
 for every 
 
 
 
 λ
 ∈
 
 C
 
 
 
 {\displaystyle \lambda \in \mathbb {C} }
 
 and every 
 
 
 
 x
 ∈
 A
 ;
 
 
 {\displaystyle x\in A;}
 
 here, 
 
 
 
 
 
 
 λ
 ¯
 
 
 
 
 
 {\displaystyle {\bar {\lambda }}}
 
 denotes the <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> of 
 
 
 
 λ
 .
 
 
 {\displaystyle \lambda .}
 
</li>
<li>
 
 
 
 (
 x
 y
 
 )
 
 ∗
 
 
 =
 
 y
 
 ∗
 
 
 
 x
 
 ∗
 
 
 
 
 {\displaystyle (xy)^{*}=y^{*}x^{*}}
 
 for all 
 
 
 
 x
 ,
 y
 ∈
 A
 .
 
 
 {\displaystyle x,y\in A.}
 
</li></ol>
In other words, a Banach *-algebra is a Banach algebra over 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 that is also a <a href="/facts/*-algebra/vcE5TWnT">*-algebra</a>.
In most natural examples, one also has that the involution is <a href="/facts/Isometry/K5wX8ah6">isometric</a>, that is,

‖
        
          x
          
            ∗
          
        
        ‖
        =
        ‖
        x
        ‖
        
        
           for all 
        
        x
        ∈
        A
        .
      
    
    {\displaystyle \|x^{*}\|=\|x\|\quad {\text{ for all }}x\in A.}

Some authors include this isometric property in the definition of a Banach *-algebra.
A Banach *-algebra satisfying 
 
 
 
 ‖
 
 x
 
 ∗
 
 
 x
 ‖
 =
 ‖
 
 x
 
 ∗
 
 
 ‖
 ‖
 x
 ‖
 
 
 {\displaystyle \|x^{*}x\|=\|x^{*}\|\|x\|}
 
 is a <a href="/facts/C*-algebra/B3tnGHIo">C*-algebra</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Approximate_identity/xjNY7eol">Approximate identity</a> – net in a normed algebra that acts as a substitute for an identity elementPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Kaplansky%27s_conjecture/uP0oXvJf">Kaplansky's conjecture</a> – Numerous conjectures by mathematician Irving KaplanskyPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Operator_algebra/4fycDNQ1">Operator algebra</a> – Branch of functional analysis</li>
<li><a href="/facts/Shilov_boundary/423ceqlR">Shilov boundary</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/B%C3%A9la_Bollob%C3%A1s/9T1eot5o">Bollobás, B</a> (1990). <a href="https://archive.org/details/linearanalysisin0000boll">Linear Analysis</a>. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-38729-9.</li>
<li><a href="/facts/Frank_Bonsall/4BzyF9ju">Bonsall, F. F.</a>; Duncan, J. (1973). Complete Normed Algebras. New York: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-06386-2.</li>
<li><a href="/facts/John_B._Conway/7aMjnAxr">Conway, J. B.</a> (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96. <a href="/facts/Springer_Verlag/nAesf6nT">Springer Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-97245-5.</li>
<li>Dales, H. G.; Aeina, P.; Eschmeier, J; Laursen, K.; <a href="/facts/George_A._Willis/TUBunqSI">Willis, G. A.</a> (2003). Introduction to Banach Algebras, Operators and Harmonic Analysis. Cambridge University Press. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FCBO9780511615429">10.1017/CBO9780511615429</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-53584-0.</li>
<li>Mosak, R. D. (1975). Banach algebras. Chicago Lectures in Mathematics. University of Chicago Press). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-226-54203-3.</li>
<li><a href="/facts/Masamichi_Takesaki/ktcTmDux">Takesaki, M.</a> (1979). Theory of Operator Algebras I. Encyclopaedia of Mathematical Sciences. Vol. 124 (1st ed.). Berlin Heidelberg: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-42248-8. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0938-0396">0938-0396</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Conway 1990, Example VII.1.8. - Conway, J. B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96. Springer Verlag. ISBN 0-387-97245-5. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Conway 1990, Example VII.1.9. - Conway, J. B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96. Springer Verlag. ISBN 0-387-97245-5. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Conway 1990, Example VII.1.9. - Conway, J. B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96. Springer Verlag. ISBN 0-387-97245-5. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Conway 1990, Theorem VII.2.2. - Conway, J. B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96. Springer Verlag. ISBN 0-387-97245-5. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">García, Miguel Cabrera; Palacios, Angel Rodríguez (1995). "A New Simple Proof of the Gelfand-Mazur-Kaplansky Theorem". Proceedings of the American Mathematical Society. 123 (9): 2663–2666. doi:10.2307/2160559. ISSN 0002-9939. JSTOR 2160559. <a href="https://www.jstor.org/stable/2160559" target="_blank">https://www.jstor.org/stable/2160559</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Takesaki 1979, Proposition 2.8. - Takesaki, M. (1979). Theory of Operator Algebras I. Encyclopaedia of Mathematical Sciences. Vol. 124 (1st ed.). Berlin Heidelberg: Springer-Verlag. ISBN 978-3-540-42248-8. ISSN 0938-0396. <a href="https://search.worldcat.org/issn/0938-0396" target="_blank">https://search.worldcat.org/issn/0938-0396</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Proof: Since every element of a commutative C*-algebra is normal, the Gelfand representation is isometric; in particular, it is injective and its image is closed. But the image of the Gelfand representation is dense by the Stone–Weierstrass theorem. <a href="/wiki/Stone%E2%80%93Weierstrass_theorem" target="_blank">/wiki/Stone%E2%80%93Weierstrass_theorem</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Banach algebra open-in-new

Banach algebra