Continuous Homomorphism From C to C

Handbook of Algebra

W. Narkiewicz , in Handbook of Algebra, 1996

3.1 Theorem

I.

Reciprocity Law. If K/k is Abelian then there exists a surjective continuous homomorphism

$\psi \colon\colon I_k\to \mathrm{Gal}(K/k),$

with kernel equal to k*N _{K / k} (I _K ), hence inducing a continuous isomorphism

$C(k)/N_{K/k}\left(C(K)\right)\cong \mathrm{Gal}(K/k),$

and if a = (a _υ)_υ∈ I_k satisfies a_υ = 1 for all υ which are either Archimedean or ramified in K/k, then

$\psi (a)=F_{K/k}(a),$

where F _{K / k} denotes the Artin map for ideles.

II.

Existence Theorem. If N is a closed subgroup of finite index of C(k) then there exists a unique Abelian extension K/k satisfying

$N_{K/k}\left(C(K)\right)=N.$

This formulation is essentially due to C. Chevalley [13], who used the convenient language of infinite extensions:

Let K be a field and let L/K be an infinite Galois extension. The Galois group G of L/K has a natural topology, the Krull topology in which the Galois groups of L/K (where M/K runs over all finite extensions contained in L) is taken as a fundamental set of open neighborhoods of the unit element. In this topology G is compact and zero-dimensional. It has been established by W. Krull [76] that there is, as in the usual Galois theory, a biunique correspondence between fields lying between K and L and closed subgroups of G. Denote by K^ab the maximal Abelian extension of K, i.e. the union (or the direct limit) of all finite Abelian extensions of K contained in a fixed algebraic closure, and let G be the Galois group of K^ab /K. Chevalley established in the number field case an isomorphism between the dual groups of the Galois group of K^ab /K and C(K)/D(K) which behaves in a nice way with regard to the arithmetical properties of K and from this Theorem 3.1 can be deduced.

It was later shown by G. Hochschild and T. Nakayama [61, 102] that the use of group cohomology leads to an essential simplification of Chevalley's proof.

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Basic Representation Theory of Groups and Algebras

In Pure and Applied Mathematics, 1988

5.3 Definition

Two extensions

$N\underset{i}{\to }H\underset{j}{\to }G$

and

$N\underset{i^{\prime }}{\to }{H}^{\prime }\underset{j^{\prime }}{\to }G$

of N by G are isomorphic if there exists an isomorphism f:H → H′ of the topological groups H and H′ such that

(1) $\begin{array}{cc}{i}^{\prime }=f\circ i,& j=j^{\prime }\circ f;\end{array}$

that is, such that the diagram

commutes.

This relation of isomorphism is clearly an equivalence relation.

Remark

It is useful to observe that the preceding definition is unaltered if we assume merely that there is a continuous homomorphism f:H → H′ satisfying (1); for this implies that f must be a surjective homeomorphism.

Indeed: Suppose f(h) = e′ (unit of H′). Then by (1) j(h) = (j′ $\circ$ f)(h) = unit of G; so h = i(n) (n ∈ N), and by (1) f(h) = i′(n). Since i′ is one-to-one and i′(n) = f(h) = e′, n must be the unit of N; so h = e (unit of H). Consequently f is one-to-one. Now let h′ ∈ H′. Since j = j′ $\circ$ f there are an element h of H and an element n of N such that h′ = f(h)i′(n). Since i′ = f $\circ$ i, we have h′ = f(h)f(i(n)) = f(hi(n)); so f is onto H′.

It remains only to show that f ⁻¹ is continuous. Assume that $f\left(h_{\alpha }\right)\to f\left(h\right)$ in H′, where {h _α} is a net of elements of H and h ∈ H. We must show that h _α → h. Since we can replace {h _α} by any subnet of it, it is enough to show that some subnet of {h _α} converges to h. Now, since f(h _α) → f(h), (1) implies that j(h _α) → j(h). By the openness of j and II. 13.2 we can pass to a subnet and choose elements k _α of H and n _α of N such that h _α = k _αi(n _α) and k _α → h. Applying f and using (1), we find that f(k _α) → f(h) and f(k _α)i′(n _α) = f(k _α i(n _α)) = f(h _α) → f(h). From these two facts we deduce that i′(n _α) → unit of H′, whence n _α → unit of N, so that h _α = k _α i(n _α) → h. This completes the proof that f^{− 1} is continuous.

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Compact Groups and Their Representations

A. Kirillov , A. KirillovJr., in Encyclopedia of Mathematical Physics, 2006

Basic Notions

By a representation of G we understand a pair $\left(\pi ,V\right)$ , where V is a complex vector space and π is a continuous homomorphism $G\to \text{Aut}\left(V\right)$ . This notation is often shortened to π or V. In this article, we only consider finite-dimensional (f.d.) representations; in this case, the homomorphism π is automatically smooth and even real-analytic.

We associate to any f.d. representation $\left(\pi ,V\right)$ of G the representation $\left({\pi }_{*},V\right)$ of the Lie algebra $\mathfrak{g}=\text{Lie}\left(G\right)$ which is just the derivative of the map $\pi :G\to \text{Aut}V$ at the unit point $e\in G$ . In terms of the exponential map, we have the following commutative diagram:

Choosing a basis in V, we can write the operators $\pi \left(g\right)$ and ${\pi }_{*}\left(X\right)$ in matrix form and consider π and ${\pi }_{*}$ as matrix-valued functions on G and $\mathfrak{g}$ . The diagram above means that

[4] $\pi \left(\text{exp}X\right)=e^{{\pi }_{*}\left(X\right)}$

Recall that if G is connected, simply connected, then every representation of $\mathfrak{g}$ can be uniquely lifted to a representation of G. Thus, classification of representations of connected simply connected Lie groups is equivalent to the classification of representations of Lie algebras.

Let $\left({\pi }_1,V_1\right)$ and $\left({\pi }_2,V_2\right)$ be two representations of the same group G. An operator $A\in \text{Hom}\left(V_1,V_2\right)$ is called an "intertwining operator," or simply an "intertwiner," if $A\circ {\pi }_1\left(g\right)={\pi }_2\left(g\right)\circ A$ for all $g\in G$ . Two representations are called "equivalent" if they admit an invertible intertwiner. In this case, using an appropriate choice of bases, we can write ${\pi }_1$ and ${\pi }_2$ by the same matrix-valued function.

Let $\left(\pi ,V\right)$ be a representation of G. If all operators $\pi \left(g\right),g\in G$ , preserve a subspace $V_1\subset V$ , then the restrictions ${\pi }_1\left(g\right)=\pi \left(g\right){|}_{V_1}$ define a "subrepresentation" $\left({\pi }_1,V_1\right)$ of $\left(\pi ,V\right)$ . In this case, the quotient space $V_2=V/V_1$ also has a canonical structure of a representation, called the "quotient representation."

A representation $\left(\pi ,V\right)$ is called "reducible" if it has a nontrivial (different from V and {0}) subrepresentation. Otherwise it is called "irreducible."

We call representation $\left(\pi ,V\right)$ "unitary" if V is a Hilbert space and all operators $\pi \left(g\right),g\in G$ , are unitary, that is, given by unitary matrices in any orthonormal basis. We use a short term "unirrep" for a "unitary irreducible representation."

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Advanced Theory

In Pure and Applied Mathematics, 1986

11.2.9 Theorem

Suppose that, for j= 1,…, n, $\mathcal{R}$ _j and $\mathcal{L}$ _j are von Neumann algebras and φ _j is an ultraweakly continuous* homomorphism from $\mathcal{R}$ _j into $\mathcal{L}$ _j . Then there is a unique ultraweakly continuous * homomorphism φ, from $\mathcal{R}$ ₁⊗…⊗ $\mathcal{R}$ _n into $\mathcal{L}$ ₁⊗…⊗ $\mathcal{L}$ _n , such that

$\phi \left(A_1\otimes \mathrm{\dots }\otimes A_n\right)={\phi }_1\left(A_1\right)\otimes \mathrm{\dots }\otimes {\phi }_n\left(A_n\right)\left(A_1\in R_1,\mathrm{\dots },A_n\in R_n\right).$

_Ifϕ _j ( $\mathcal{R}$ _j ) = $\mathcal{L}$ _j , for each j, then φ( $\mathcal{R}$ ₁⊗…⊗ $\mathcal{R}$ _n ) = $\mathcal{L}$ ₁⊗…⊗ $\mathcal{L}$ _n .

Proof. We recall first that, if $\mathcal{R}$ and × are von Neumann algebras (acting on Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$ , respectively), and η: $\mathcal{R}$ → $\mathcal{L}$ is an ultraweakly continuous * homomorphism, then η( $\mathcal{R}$ ) is a von Neumann subalgebra of $\mathcal{L}$ . This is a special case of lemma 10.1.10 (with $\mathcal{U}$ = $\mathcal{U}$ ⁻= $\mathcal{R}$ , and $\overline{\eta }$ = η); it is also a straightforward consequence of the Kaplansky density theorem, together with the fact (Corollary 10.1.8) that η maps the closed unit ball ( $\mathcal{R}$ )₁ onto (η( $\mathcal{R}$ ))₁.

Upon replacing $\mathcal{L}$ _j by its von Neumann subalgebra φ _j ( $\mathcal{R}$ _j ), we may assume that $\mathcal{L}$ _j = φ _j ( $\mathcal{R}$ _j ) (j= 1,…, n). From Theorem 11.1.3 there is a * homomorphism φ₀, from the represented C ^*-algebra $\mathcal{R}$ ₁⊗…⊗ $\mathcal{R}$ _n onto $\mathcal{L}$ ₁⊗…⊗ × _n , such that

${\phi }_0\left(A_1\otimes \mathrm{\dots }\otimes A_n\right)={\phi }_1\left(A_1\right)\otimes \mathrm{\dots }\otimes {\phi }_n\left(A_n\right)\left(A_1\in R_1,\mathrm{\dots },A_n\in R_n\right).$

It now suffices to prove that φ₀ extends to an ultraweakly continuous * homomorphism φ, from $\mathcal{R}$ ₁⊗…⊗ $\mathcal{R}$ _n onto $\mathcal{L}$ ₁⊗…⊗ $\mathcal{L}$ _n . From lemma 10.1.10, it is enough to show that φ₀ is ultraweakly continuous.

Suppose that ω _j is a normal state of $\mathcal{L}$ _j (j= 1,…, n), and ω is the normal product state ω₁⊗…⊗ ω _n of $\mathcal{L}$ ₁⊗…⊗ $\mathcal{L}$ _n . Since φ _j is ultraweakly continuous, ω _j $\mathcal{L}$ φ _j is a normal state σ _j of $\mathcal{R}$ _j , and σ₁⊗…⊗ Σ _n is a normal product state σ of $\mathcal{R}$ ₁⊗…⊗ $\mathcal{R}$ _n . When A ₁∈ $\mathcal{R}$ ₁,…, A _n ∈ $\mathcal{R}$ _n ,

$\begin{array}{l}\sigma \left(A_1\otimes \mathrm{\dots }\otimes A_n\right)=\sigma \left(A_1\right){\sigma }_2\left(A_2\right)\mathrm{\dots }{\sigma }_n\left(A_n\right)\\ ={\omega }_1\left({\phi }_1\left(A_1\right)\right){\omega }_2\left({\phi }_2\left(A_2\right)\right)\mathrm{\dots }{\omega }_n\left({\phi }_n\left(A_n\right)\right)\\ =\omega \left({\phi }_1\left(A_1\right)\otimes \mathrm{\dots }\otimes {\phi }_n\left(A_n\right)\right)\\ =\omega \left({\phi }_0\left(A_1\otimes \mathrm{\dots }\otimes A_n\right)\right)\end{array}$

By linearity and norm continuity of φ₀, it follows that σ(A) = ω(φ₀(A)) for each A in $\mathcal{R}$ ₁⊗…⊗ $\mathcal{R}$ _n .

The preceding paragraph shows that the linear functional ω × φ₀ on $\mathcal{R}$ ₁⊗…⊗ $\mathcal{R}$ _n is ultraweakly continuous (in fact, the restriction of a normal product state on $\mathcal{R}$ ₁⊗…⊗ $\mathcal{R}$ _n ), whenever ω is a normal product state of $\mathcal{L}$ ₁⊗…⊗ × _n . By Proposition 11.2.8, each normal state ρ of $\mathcal{L}$ ₁⊗…⊗ $\mathcal{L}$ _n is the limit of a norm convergent sequence {ρ _n }, each ρ _n being a finite linear combination of normal product states. The linear functionals ρ _n $\mathcal{L}$ φ₀ on $\mathcal{R}$ ₁⊗…⊗ $\mathcal{R}$ _n are ultraweakly continuous, by the above argument, and converge in norm to ρ $\mathcal{L}$ φ₀. By Theorem 10.1.15 (1), ρ $\mathcal{L}$ φ₀ is ultraweakly continuous; and hence, so is φ₀.

The * homomorphism φ occurring in Theorem 11.2.9 is denoted by φ₁⊗…⊗ φ _n .

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Automorphism Groups

In C*-Algebras and their Automorphism Groups (Second Edition), 2018

7.4.1

A $C^{⁎}$ -dynamical system (or just a dynamical system) is a triple $(A,G,\alpha )$ consisting of a $C^{⁎}$ -algebra A, a locally compact group G , and a continuous homomorphism α of G into the group Aut(A) of automorphisms (i.e., ^⁎-automorphisms) of A equipped with the topology of pointwise convergence. This means that, for each x in A, the function $\alpha (x):G\to A$ defined by $t\to {\alpha }_t(x)$ is continuous. We are mainly interested in the case where both G and A are separable and refer to this as a separable dynamical system.

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Stable Operations in Generalized Cohomology

J. Michael Boardman , in Handbook of Algebraic Topology, 1995

PROOF

We first consider the uncompleted tensor product A ⊗ B, made into an E ^*-algebra in the standard way, filtered as in (4.15) by the ideals

$F^{a,b}\left(A\otimes B\right)=\mathrm{Im}\left[\left(F^{a}A\otimes B\right)\oplus \left(A\otimes F^{b}B\right)\to A\otimes B\right].$

We define continuous injections i: A → A ⊗ B and j: B → A ⊗ B by ix = x ⊗ 1 and jy = 1 ⊗ y . Given continuous homomorphisms f: A → C and g: B → C, where C is any object in FAlg, there is a unique homomorphism of algebras h: A ⊗ B → C satisfying h о i = f and h о j = g, defined by h(x ⊗ y) = (fx)(gy), thanks to the commutativity of C. It is also continuous: given F^CC ⊂ C, choose F^a A and F^b B such that f(F^a A) ⊂ F^c C and g(F^b B) ⊂ F^c C; then h(F^{a, b} (A ⊗ B)) ⊂ F^c C. Because A ⊗ B is rarely complete, we complete it, and the homomorphism h, to obtain the desired unique algebra homomorphism ${\displaystyle \widehat{h}}:A{\displaystyle \widehat{\otimes }}B\to C$ in FAlg.

Although E ^*(–)^ does not in general take products in Ho to coproducts in FAlg, it does in the favorable cases when we have the Künneth homeomorphism $E^{*}\left(X\times Y\right)\cong E^{*}\left(X\right)\otimes E^{*}\left(Y\right)$ as in Definition 4.17.

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Real Reductive Groups I

In Pure and Applied Mathematics, 1988

0.3 The structure of compact Lie groups

0.3.1

Let G be a compact Lie group with Lie algebra g. Let g _C denote the complexification of g. Then g _C is a reductive Lie algebra over C. In fact, if ( , ) is any positive non-degenerate symmetric bilinear form on g then we define a new form on g, 〈 , 〉, as follows:

$〈X,Y〉=\underset{G}{\int }\left(\text{Ad}\left(g\right)X,\text{Ad}\left(g\right)Y\right)dg\text{for}X,Y,\in g.$

Here (as usual) dg denotes normalized invariant measure on G. The invariance of dg immediately implies that

$〈\text{Ad}\left(g\right)X,\text{Ad}\left(g\right)Y〉=〈X,Y〉\text{for}g\in G\text{and}X,Y\in g,$

By differentiating this formula one sees that 〈 , 〉 is an invariant form on g. Thus, if u is an ideal of g then the orthogonal complement to u is also an ideal of g. Hence, dimension considerations imply that g is a direct sum of 1-dimensional and simple ideals. This clearly implies that g is reductive.

Recall that the Killing form of g, B, is defined by the following formula:

$B\left(X,Y\right)=\text{tr ad}X\text{ ad}Y\text{for}X,Y\in g.$

Since ad X is skew adjoint relative to 〈, 〉 for X ∈ g it is clear that B(X, X) ≤ 0 for X ∈ g. Also, B(X, X) = 0 if and only if ad X = 0. Thus, g is semisimple if and only if B is negative definite. The converse is also true.

Theorem

If g is a Lie algebra over R with negative definite Killing form then any connected Lie group with Lie algebra g is compact.

This theorem is known as Weyl's theorem. For a proof see, for example, Helgason [1, Theorem 6.9, p. 133].

0.3.2

In this book a commutative compact, connected Lie group will be called a torus. Let T be a torus with Lie algebra t. If we look upon t as a Lie group under addition then exp is a covering homomorphism of t onto T. The kernel of exp is a lattice, L, in t. That is, L is a free Z module of rank equal to dim t.

Let T ^∧ denote the set of all continuous homomorphisms of T into the circle. If μ ∈ T ^∧ then the differential of μ (which we will also denote by μ) is a linear map of t into i R such that μ(L) ⊂ 2πi Z. If μ is a linear map of t into i R such that μ(L) ⊂ 2πi Z then μ is called integral. If μ is an integral linear form on t then we define for t = exp(X), t ^μ = exp(μ(X)). This sets up an identification of integral linear forms on t and characters of T.

0.3.3

Let G be a compact, connected Lie group. Then a maximal torus of G is (as the name implies) a torus contained in G but not properly contained in any sub-torus of G. Fix a maximal torus, T, of G. Then t _C is a Cartan subalgebra of g _C . The elements of Φ(g _C , t _C )are integral on t and thus define elements of T ^∧. Thus, we will look upon roots as characters of T. We now list some properties of maximal tori that will be used in this book.

(1)

A maximal torus of G is a maximal abelian subgroup of G (Helgason [1, p.287]).

(2)

If T and S are maximal tori of G then there exists an element g ∈ G such that S = gTg ^–1 (Helgason [1, p.248]).

(3)

Every element of G is contained in a maximal torus of G. That is, the exponential map of G is surjective. (Helgason [1, p.135].)

(4)

If T is a maximal torus of G then G/T is simply connected. (This follows from say Helgason [1, Cor.2.8, p.287].)

Let T be a maximal torus of G. Let N(T) denote the normalizer of T in G (the elements g of G such that gTg ^–1 = T). Let W(G, T) denote the group N(T)/T. Then W(G, T) is called the Weyl group of G with respect to T. If g ∈ s ∈ W(G, T) then we set sH = Ad(g)H for H ∈ t. This defines an action of W(G, T) on t.

(5)

Under this action W(G, T) = W(g _C , t _C ) (Helgason [1, Cor.2.13, p.289]).

0.3.4

Let g be a semisimple Lie algebra over C. Then a real form of g, u, will be called a compact form if u has a negative definite Killing form. The following result is due to Weyl. Combined with Theorem 0.3.1 it is the basis of what he called the "unitarian trick".

Theorem

If h is a Cartan subalgebra of g then there exists a compact form, u, of g such that u ∩ h is maximal abelian in u. ( Jacobson, [1, p. 147].)

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Advanced Theory

In Pure and Applied Mathematics, 1986

10.1.12 Theorem.

If π is a representation of aC ^*-algebra $\mathfrak{A}$ , and φ is the universal representation of $\mathfrak{A}$ , there is a projection P in the center of φ( $\mathfrak{A}$ )⁻, and a* isomorphism α from the von Neumann algebra φ( $\mathfrak{A}$ )⁻ P onto π( $\mathfrak{A}$ )⁻, such that π(A) = α(φ(A)P) for each A in $\mathfrak{A}$ .

Proof. Since φ is a faithful representation, π $\mathcal{J}$ φ⁻¹ is a * homomorphism β from φ( $\mathfrak{A}$ ) onto π( $\mathfrak{A}$ ), and is therefore norm decreasing. With ω an ultraweakly continuous linear functional on π( $\mathfrak{A}$ ), the linear functional ω ∘ β on φ( $\mathfrak{A}$ ) is bounded, and is therefore ultraweakly continuous by Proposition 10.1.1. From this, β is ultraweakly continuous, and so extends to an ultraweakly continuous * homomorphism $\mathcal{J}$ from φ( $\mathfrak{A}$ )⁻ onto π( $\mathfrak{A}$ )⁻, by Lemma 10.1.10. The kernel of $\overline{\beta }$ is an ultraweakly closed two-sided ideal in φ( $\mathfrak{A}$ )⁻, and by Theorem 6.8.8 it has the form φ( $\mathfrak{A}$ )⁻, for some projection Q in the center of φ( $\mathfrak{A}$ )⁻. With P defined as I − Q,

$\overline{\beta }\left(A\right)=\overline{\beta }\left(AP+AQ\right)=\overline{\beta }\left(AP\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(A\in \phi {\left(\mathfrak{A}\right)}^{-}),$

so the restriction, α = $\mathcal{J}$ | φ( $\mathfrak{A}$ )⁻ P, has the same range, π( $\mathfrak{A}$ )⁻, as $\mathcal{J}$ . Moreover, α is one-to-one, and is therefore a * isomorphism from φ( $\mathfrak{A}$ )⁻ P onto π( $\mathfrak{A}$ )⁻, since the kernel φ( $\mathfrak{A}$ )⁻ Q of $\mathcal{J}$ meets φ( $\mathfrak{A}$ )⁻ P only at 0. Finally, for each A in $\mathfrak{A}$ ,

$\begin{array}{c}\alpha \left(\phi \left(A\right)P\right)=\overline{\beta }\left(\phi \left(A\right)P=\overline{\beta }(\phi \left(A\right)P+\phi \left(A\right)Q\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\overline{\beta }\left(\phi \left(A\right)\right)=\stackrel{}{\beta }\left(\phi \left(A\right)\right)=\pi \circ {\phi }^{-1}\left(\phi \left(A\right)\right)=\pi \left(A\right).\end{array}$

The conclusions of the preceding theorem can be expressed conveniently by means of a diagram. With φ the universal representation of a C ^*-algebra $\mathfrak{A}$ , and π another representation, we can choose a central projection P in φ( $\mathfrak{A}$ )⁻ and a * isomorphism α from φ( $\mathfrak{A}$ )⁻ P onto π( $\mathfrak{A}$ )⁻, and set up the following commutative diagram, in which l denotes an inclusion mapping, α₀ is the * isomorphism α | φ( $\mathfrak{A}$ ) P from φ( $\mathfrak{A}$ ) P onto π( $\mathfrak{A}$ ), and ψ is the * homomorphism A → AP from φ( $\mathfrak{A}$ ) onto φ( $\mathfrak{A}$ ) P:

From Remark 7.4.4, α is ultraweakly bicontinuous, and therefore the same is true of α₀. Moreover, ψ is ultraweakly continuous, and is a * isomorphism (and hence isometric) if π is a faithful representation.

The conclusions of Theorem 10.1.12 will be discussed further (Remark 10.3.2) in connection with quasi-equivalence and disjointness of representations. In the meantime, we use the theorem to obtain the following characterization of ultraweak continuity of linear mappings between operator algebras.

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Phantom Maps

C.A. McGibbon , in Handbook of Algebraic Topology, 1995

PROOF OF THEOREM 5.1

In part i) notice that if φ : X → Y is a phantom map, then so is the composite $\widehat{e}\phi :X\to \widehat{Y}$ . But Ph $\left(X,\widehat{Y}\right)=*$ because this is isomorphic to the lim ¹ term of a tower of profinite groups and continuous homomorphisms which, in turn, is trivial by Proposition 4.3. Going the other way, if $\widehat{e}\phi :X\to \widehat{Y}$ is null homotopic, then so is its restriction to any finite skeleton of X. But this forces φ | X_n to be null-homotopic by a basic result of Sullivan [66, Theorem 3.2].

In part ii), let X_τ , denote the homotopy fiber of the rationalization map r : X → X_o . This fiber is a space whose homotopy groups are torsion, as are its reduced integral homology groups. The fibration sequence

$X_{\tau }\stackrel{i}{\to }X\stackrel{r}{\to }{X}_o$

happens to be a cofibration sequence as well [42]. Now if φ : X → Y is a phantom map, then so is the composite φi : X_τ → Y. But Ph(X_τ, Y) = * by Example 3.15 and so φi must be null. Thus φ factors through the rationalization X_o as claimed. Going in the other direction, first notice that if A_o is the rationalization of a finitely generated abelian group and $\widehat{B}$ is the profinite completion of another finitely generated abelian group then

$\text{Hom}\left(A_o,\widehat{B}\right)=\text{Ext}\left(A_o,\widehat{B}\right)=0.$

For a proof, see [19, Chapter IX]. Together with basic obstruction theory, this implies that every map from X _o to $\widehat{Y}$ is null homotopic. Therefore if φ factors through X_o it must be a phantom map by this observation and part.

The theorem of Sullivan, mentioned in the proof, says that if K is a finite complex and Y is as in 5.1, then two maps f, g : K → Y are homotopic if and only if $\widehat{e}f\simeq \widehat{e}g$ . This implies that for a domain Z, not necessarily of finite type, the kernel of ${\widehat{e}}_{*}:\left[Z,Y\right]\to \left[Z,\widehat{Y}\right]$ is the set of all phantom maps of the second kind. In particular this means that in Theorem 5.1, the set [X_o , Y] consists solely of phantom maps of the second kind. However, notice that the only phantom map (of the first kind) in [X_o , Y] is the trivial one! This follows from Theorem 3.3 since the suspension of a rational space is a bouquet of rational spheres.

Both conclusions in Theorem 5.1 fail to hold if the finite type hypothesis on Y is dropped. To see this in part i), recall that if Y were a rational space, then its profinite completion would have the homotopy type of a point. If 5.1 still held this would mean that every map into a rational space is a phantom map, which is obviously nonsense. In part ii) consider the universal phantom map out of ℝP ^∞. It is essential even though this domain has the rational homotopy type of a point. Thus the description of Ph(X, Y) as a quotient of [X_o , Y] fails to hold in this generality.

It is apparent from Theorem 5.1 that, under the appropriate restrictions on the spaces involved, the set [X_o , Y] serves as an upper bound on Ph(X, Y). The following theorem of Zabrodsky describes this upper bound in terms of ordinary rational invariants.

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Topology, General

Taqdir Husain , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

VIII Marriage of Topology and Algebra

Algebraic operations such as addition, subtraction, multiplication, and division play an important role in algebra. Topology, however, is not primarily concerned with them. When algebra and topology are married in the sense that a set with algebraic operations is endowed with a topology, new and interesting theories emerge. For instance, in the field of functional analysis (a major branch of mathematics), the topics of topological groups, topological vector spaces, and topological algebras, among others, are the outcome of this marriage.

Among many topics in algebra, one specifically comes across the following algebraic structures: semigroups, groups, vector spaces, and algebras. By endowing them with topologies so that the underlying algebraic operations are continuous, one obtains interesting areas of mathematics.

VIII.A Topological Semigroups and Groups

A set S with an associative binary operation [i.e., for all x, y  ∈ S, x ∘ y  ∈ S and x ∘ y ∘ z  =   (x ∘ y) ∘ z  = x ∘ (y ∘ z), in which "∘" denotes the binary operation] is called a semigroup. A semigroup S with a topology T is called a topological semigroup (TS) if the map (x, y)   → x ∘ y of S  × S into S is continuous.

A semigroup G is called a group if G has an identity e(e ∘ x  = x ∘ e  = x for x  ∈ G) and each x  ∈ G has an inverse x ⁻¹ (x ⁻¹ ∘ x  = x ∘ x ⁻¹  = e). According to whether "∘" is "+" or "×," the group is called additive or multiplicative. If x ∘ y  = y ∘ x, for all x, y  ∈ G, the group G is called Abelian or commutative. A map f of a group G into a group H is called a homomorphism if f(xy)   = f(x)f(y). A bijective homomorphism of a group G into a group H is called an isomorphism.

A group G with a topology is called a topological group (TG) if the map (x, y)   → xy ⁻¹ of G  × G into G is continuous.

Each topological group is a uniform space, hence completely regular. If the topology satisfies the T₀-axiom, the topological group becomes a Tychonoff space, hence Hausdorff. A Hausdorff topological group is metrizable iff the neighborhood system at its identity has a countable base.

The class of all topological groups satisfies arbitrary productive properties. Moreover, the closure of each subgroup (i.e., a subset H of G such that H by its own right is a group under induced algebraic operations) is a topological group and each open subgroup is closed.

Let H be an invariant (xH  = Hx for all x  ∈ G) subgroup of a group G. To each x  ∈ G we associate xH, called the coset. Let G/H denote the set of all cosets. With multiplication (xH)(yH)   = xyH and identity H, G/H becomes a group called the quotient or factor group. The map ϕ: x  → xH of G onto G/H is called the quotient or canonical map; ϕ is a homomorphism because ϕ(xy)   =   ϕ(x)ϕ(y). If G is a topological group, we can endow G/H with the quotient topology (see Section III), making ϕ continuous. It turns out that ϕ is a continuous, open, and surjective homomorphism of the topological group G into the topological group G/H. The quotient topology on G/H is Hausdorff iff H is closed.

VIII.A.1 Examples

$\mathbb{R}$ ⁿ (n  ≥   1) with coordinatewise addition (x ₁,   …   ,x _n )+(y ₁,   …   ,y _n )   =   (x ₁  + y ₁,   …   ,x _n   + y _n ) is an additive Abelian group and has identity (0, …, 0). With the euclidean topology defined in SectionI.A, $\mathbb{R}$ ⁿ is an Abelian additive topological group. In particular, the set $\mathbb{R}$ of all real numbers is an Abelian additive topological group in which $\mathbb{Z}$ , the subgroup of all integers, is a closed invariant subgroup. Thus, the quotient group $\mathbb{R}$ / $\mathbb{Z}$ is a topological group. We denote $\mathbb{R}$ / $\mathbb{Z}$ by $\mathbb{T}$ . $\mathbb{T}$ is actually isomorphic with the circle group $\left\{{\text{e}}^{\text{i}\text{t}}:0\le \text{t}\le 2\pi ,\text{i}=\sqrt{-1}\right\}$ . Hence, $\mathbb{T}$ is a compact Abelian topological group.

If G is an Abelian topological group, the set G ′ of all continuous homomorphisms ϕ: G  → $\mathbb{T}$ is called the dual group of G. We can endow G′ with the compact-open topology T _c (see Section VII). If G is a Hausdorff locally compact Abelian topological group, then (G′, T _c) is also a Hausdorff locally compact Abelian topological group. Repeating the process, we see that G″ (the dual group of G′) with the compact-open topology is also a locally compact Abelian topological group. A celebrated result called the Pontrjagin duality theorem tells us that G and G″ are isomorphic and homeomorphic. Furthermore, if G is compact (discrete), G′ is discrete (compact).

If algebraic operations in a topological group with some additional structure are differentiable instead of being only continuous, it leads to the theory of Lie groups, which form an important branch of mathematics.

VIII.B Topological Vector Spaces, Banach Spaces, and Hilbert Spaces

An algebraic system that is richer than groups is vector space. All scalars involved in this section are either real or complex numbers.

An Abelian additive group E is called a real or complex (depending on which scalars are used throughout) linear or vector space if for all x  ∈ E and scalar λ, λx  ∈ E satisfies the following: λ(x  + y)   =   λx  +   λy; (λ   +   μ)x  =   λx  +   μx; λ(μx)   =   (λμ)x for all scalars λ, μ and all x, y  ∈ E; 1x  = x and 0x  =   0, the identity of the additive group E. If F  ⊂ E and F is a vector space in its own right over the same scalars as those of E, then F is called a linear subspace of E. A subset C of E is called circled if for scalars λ, ∣λ∣   ≤   1 and x  ∈ C, λx  ∈ C. A subset C of a vector space E is called convex if for all x, y  ∈ C and 0   ≤   λ   ≤   1, λx  +   (1   −   λ)y  ∈ C. C is absorbing if for all x  ∈ E there is α₀  >   0 such that for λ, ∣λ∣   ≥   α₀, x  ∈   λ C.

A linear or vector space E endowed with a topology T is called a topological linear or vector space (TVS) if the maps (x, y)   → x  + y and (λ, x)   →   λx of E  × E into E and of K  × E into E, respectively, are continuous, in which the field K of real numbers or complex numbers is endowed with its natural topology. If there exists a base of convex neighborhoods of identity 0   in E, then E is called a locally convex (LC) space.

Clearly, each topological vector space is an Abelian additive topological group, hence it is a uniform space and, therefore, completely regular. Hence, a Hausdorff topological vector space is a Tychonoff space. As for topological groups, a Hausdorff topological vector space is metrizable iff the neighborhood system at 0   has a countable base. Among the metrizable topological vector spaces, there is a very distinguished and useful subclass of spaces called the normable spaces defined below.

Let E be a real (or complex) vector space. A map p: E  → K is called a functional. If p(x)   ≥   0 for all x  ∈ E such that p(λx)   =   ∣λ∣p(x) and p(x  + y)   ≤ p(x)   + p(y), then p is called a seminorm. If p(x)   =   0 ⇔ x  =   0, then p is called a norm, denoted p(x)   =   ∥x∥. A seminorm p on a vector space E defines a pseudometric d(x, y)   = p(x  − y). If p is a norm, d becomes a metric. If the topology of a topological vector space is induced by a norm, it is called a normable topological vector space. If a normable topological vector space is complete in its induced metric topology, it is called a Banach space (BAS). Each Banach space is a complete metric locally convex space called a Fréchet space (F). A Hausdorff locally convex space is called a barreled space (BS) if, in it, each barrel (convex, circled, absorbing, and closed set) is a neighborhood of 0. Each Baire locally convex (BLC) space is a barreled space, and each Fréchet space is a Baire space.

A map f of a vector space E into another vector space F is called linear if f(αx  +   βy)   =   α f(x)   +   βf(y) for all x, y  ∈ E and scalars α, β. A functional f on a normed space E is called bounded if there is a real number M  >   0 such that ∣f(x)∣   ≤ M∥x∥ for all x  ∈ E. An interesting but simple fact is that a linear functional on a normed space is bounded iff it is continuous.

To exhibit the richness that results from the marriage of topology and vector spaces, we cite a few prototypical results.

The first is the so-called Hahn–Banach extension theorem: Let F be a linear subspace of a vector space E. Let p be a seminorm on E, and f a linear functional on F such that ∣f(x)∣   ≤ p(x) for all x  ∈ F. Then there exists a linear functional $\tilde{\text{f}}$ on E such that $\tilde{\text{f}}\left(\text{x}\right)=\text{f}\left(\text{x}\right)$ for all x  ∈ F and $\left|\tilde{\text{f}}\left(\text{x}\right)\right|\le \text{p}\left(\text{x}\right)$ for all x  ∈ E. It is worth comparing this result with Tietze's extension theorem given in Section IV.

The heart of functional analysis, especially that of topological vector spaces, lies in the so-called twins of functional analysis, popularly known as the open-mapping and closed-graph theorems.

VIII.B.1 Open-Mapping Theorem

Each continuous linear surjective map of a Fréchet (in particular, Banach) space onto a barreled (in particular, Fréchet or Banach) space is open.

VIII.B.2 Closed-Graph Theorem

Each linear map of a barreled (in particular, Fréchet or Banach) space into a Fréchet (or Banach) space with closed graph is continuous.

Another important result in topological vector spaces is as follows. If {f _n } is a sequence of continuous linear functionals on a barreled (or Fréchet or Banach) space E such that f(x)   =   lim _n f _n (x), x  ∈ E, then f is also linear and continuous. (This is called the Banach–Steinhaus theorem.)

If E is a real or complex topological vector space, the set E′ of all continuous linear functionals on E is called the dual of E. If E is a normed space, so is E′ with the norm ∥f∥   =   the least upper bound of {∣f(x)∣: ∥x∥   ≤   1}. Actually E′ is a Banach space regardless of E being a complete or incomplete normed space. In special cases E′ can be determined by the points of E, as shown below.

A normed space E is called a pre-Hilbert or an inner product space if the norm ∥·∥ satisfies the so-called parallelogram law: ∥x  + y∥²  +   ∥x  − y∥²  =   2(∥x∥²  +   ∥y∥²). The parallelogram law implies the existence of a bilinear functional 〈,〉: E  × E  → K satisfying the following properties: 〈x, x〉   ≥   0; 〈x, x〉   =   0 ⇔ x  =   0; 〈x  + y, z〉   =   〈x, z〉   +   〈y, z〉; and 〈x, y〉   =   〈y, x〉 (or $〈\overline{\text{y},\text{x}}〉$ for the complex scalars). We see that $\Vert \text{x}\Vert =+\sqrt{〈\text{x},\text{x}〉}$ gives a norm. If a pre-Hilbert space is complete, it is called a Hilbert space (H). For a Hilbert space H, its dual H′ can be identified with the points of H as demonstrated by the so-called Riesz representation theorem: If H is a Hilbert space, then for each f  ∈ H′ there exists a unique element y _f   ∈ H such that f(x)   =   〈x, y _f 〉, x  ∈ H with ∥y _f ∥   =   ∥f∥. (See Table II for the interrelation of various topological spaces.)

Hereafter, we assume all topological vector spaces to be Hausdorff. For any vector space E over $\mathbb{R}$ or $\mathbb{C}$ , E ^* denotes the set of all real or complex linear functionals on E, called the algebraic dual of E. If E is a TVS, E′ denotes the set of all continuous linear functionals on E, called the topological dual or simply the dual of E as mentioned above. If E is LC and E  ≠   {0}, then E′   ≠   {0} and clearly E′   ⊂ E ^*  ⊂ $\mathbb{R}$ ^E (or $\mathbb{C}$ ^E ), where $\mathbb{R}$ ^E carries the pointwise convergence topology. The topology induced from $\mathbb{R}$ ^E to E′ is called the weak-star or w ^* -topology. Under this topology all maps f  → f(x) (for each fixed x, i.e., evaluation maps) are continuous.

Similarly, the coarsest topology on E which makes all the maps x  → f(x) (for any fixed f  ∈ E′) continuous is called the weak topology σ(E, E′) on E, which is coarser than the initial topology of E. The space E with the weak topology σ(E, E′) becomes an LC space. The finest locally convex topology on E that gives the same dual E′ of E is called the Mackey topology τ(E, E′), which is finer than the initial topology T of E. Thus we have σ(E, E′)   ⊂ T  ⊂   τ(E, E′). An LC space with the Mackey topology [i.e., T  =   τ(E, E′)] is called a Mackey space. Every barreled (hence Fréchet, Banach, or Hilbert) space is a Mackey space. However, there exist Mackey spaces which are not Fréchet spaces. The weak, weak^*, and Mackey topologies are extensively used in functional analysis. Here we list only a few samples of their usage.

Let E be a Banach space; then for each weakly compact subset A of E, the convex closure ${\overline{\text{c}}}_{\text{o}}\left(\text{A}\right)$ of A is also weakly compact, where ${\overline{\text{c}}}_{\text{o}}\left(\text{A}\right)$ is the intersection of all closed convex subsets of E containing A. Further, a weakly closed subset A of E is weakly sequentially compact if and only if A is weakly compact. This is known as the Eberlein theorem. Note that in general topological spaces sequential compactness is not equivalent to compactness. Furthermore, if A is a closed convex subset of E such that for each f  ∈ E′ there is x _f   ∈ A with ∣f(x _f )∣   =   the least upper bound of {∣f(x)∣: x  ∈ A}, then A is weakly compact. This is known as the James theorem.

As pointed out earlier, the dual E′ of a normed space E is a Banach space with the norm ∥f∥. The w ^*-topology on E′ is coarser than this norm topology. The unit ball {f  ∈ E′: ∥f∥   ≤   1} of E′ is w ^*-compact (Alaoglu's theorem) but not norm compact in general. Actually, the unit ball of a normed space E is norm compact iff E is finite-dimensional, i.e., E is homeomorphic to $\mathbb{R}$ ⁿ (or $\mathbb{C}$ ⁿ ) for some finite positive integer n.

Since the dual E′ of a normed space E is a normed (actually Banach) space, we can consider the dual E″ of E′. E″ is called the bidual of E. Clearly E″ is also a Banach space. There is a natural embedding of E in E″. Put x″(f)   = f(x), f  ∈ E′, for each x  ∈ E. Then it can be verified that x″   ∈ E″ for each x  ∈ E and so x  → x″ gives a mapping of E into E″ such that ∥x″∥   =   ∥x∥. Thus E is embedded into E″ isometrically. In general this embedding is not surjective, i.e., E  ⊂ E″, E  ≠ E″. Whenever E  = E″, E is called reflexive. Since E″ is a Banach space, it follows that a reflexive normed space must be a Banach space. By virtue of the Riesz representation theorem, each Hilbert space is reflexive. Here are some examples of reflexive and nonreflexive spaces. Let 1   ≤ p  <   ∞ be a given real number. Let ℓ _p   =   {{a _i }: ∑^∞ _i=1∣a _i ∣ ^p   <   ∞}. Then ℓ _p is a Banach space with the norm ${\Vert \left\{{\text{a}}_{\text{i}}\right\}\Vert }_{\text{p}}={\left\{{\displaystyle {\sum }_{\text{i}=1}^{\infty }{\left|{\text{a}}_{\text{i}}\right|}^{\text{p}}}\right\}}^{\frac{1}{\text{p}}}$ . For all p's, 1   < p  <   ∞, ℓ _p is a reflexive Banach space; note that ℓ₂, being a Hilbert space, is reflexive, but ℓ₁ is not reflexive. Note that for 1   < p  <   ∞, ℓ′ _p   =   ℓ _q , where (1/p)   +   (1/q)   =   1 and ℓ′₁  =   ℓ_∞, the space of all bounded sequences.

For a Tychonoff space (X, T), C(X)   =   (C(X), T _c) denotes the set of all real or complex continuous functions on X, endowed with the compact-open topology T _c (see Section VII). One sees that C(X) is an LC space. It is not, in general, a metrizable, or a complete, or a barreled or a Fréchet or a Banach space. As shown above, if X is compact, then C(X) is a Banach space. Conversely, if C(X) is a Banach space, then indeed X is compact. This immediately suggests an interplay of topological properties of X and C(X).

To display this duet between X and C(X) briefly, we need the following concept. Let L  ∈ C′(X), the dual of C(X). The smallest compact subset A of X such that for all f  ∈ C(X) with f(A)   =   0 implies L(f)   =   0 is called the support of L, denoted suppL. If B′ is a subset of C′(X), then suppB′   =   Cl{∪ (suppL: L  ∈ B′)} is the support of B′. Now we have the following:

•

C(X) is a Banach space iff X is compact.

•

C(X) is metrizable iff X is hemicompact.

•

C(X) is complete iff X is a k _r -space.

•

C(X) is a Fréchet space iff X is a hemicompact, k _r -space.

•

C(X) is a Mackey space iff suppB′ is compact for each convex circled w*-compact subset B′ of C′(X).

•

C(X) is a barreled space iff X is a μ-space [i.e., for each w*-bounded subset B′ of C′(X), suppB′ is compact].

VIII.C Topological Algebras

An algebraic structure richer than vector spaces is what is known as algebra. A real or complex vector space A is called an algebra if there is some multiplication defined on A, that is, for all x, y  ∈ A, xy  ∈ A satisfying the following axioms: x(y  + z)   = xy  + xz;(x  + y)z  = xz  + yz; λ(xy)   =   (λx)y  = x(λy); x(yz)   =   (xy)z  = xyz for all x, y, z  ∈ A and scalar λ. An algebra A is called commutative if xy  = yx for all x, y  ∈ A. A has an identity e if ex  = xe  = x for all x  ∈ A. An element x  ∈ A is called invertible if there is y  ∈ A such that xy  = yx  = e. y is called the inverse of x and written y  = x ⁻¹. An algebra in which each nonzero element has an inverse is called a division algebra. An operation *:x →x* of A onto A is called an involution if $\left(\text{x}+\text{y}\right)*=\text{x}*+\text{y}*,\left(\lambda \text{x}\right)*=\overline{\lambda }\text{x}*,\left(\text{x}\text{y}\right)*=\text{y}*\text{x}*$ , and x ^**  = x.

An algebra A with a topology T is called a topological algebra (TA) if the maps (x, y)   → x  + y, (λ, x)   →   λx, and (x, y)   → xy are continuous. It is clear that each topological algebra is a topological vector space. Hence, the results pertaining to topological vector spaces can be used for topological algebras. If the topology is given by a norm on an algebra, it is called a normed algebra, provided that ∥xy∥   ≤   ∥x∥ ∥y∥. A complete normed algebra is called a Banach algebra (BAA). A Banach algebra with an involution * is called a B*-algebra if ∥xx*∥   =   ∥x∥² for all x. We deal with these algebras in the next section.

If the topology of a topological algebra is locally convex, it is called a locally convex algebra. A complete metric locally convex algebra is called a B ₀-algebra (B₀-A). Clearly, each Banach algebra is a B₀-algebra.

Examples of nonnormed algebras also abound in the literature. For example, for any Tychonoff space (X, T), the set C(X) of all continuous real or complex functions forms an algebra with pointwise operations, i.e.,

$\left(\text{f}+\text{g}\right)\left(\text{x}\right)=\text{f}\left(\text{x}\right)+\text{g}\left(\text{x}\right),\left(\lambda \text{f}\right)\left(\text{x}\right)=\lambda \text{f}\left(\text{x}\right),\left(\text{f}\text{g}\right)\left(\text{x}\right)=\text{f}\left(\text{x}\right)\text{g}\left(\text{x}\right),$

where λ is a real or complex scalar. With the compact-open topology T _c , C(X) is actually a locally convex algebra. Among many applications associated with C(X), there is a celebrated result called the Stone–Weierstrass theorem: Let A be a subalgebra of C(X) such that

i

for all x, y  ∈ X, x  ≠ y, there is f  ∈ A with f(x)   ≠ f(y), (i.e., A separates points of X),

ii

for each x  ∈ X, there is f  ∈ A with f(x)   ≠   0, and

iii

for each f  ∈ A, the complex conjugate $\overline{\text{f}}\left[\text{i}.\text{e}.,\overline{\text{f}}\left(\text{x}\right)=\overline{\text{f}\left(\text{x}\right)}\right]\text{is}\text{in}\text{A}$ .

Then A is dense in C(X) [i.e.}, Ā   = C(X)]. If, in addition, A is a closed subalgebra of C(X), then A  = C(X). Note that condition (iii) is redundant for real algebras. From this theorem, one derives the classical Weierstrass theorem: Each continuous real function on [a, b], −∞   < a  < b  <   ∞, can be approximated by polynomials.

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