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.

(II) Cⁿ with the coordinatewise addition, scalar multiplication and the inner product <(z₁, ... , z_n),(w₁, ... , w_n)> = Sⁿ_i=1 z_i \overline{w_i} is a Hilbert space.

(III) The space C² (see (II)) with the product (a,b)(a',b')=(aa',ab'+a'b) is a unital commutative Banach algebra.

(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 e X ; poinwise scalar multiplication (l f)(x) = l f(x),l e C,x e X; and supremum norm ||f||= sup{|f(x)|;x e 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, c₀, is a closed *-subalgebra of c.

(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.

(VI) If X is a locally compact Hausdorf space, then the set C₀(X) of all continuous complex valued functions on X vanishing at infinity (i.e. for each e >0, the set {x e X ; |f(x)| e} is compact) is a closed *-subalgebra of l(X) and so is a commutative C*-algebra. C₀(X) is unital iff X is compact. Each non-unital commutative C*-algebra is of this form (cf. [MUR]).

(VII) If X is a compact Hausdorf space, then the set C(X) of all continuous complex functions on X is exactly C₀(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).

(VIII) Let D denote the closed unit disc {z e C, |z| 1}. Suppose that A(D) denoted the set of all elements of C(D) which are analytic on the interior of D. A(D) is a closed subalgebra of C(D) and so is a unital commutative Banach algebra. We call this the disc algebra.

(IX) Let (W ,m) be a measure space and L^p(W,m) for 1 p < be the set of all complex valued measurable functions f on W (we assume f is equal to g if f=g a.e. [m]) for which ||f||_p = (_W |f|^p dm)^1/p < . L^p(W,m) with the norm ||.||_pis a Banach space and is a Hilbert space iff p=2. L^p(W,m) denoted by l^p(W) if m is counting measure. In particular, l^p(N) denoted by l^p.
Let H=l², (a_n) be a bounded sequence of complex numbers, and (x_n) be the (usual) standard orthonormal basis of H, that is, (x_n)(m)= d_nm, n,m e N (d denoted the kronecker delta), so that z = S_n<z,x_n> x_n for any z in H. Then the operator T e B(H) defined by T x_n = a_n x_n+1 is called a weighted shift with the weights (a_n). If a_n = 1 for all n, then T is called unilateral shift operator.It is straightforward to show that ||T|| = sup_n | a_n|, r(T) = lim_k sup_n |P_i=1^k-1a_n+i|^1/k and T* x₁ = 0 and T*x _n = \overline{a_n}x_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 Holder inequality.) The l^p, 1 p < , with the involution f \mapsto \overline {f} is an involutive Banach algebra.

(X) The Banach space L¹([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 m a left invariant Haar measure on G, i.e. a Borel measure satisfying the following conditions:
(a) m(xE) = m(E), for every x e E and every measurable E . G(B).
(b) m(U) > 0, for every non-void open set U G.
(c) m(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^-1s)dm(t) (s e G), L¹(G) is a commutative Banach algebra which called the group algebra of G.
In particular, we can cansider L¹(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 l¹(G) and is called discrete group algebra.

(XII) Let S be a semi-group and a a positive real-valued function on S such that a(st) a(s)a(t) (s,t e S). If l¹(S,a) is the set of all complex-valued functions f on S for which S_{se S}|f(s)||a(s)| < , then l¹(S,a) with the usual pointwise addition and scalar multiplication and the product (convolution) (f_*g)(s) = S_tu=sf(t)g(u) (if tu=s has no solutions, we assume (f_*g)(s)=0), and with the norm ||f|| = S_seS|f(s)|a(s) is a Banach algebra.
If a(s) = 1, l¹(S,a) = l¹(S) is called discrete semi-group algebra, Moreover if S=G is a group then l¹(S) is the same discrete group algebra l¹(G).

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

(XIV) If (W,m) is a measure space, then B(W) that is the set of all bounded complex valued measurable functions on W is a closed subalgebra of l(W) and L(W,m) (again we identify almost everywhere equal functions).

(XV) The algebra C^m([0,1]) of the complex valued m times continuously differentiable on [0,1] with the norm ||f|| = S_k=0^m 1/K! sup_{x e [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 e [0,1]. Hence C^m([0,1]) is semi-simple.

(XVI) If W is the set of all complex-valued functions f defined on the interval [0,2p] of the form f(t) = S_{k e Z} a_k exp(ikt) (t e [0,2p]), where the a_k e C and S_k|a_k| < . The set W with the usual pointwise operations and with the norm ||f|| = S_{k e Z}|a_k| is a commutative Banach algebra and called the Wiener algebra. There is an isometric isomorphism between l¹(Z) and W given by f --> \tilde{f} where \tilde{f}(t) = S_{k e Z}f(k) exp(ikt) (t e [0,2p]).

(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{||Tx||; ||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(Tx) is a normed algebra (Banach algebra, if X is a Banach space).

(XVIII) In (IV) if X=H is a Hilbert space, then B(H) with the involution T ---> T* being defined by <T*x , y> = <x , Ty> (x,y e H is a C*-algebra. Each C*-algebra is isometrically isomorphic to a norm closed *-subalgebra of B(H) for a Hilbert space H.

(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, (Tx_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).

(XX) Identifying M_n(C), the algebra of all n \times n matrices with entries in C, with B(Cⁿ) = K(Cⁿ). So it is a unital non-commutative C*-algebra.

(XXI) Let H be a Hilbert space and x y is the (one-rank) operator given by <x y>z = <z,y>x. Suppose that (e_i)_{i e I}, (f_i) _{ie I} are othonormal bases for H and (l_i)_{i e I} is a family of complex numbers indexed by the same set I. The operator T = S_{i e I} l_ie_i f_i is well-defined and belongs to B(H) iff (l_i) is bounded and then ||T|| = sup{|l_i|; i e I}.

(XXIa) An operator T is called of finite rank n if n=dim T(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 S_{i e I} l_i e_i f_i e B(H) such that l_i = 0 for all i except finitely many i.

(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 = S_{i e I}l_ie_i f_i e B(H) such that the l_i are positive (the l_i² are the eigenvalues of T*T). This sum has either a finite or a denumerably infinite number of terms; in the last case, l_i --> 0.

(XXIc) The set S(H) of all operators T for which S_{i e I}||Te_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 = (S_{i e I}||Te_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 (<Te_i, Se_i>)_{i e I} is summable. Its sum (T,S) 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 = S_i l_ie_i f_i for which S_il_i² < . In addition ||T||₂ = (S_il_i²)^1/2.

(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 S_{i e I}<|T|e_i,e_i> < where |T| is the absolute value of T in the C*-algebra B(H). If ||T||₂ = S_{i e 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 = S_{i e I} l_ie_i f_i for which S_i l_i < . Moreover, ||T||₁ = S_i l_i.

(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_{|l| 1}|p(l)| is a normed algebra.

(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|| = S_k=1ⁿ |a_k| (p(x) = S_k=1ⁿ a_k x^k) is a finite dimensional Banach algebra.

(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.

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