Notation

In this site we use X^{#} for the topological dual of a normed spaceX, S′ for the commutant of a subset S of B(H) and T^{∗} for the Hilbert adjoint of an operator T in B(H) for any Hilbert space H.

Main Examples

(I) The set of complex numbers C with usual addition, multiplication and the absolute value as a norm is a unital commutative Banach algebra.

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(II) C^{n} with the coordinatewise addition, scalar multiplication and the inner product

< (z_{1}, …, z_{n}), (w_{1}, …, w_{n}) >= ∑_{i=1}^{n} z_{i}\overline{w_{i}}  (1)

is a Hilbert space.

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(III) The space C^{2} (see (II)) with the product (a, b)(a′, b′) = (aa′, ab′+a′b) is a unital commutative Banach algebra.

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(IV) Let X be a non-empty set and Y is a normed (Banach) space. Then the set l^{∞}(X, Y) of all bounded mappings of X into Y with the pointwise addition (f + g)(x) = f(x) + g(x), x ∈ X; poinwise scalar multiplication (λf)(x) = λf(x), λ ∈ C, x ∈ X; and supremum norm ∥f∥ = sup{|f(x)|; x ∈ X} is a normed (Banach) space. If Y is normed algebra then l^{∞}(X, Y) with the pointwise product (fg)(x) = f(x)g(x) is a normed algebra. We denote l^{∞}(E, C) with l^{∞}(E) that is a unital cammutative C^{∗}-algebra under the involution f^{∗} = \overline{f}, the conjugate of f. Also l^{∞}(N) is denoted by l^{∞}. The set of all convergent sequences of complex numbers, c, is a closed *- subalgebra of l^{∞} and the set of all elements of c converging to zero, c_{0}, is a closed *-subalgebra of c.

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(V) If X is a topological space, then the set C_{b}(X) of all bounded continuous complex valued functions on X is a closed *-subalgebra of l^{∞}(X) containing the constant function 1. So C_{b}(X) is a unital commutative C^{∗}- algebra.

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(VI) If X is a locally compact Hausdorff space, then the set C_{0}(X) of all continuous complex valued functions on X vanishing at infinity (i.e. for each ε > 0, the set {x ∈ X; |f(x)| ≥ ε} is compact) is a closed *-subalgebra of l^{∞}(X) and so is a commutative C^{∗}-algebra. C_{0}(X) is unital iff X is compact. Each non-unital commutative C^{∗}-algebra is of this form (cf. [Mur]).

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(VII) If X is a compact Hausdorff space, then the set C(X) of all continuous complex functions on X is exactly C_{0}(X) and so is a unital commutative C^{∗}-algebra. Each unital commutative C^{∗}-algebra is of this form (cf. [Mur]). By ([K&R1, Th. 5.3.1]), An abelian W^{∗}-algebra is isometrically ∗-isomorphic to C(X) for some extremely disconnected compact Hausdorff space X. (A topological space is called extremely disconnected or Stonean if the closure of any open set is open).

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(VIII) Let Δ denote the closed unit disc {z ∈ C, |z| ≤ 1}. Suppose that A(Δ) denoted the set of all elements of C(Δ) which are analytic on the interior of Δ. A(Δ) is a closed subalgebra of C(Δ) (Since if f_{n} ∈ A(Δ) and (f_{n}) converges to f ∈ C(Δ) in the norm of C(Δ) and γ is a simple closed path in the interior of Δ, then lim_{n→∞} ∫_{γ} f_{n}(z)dz = ∫_{γ} f(z)dz but by Cauchy’s theorem ∫_{γ} f_{n}(z)dz = 0(n ∈ N). So ∫_{γ} f(z)dz = 0. Now Morera’s theorem implies that f is analytic in the interior of Δ), and so it is a unital commutative Banach algebra. We call this the disc algebra.

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(IX) Let (Ω, μ) be a measure space and L^{p}(Ω, μ) for 1 ≤ p < ∞ be the set of all complex valued measurable functions f on Ω (we assume f is equal to g if f = g a.e.[μ]) for which ∥f∥_{p} = (∫_{Ω} |f|^{p}dμ)^{\frac{1}{2}} < ∞. L^{p}(Ω, μ) with the norm ∥.∥_{p} is a Banach space and is a Hilbert space iff p = 2. L^{p}(Ω, μ) denoted by l^{p}(Ω) if μ is counting measure. In particular, l^{p}(N) denoted by l^{p}. Let H = l^{2}, (α_{n}) be a bounded sequence of complex numbers, and (ξ_{n}) be the (usual) standard orthonormal basis of H, that is, (ξ_{n})(m) = δ_{nm}, n, m ∈ N (δ denoted the kronecker delta), so that ζ = ∑_{n=1}^{∞} < ζ, ξ_{n} > ξ_{n} for any ζ ∈ H. Then the operator T ∈ B(H) defined by T ξ_{n} = α_{n}ξ_{n+1} is called a weighted shift with the weights (α_{n}). If α_{n} = 1 for all n, then T is called unilateral shift operator. It is straightforward to show that ∥T ∥ = sup_{n} |α_{n}|,

r(T) = lim_{k} sup_{n}| ∏_{i=0}^{k−1} α_{n+i}|^{1/k} and T^{∗}ξ_{1} = 0 and T^{∗}ξ_{n} = \overline{α_{n}}ξ_{n−1}.

If 1 ≤ p < ∞, then l^{p} can be regarded as a commutative Banach algebra with coordinatewise multiplication. (For p > 1, ∥fg∥_{p} ≤ ∥f∥_{p}∥g∥_{p} is a conclusion of Hőlder inequality.) The l^{p}, 1 ≤ p < ∞, with the involution f ⟼ \overline{f} is an involutive Banach algebra.

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(X) The Banach space L^{1}([0, 1]) with the product (fg)(x) = ∫_{0}^{x} f(x − y)g(y)dy is a non-unital commutative Banach algebra. It is called Volterra algebra.

(XI) Let G be a locally compact group and μ a left invariant Haar measure on G, i.e. a Borel measure satisfying the following conditions.

(a) μ(xE) = μ(E), for every x ∈ E and every measurable E ⊆ G.

(b) μ(U) > 0, for every non-void open set U ⊆ G.

(c) μ(K) < ∞, for every compact set K ⊆ G. With the notation IX, and under the product given by the convolution (f ∗ g)(s) = ∫_{G} f(t)g(t^{−1}s)dμ(t) (s ∈ G), L^{1}(G) is a commutative Banach algebra which called the group algebra of G. In particular, we can cansider L^{1}(R), where the Lebesgue measure is an invariant Haar measure on R. Also if G be an (algebraic) group, then G with the discrete topology is a locally compact group. A left invariant Haar measure on G is the counting measure on G. The corresponding group algebra, denoted by l^{1}(G) and is called discrete group algebra.

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(XII) Let S be a semi-group and α a positive real-valued function on S such that α(st) ≤ α(s)α(t) (s, t ∈ S). If l^{1}(S, α) is the set of all complexvalued functions f on S for which ∑_{s∈S}|f(s)||α(s)| < ∞, then l^{1}(S, α) with the usual pointwise addition and scalar multiplication and the product (convolution) (f ∗ g)(s) = ∑_{tu=s}f(t)g(u) (if tu = s has no solutions, we assume (f ∗ g)(s) = 0), and with the norm ∥f∥ = ∑_{s∈S}|f(s)|α(s)is a Banach algebra. If α(s) = 1, l^{1}(S, α) = l^{1}(S) is called discrete semi-group algebra, Moreover if S = G is a group then l^{1}(S) is the same discrete group algebra l^{1}(G).

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(XIII) Let (Ω, μ) be a measure space. Then the set L^{∞}(Ω, μ) consisting of all complex valued measurable functions f on Ω (with identifying functions which are almost everywhere equal) for which ∥f∥_{∞} = inf{λ; μ{x ∈ Ω; |f(x)| > λ} = 0} < ∞ with the essential norm ∥.∥_{∞} and pointwise operations is a unital commutative Banach algebra.

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(XIV) If (Ω, μ) is a measure space, then B_{∞}(Ω) that is the set of all bounded complex valued measurable functions on Ω is a closed subalgebra of l^{∞}(Ω) and L^{∞}(Ω, μ) (again we identify almost everywhere equal functions).

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(XV) The algebra C^{m}([0, 1]) of the complex valued m times continuously differentiable on [0, 1] with the norm ∥f∥ = ∑_{k=0}^{m} \frac{1}{K!} sup_{x∈[0,1]} |f^{(k)}(x)| is a unital commutative Banach algebra. Its maximal ideals are precisely the I_{z} = {f; f(z) = 0} where z ∈ [0, 1]. Hence C^{m}([0, 1]) is semi-simple.

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(XVI) Suppose W is the set of all complex-valued functions f defined on the interval [0, 2π] of the form f(t) = ∑_{k∈Z}α_{k} exp(ikt) (t ∈ [0, 2π]), where the α_{k} ∈ C and ∑_{k}|α_{k}| < ∞. The set W with the usual pointwise operations and with the norm ∥f∥ = ∑_{k∈Z}|α_{k}| is a commutative Banach algebra and called the Wiener algebra. There is an isometric isomorphism between l^{1}(Z) and W given by f ⟶ f̃ where f̃(t) = ∑_{k∈Z}f(k) exp(ikt) (t ∈ [0, 2π]).

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(XVII) Let X and Y are normed spaces. Then the set of all bounded linear mappings (bounded operators) from X into Y with the operator norm ∥T ∥ = sup{∥T x∥; ∥x∥ ≤ 1} and with the pointwise addition and scalar multiplication is a normed space. It is Banach iff Y is Banach. If Y = X, the space B(X, X) = B(X) with the product (ST)x = S(T x) is a normed algebra ( Banach algebra, if X is a Banach space).

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(XVIII) In (XV II) if X = H is a Hilbert space, then B(H) with the involution T ⟼ T^{∗} being defined by < T^{∗}x, y >=< x, T y > (x, y ∈ H) is a C^{∗}-algebra. Each C^{∗}-algebra is isometrically isomorphic to a norm closed ∗-subalgebra of B(H) for a Hilbert space H.

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(XIX) An operator from normed space X into normed space Y is called compact if T (U) is relatively compact in Y, where U is open unit ball of X; or equivalently for each bounded sequence (x_{n}) in X, (T x_{n}) has a convergent subsequent in Y. The set of all compact operators from X into Y is denoted by K(X, Y) that is a subspace of B(X, Y). If X is a Banach space, K(X) = K(X, X) is a closed two-sided ideal of B(X).

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(XX) Identifying M_{n}(C), the algebra of all n × n matrices with entries in C, with B(C^{n}) = K(C^{n}). So it is a unital non-commutative C^{∗}-algebra.

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(XXI) Let H be a Hilbert space and x\overline{⊗}y is the (one-rank) operator given by (x\overline{⊗}y)z =< z, y > x. Suppose that (e_{i})_{i∈I}, (f_{i})_{i∈I} are othonormal bases for H and (λ_{i})_{i∈I} is a family of complex numbers indexed by the same set I. The operator T = ∑_{i∈I}λ_{i}e_{i}\overline{⊗}f_{i} is well-defined and belongs to B(H) iff (λ_{i}) is bounded and then ∥T ∥ = sup{|λ_{i}|; i ∈ I}.

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(XXIa) An operator T is called of finite rank n if n = dimT (H) < ∞. The set F (H) of all finite rank operators is a self-adjoint two-sided ideal of B(H). It is consisting of all operators as ∑_{i∈I}λ_{i}e_{i}\overline{⊗}f_{i} ∈ B(H) such that λ_{i} = 0 for all i except finitely many i.

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(XXIb) The two-sided ideal of the compact operators K(H) is self-adjoint and F (H) is norm-dense in K(H). K(H) is consisting of all operators as T = ∑_{i∈I}λ_{i}e_{i}\overline{⊗}f_{i} ∈ B(H) such that the λ_{i} are positive (the λ_{i}^{2} are the eigenvalues of T^{∗}T). This sum has either a finite or a denumerably infinite number of terms; in the last case, λ_{i} → 0.

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(XXIc) The set S(H) of all operators T for which ∑_{i∈I}∥T e_{i}∥^{2} < ∞ is a self-adjoint ideal of B(H). These operators are called Hilbert-Schmidt operators on H. The algebra S(H) with the Hilbert-Schmidt norm ∥T ∥_{2} = (∑_{i∈I}∥T e_{i}∥^{2})^{1/2} is a Banach algebra. It contains operators of finite rank as a dense subset. For any pair of operators T and S in S(H), the family (< T e_{i}, Se_{i} >)_{i∈I} is summable. Its sum (A, B) defines an inner product in S(H) and (T, T)^{1/2} = ∥T ∥_{2}. So S(H) is a Hilbert space (independent on the choice basis (e_{i})). S(H) ⊆ K(H). S(H) consists of precisely those compact operators T = ∑_{i}λ_{i}e_{i}\overline{⊗}f_{i} for which ∑_{i}λ_{i}^{2} < ∞. In addition ∥T ∥_{2} =

(∑_{i}λ_{i}^{2})^{1/2}.

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(XXId) The set of all products of two Hilbert-Schmidt operators is denoted by N(H) and its elements are called trace-class operators. This set is a self-adjoint two-sided ideal of B(H) and coincides with the set of those operators T for which ∑_{i∈I} < |T |e_{i}, e_{i} >< ∞ where |T | is the absolute value of T in the C^{∗}-algebra B(H). If ∥T ∥_{1} = ∑_{i∈I} < |T |e_{i}, e_{i} >, then N(H) with this norm is a Banach algebra. F (H) is a dense subset of N(H). N(H) is contained in K(H) and contains S(H). The elements of N(H) are precisely the compact operators T = ∑_{i∈I}λ_{i}e_{i}\overline{⊗}f_{i} for which ∑_{i}λ_{i} < ∞. Moreover,

∥T ∥_{1} = ∑_{i}λ_{i}.

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(XXII) The set C[z] of all polynomials in an indeterminate z with complex coefficients under usual operations on polynomials and with the norm ∥p∥ = sup_{|λ|≤1} |p(λ)| is a normed algebra.

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(XXIII) The set of all formal polynomials of degree at most n with the usual addition, scalar multiplication and product (but together with the convention that x^{k} = 0 if k > n) and with the norm ∥p∥ = ∑_{k=1}^{n}|α_{k}| (p(x) =

∑_{k=1}^{n}α_{k}x^{k}) is a finite dimensional Banach algebra.

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(XXIV) The algebra C([0, 1]) with the supremum norm ∥.∥ and multiplication (f ∗ g)(t) = ∫_{0}^{t} f(s)g(t − s)ds is a Banach algebra.