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Lemma 1.4.5. Let \( A \) be a Banach algebra and \( I \) a proper modular ideal of A. If \( u \in A \) is an identity modulo \( I \), then\n\n\[ I \cap \{ x \in A : \parallel x - u\parallel < 1\} = \varnothing . \]\n\nIn particular, \( \bar{I} \) is also a proper ideal, and every maximal modular ideal of \( A \) is clo... | Proof. Let \( {A}^{\prime } \) be defined to be \( A \) if \( A \) has an identity \( e \) and \( {A}^{\prime } = {A}_{e} \), the unitisation of \( A \), otherwise. If \( x \in A \) is such that \( \parallel x - u\parallel < 1 \), then \( e - \left( {u - x}\right) \) is invertible in \( {A}^{\prime } \) by Lemma 1.2.6.... | Yes |
Proposition 1.4.7. A closed linear subspace \( I \) of \( {L}^{1}\left( G\right) \) is an ideal in \( {L}^{1}\left( G\right) \) if and only if \( I \) is two-sided translation invariant. | Proof. Suppose that \( I \) is two-sided translation invariant. We have to show that \( g * f \in I \) and \( f * g \in I \) for each \( f \in I \) and \( g \in {L}^{1}\left( G\right) \) . Let \( \varphi \in {L}^{\infty }\left( G\right) \) be such that \( {\int }_{G}f\left( x\right) \varphi \left( x\right) {dx} = 0 \) ... | Yes |
Lemma 1.4.8. Let \( I \) be a closed ideal of a normed algebra \( A \). (i) Suppose that \( A \) has a (bounded) left approximate identity. Then \( A/I \) has a (bounded) left approximate identity. | Proof. (i) Clearly, if \( {\left( {e}_{\lambda }\right) }_{\lambda } \) is a (bounded) left approximate identity for \( A \), then \( {\left( {e}_{\lambda } + I\right) }_{\lambda } \) is a (bounded) left approximate identity for \( A/I \) . | Yes |
Lemma 1.4.9. Let \( I \) and \( J \) be closed ideals with bounded left approximate identities. Then the ideals \( I \cap J \) and \( \overline{I + J} \) both have bounded left approximate identities. | Proof. Let \( {\left( {u}_{\lambda }\right) }_{\lambda } \) and \( {\left( {v}_{\mu }\right) }_{\mu } \) be bounded left approximate identities for \( I \) and \( J \), respectively. Then \( {\left( {u}_{\lambda }{v}_{\mu }\right) }_{\lambda ,\mu } \) is a bounded left approximate identity for \( I \cap J \) since \( {... | Yes |
Theorem 1.4.12. Let \( A \) be a faithful commutative Banach algebra. Then the mapping \( L : x \rightarrow {L}_{x} \) is a continuous isomorphism of \( A \) onto the ideal \( L\left( A\right) = \) \( \left\{ {{L}_{x} : x \in A}\right\} \) of \( M\left( A\right) \) . If \( A \) has an approximate identity bounded by \(... | Proof. It is obvious that \( x \rightarrow {L}_{x} \) is a norm decreasing homomorphism of \( A \) into \( M\left( A\right) \) . The range of \( A \) is an ideal of \( M\left( A\right) \) since, for \( T \in M\left( A\right) \) and \( x, y \in A \) ,\n\n\[ \n\left( {{L}_{x}T}\right) y = x\left( {Ty}\right) = \left( {Tx... | Yes |
Let \( X \) be a locally compact Hausdorff space. We show that the multiplier algebra of \( {C}_{0}\left( X\right) \) can be canonically identified with \( {C}^{b}\left( X\right) \) . | Clearly, any \( f \in {C}^{b}\left( X\right) \) defines a multiplier \( {T}_{f} \) of \( {C}_{0}\left( X\right) \) by \( {T}_{f}g = {fg}, g \in \) \( {C}_{0}\left( X\right) \), and \( \begin{Vmatrix}{T}_{f}\end{Vmatrix} \leq \parallel f{\parallel }_{\infty } \) . Conversely, let \( T \) be an arbitrary multiplier of \(... | Yes |
Let \( G \) be a locally compact Abelian group. We determine the multiplier algebra of \( {L}^{1}\left( G\right) \) . Since \( {L}^{1}\left( G\right) \) is an ideal in \( M\left( G\right) \), it is immediate that for every \( \mu \in M\left( G\right) \), the convolution operator \( {T}_{\mu } : {L}^{1}\left( G\right) \... | It is less evident that every multiplier of \( {L}^{1}\left( G\right) \) arises in this way. To see this, let \( T \in M\left( {{L}^{1}\left( G\right) }\right) \) be given and view \( T \) as a continuous linear mapping from \( {L}^{1}\left( G\right) \) into \( M\left( G\right) \) . Let \( {\left( {u}_{\alpha }\right) ... | Yes |
Proposition 1.5.1. Let \( A \) and \( B \) be algebras. On the vector space \( A \otimes B \) there exists a unique product with respect to which \( A \otimes B \) is an algebra, the algebraic tensor product and which satisfies \( \left( {a \otimes b}\right) \left( {c \otimes d}\right) = {ac} \otimes {bd} \) for all \(... | Proof. Given \( a \in A \) and \( b \in B \), there exists a unique linear operator \( \lambda \left( {a, b}\right) \) on \( A \otimes B \) such that\n\n\[ \lambda \left( {a, b}\right) \left( {c \otimes d}\right) = {ac} \otimes {bd}\;\left( {c \in A, d \in B}\right) . \]\n\nThe mapping \( \left( {a, b}\right) \rightarr... | Yes |
Lemma 1.5.2. The projective tensor norm on \( A \otimes B \) is an algebra norm. | Proof. Let \( x = \mathop{\sum }\limits_{{i = 1}}^{n}{a}_{i} \otimes {b}_{i} \) and \( y = \mathop{\sum }\limits_{{j = 1}}^{m}{c}_{j} \otimes {d}_{j} \) . Then\n\n\[ \n{xy} = \mathop{\sum }\limits_{{i = 1}}^{n}\mathop{\sum }\limits_{{j = 1}}^{m}{a}_{i}{c}_{j} \otimes {b}_{i}{d}_{j} \n\]\n\nand\n\n\[ \n\mathop{\sum }\li... | Yes |
Lemma 1.5.3. Let \( A \) and \( B \) be Banach algebras having left approximate identities bounded by \( M \) and \( N \), respectively. Then \( A{\widehat{ \otimes }}_{\pi }B \) has a left approximate identity bounded by \( {MN} \) . | Proof. Let \( {\left( {u}_{\lambda }\right) }_{\lambda } \) and \( {\left( {v}_{\mu }\right) }_{\mu } \) be left approximate identities bounded \( M \) and \( N \) of \( A \) and \( B \), respectively. Let \( x = \mathop{\sum }\limits_{{j = 1}}^{\infty }{a}_{j} \otimes {b}_{j} \in A{\widehat{ \otimes }}_{\pi }B \) such... | Yes |
Proposition 1.5.5. Let \( G \) and \( H \) be locally compact groups. Then there exists an isometric \( * \) -isomorphism \( \phi \) from \( {L}^{1}\left( G\right) {\widehat{ \otimes }}_{\pi }{L}^{1}\left( H\right) \) onto \( {L}^{1}\left( {G \times H}\right) \) such that\n\n\[ \phi \left( {f \otimes g}\right) \left( {... | Proof. We define a linear mapping \( \psi \) from \( {L}^{1}\left( {G \times H}\right) \) onto \( {L}^{1}\left( {G,{L}^{1}\left( H\right) }\right) \) by \( \left\lbrack {\psi \left( F\right) \left( x\right) }\right\rbrack \left( y\right) = F\left( {x, y}\right) \) . It is easy to check that \( \psi \left( {F}^{ * }\rig... | Yes |
Lemma 2.1.1. Let \( A \) be a real or complex algebra with identity \( e \), and let \( \varphi \) be a linear functional on \( A \) satisfying\n\n\[ \varphi \left( e\right) = 1\text{ and }\varphi \left( {x}^{2}\right) = \varphi {\left( x\right) }^{2} \]\n\nfor all \( x \in A \) . Then \( \varphi \) is multiplicative. | Proof. By assumption we have\n\n\[ \varphi \left( {x}^{2}\right) + \varphi \left( {{xy} + {yx}}\right) + \varphi \left( {y}^{2}\right) = \varphi \left( {{x}^{2} + {xy} + {yx} + {y}^{2}}\right) \]\n\n\[ = \varphi \left( {\left( x + y\right) }^{2}\right) = {\left( \varphi \left( x\right) + \varphi \left( y\right) \right)... | Yes |
Theorem 2.1.2. Let \( A \) be a unital Banach algebra. For a linear functional \( \varphi \) on \( A \) the following conditions are equivalent.\n\n(i) \( \varphi \) is nonzero and multiplicative.\n\n(ii) \( \varphi \left( e\right) = 1 \) and \( \varphi \left( x\right) \neq 0 \) for every invertible element \( x \) of ... | Proof. If \( \varphi \) is nonzero and multiplicative, then \( \varphi \left( e\right) = 1 \) and \( 1 = \varphi \left( x\right) \varphi \left( {x}^{-1}\right) \) whenever \( x \) is invertible. Thus (i) \( \Rightarrow \) (ii). Also,(ii) \( \Rightarrow \) (iii) is obvious since if \( \lambda \in {\rho }_{A}\left( x\rig... | Yes |
Lemma 2.1.5. Let \( A \) be a Banach algebra. Every \( \varphi \in \Delta \left( A\right) \) is a bounded linear functional on \( A \) and \( \left| {\varphi \left( x\right) }\right| \leq {r}_{A}\left( x\right) \) holds for all \( x \in A \) . In particular, \( \parallel \varphi \parallel \leq 1 \) and \( \parallel \va... | Proof. We can assume that \( A \) has an identity \( e \) . If \( x \in A \) and \( \lambda \in \mathbb{C} \) are such that \( \left| \lambda \right| > {r}_{A}\left( x\right) \), then \( {r}_{A}\left( {\left( {1/\lambda }\right) x}\right) < 1 \) and hence \( {\lambda e} - x = \lambda \left( {e - \left( {1/\lambda }\rig... | Yes |
We claim that \( f \circ g \in A \). | For that notice first that if polynomials\n\n\[ p\left( z\right) = \mathop{\sum }\limits_{{j = 0}}^{n}{a}_{j}{z}^{j}\text{ and }q\left( z\right) = \mathop{\sum }\limits_{{k = 0}}^{m}{b}_{k}{z}^{k}, \]\n\n\( {a}_{j},{b}_{k} \in \mathbb{C} \), are given then, for any \( z \in \mathbb{D} \), \n\n\[ \left( {{\left. p\right... | Yes |
Define a bounded linear operator \( T \) on \( C\left\lbrack {0,1}\right\rbrack \) by\n\n\[ \n{Tf}\left( t\right) = {\int }_{0}^{t}f\left( s\right) {ds},\;f \in C\left\lbrack {0,1}\right\rbrack ,\;t \in \left\lbrack {0,1}\right\rbrack .\n\]\n\nLet \( A \) be the norm closure in \( \mathcal{B}\left( {C\left\lbrack {0,1}... | A straightforward induction argument shows\n\n\[ \n\left| {{T}^{n}f\left( t\right) }\right| \leq \parallel f{\parallel }_{\infty }\frac{{t}^{n}}{n!}\n\]\n\nfor all \( t \in \left\lbrack {0,1}\right\rbrack \) and \( n \in \mathbb{N} \) . Hence\n\n\[ \n{\begin{Vmatrix}{T}^{n}f\end{Vmatrix}}_{\infty } \leq \frac{1}{n!}\pa... | Yes |
Theorem 2.1.8. For a commutative Banach algebra \( A \), the mapping\n\n\[ \varphi \rightarrow \ker \varphi = \{ x \in A : \varphi \left( x\right) = 0\} \]\n\nis a bijection between \( \Delta \left( A\right) \) and \( \operatorname{Max}\left( A\right) \), the set of all maximal modular ideals in \( A \) . | Proof. For \( \varphi \in \Delta \left( A\right) \), ker \( \varphi \) is an ideal and a closed linear subspace of codi-mension one in \( A \) . To verify that \( \ker \varphi \) is modular simply choose \( u \in A \) such that \( \varphi \left( u\right) = 1 \) . Then, for any \( x \in A \),\n\n\[ \varphi \left( {{ux} ... | Yes |
Corollary 2.1.10. Let \( \phi \) be a homomorphism from a commutative Banach algebra \( A \) into a semisimple commutative Banach algebra \( B \) . Then \( \phi \) is continuous. | Proof. By the closed graph theorem it suffices to show that if \( {x}_{n} \in A, n \in \mathbb{N} \) , are such that \( {x}_{n} \rightarrow 0 \) and \( \phi \left( {x}_{n}\right) \rightarrow b \) for some \( b \in B \), then \( b = 0 \) . Let \( \varphi \in \Delta \left( B\right) \) . Then \( \varphi \circ \phi \in \De... | Yes |
Corollary 2.1.11. On a semisimple commutative Banach algebra all Banach algebra norms are equivalent. | Proof. Suppose \( A \) is a semisimple commutative Banach algebra, and let \( \parallel \cdot {\parallel }_{1} \) and \( \parallel \cdot {\parallel }_{2} \) be two Banach algebra norms on \( A \) . The statement follows by applying Corollary 2.1.10 with \( \phi \) the identity mappings \( \left( {A,\parallel \cdot {\pa... | Yes |
Corollary 2.1.12. Every involution on a semisimple commutative Banach algebra \( A \) is continuous. | Proof. Let \( \parallel \cdot \parallel \) be the given norm an \( A \) . We define a new norm \( \left| \cdot \right| \) on \( A \) by \( \left| x\right| = \begin{Vmatrix}{x}^{ * }\end{Vmatrix} \) . It is clear that \( \left| \cdot \right| \) is submultiplicative. If \( {x}_{n} \in A, n \in \mathbb{N} \) , form a Cauc... | Yes |
Corollary 2.1.13. The algebra \( {C}^{\infty }\left\lbrack {0,1}\right\rbrack \) admits no Banach algebra norm. | Proof. Suppose there is a Banach algebra norm \( \parallel \cdot \parallel \) on \( {C}^{\infty }\left\lbrack {0,1}\right\rbrack \) . Applying Corollary 2.1.10 to the identity mapping from \( {C}^{\infty }\left\lbrack {0,1}\right\rbrack \) into \( C\left\lbrack {0,1}\right\rbrack \) we see that there exists \( c > 0 \)... | Yes |
Theorem 2.2.5. Let \( A \) be a commutative Banach algebra. For each \( x \in A \) ,\n\n\[ \n{\sigma }_{A}\left( x\right) \smallsetminus \{ 0\} \subseteq \widehat{x}\left( {\Delta \left( A\right) }\right) = \{ \varphi \left( x\right) : \varphi \in \Delta \left( A\right) \} \subseteq {\sigma }_{A}\left( x\right) .\n\]\n... | Proof. Suppose first that \( A \) has an identity \( e \) . Then \( \varphi \left( x\right) \in {\sigma }_{A}\left( x\right) \) for every \( \varphi \in \Delta \left( A\right) \) (see Theorem 2.1.2). Conversely, if \( \lambda \in {\sigma }_{A}\left( x\right) \), then\n\n\[ \nI = \left( {{\lambda e} - x}\right) A\n\]\n\... | Yes |
Theorem 2.2.7. Let \( A \) be a commutative Banach algebra and \( \Gamma \) the Gelfand representation of \( A \) . (i) \( \Gamma \) maps \( A \) into \( {C}_{0}\left( {\Delta \left( A\right) }\right) \) and is norm decreasing. (ii) \( \Gamma \left( A\right) \) strongly separates the points of \( \Delta \left( A\right)... | Proof. (i) Since, by Theorem \( {2.2.3},\Delta \left( {A}_{e}\right) \) is the one-point compactification of \( \Delta \left( A\right) \) and \( \widehat{x}\left( {\varphi }_{\infty }\right) = 0 \) for \( x \in A \), we have \( \widehat{x} \in {C}_{0}\left( {\Delta \left( A\right) }\right) \) . Moreover, by Theorem 2.2... | Yes |
Let \( X \) be a locally compact Hausdorff space. The closed ideals in \( {C}_{0}\left( X\right) \) have been completely determined in Theorem 1.3.6. In particular, \[ x \rightarrow {M}_{x} = \left\{ {f \in {C}_{0}\left( X\right) : f\left( x\right) = 0}\right\} \] sets up a one-to-one correspondence between the points ... | Indeed, given \( x \in X \) and an open neighbourhood \( V \) of \( x \), by Urysohn’s lemma there exists \( f \in \) \( {C}_{0}\left( X\right) \) such that \( f\left( x\right) \neq 0 \) and \( {\left. f\right| }_{X \smallsetminus V} = 0 \), and hence \( V \) contains the Gelfand neighbourhood \( \left\{ {y : \left| {{... | Yes |
Let \( A = {C}^{n}\left\lbrack {a, b}\right\rbrack \), and for each \( t \in \left\lbrack {0,1}\right\rbrack \) define \( {\varphi }_{t} \in \Delta \left( A\right) \) by \( {\varphi }_{t}\left( f\right) = f\left( t\right) \). We claim that \[ \phi : \left\lbrack {a, b}\right\rbrack \rightarrow \Delta \left( A\right) ,\... | Obviously, \( \phi \) is injective and continuous. Let \( M \) be any maximal ideal in \( A \). Then, by the same reasoning as in the proof of Theorem 1.3.6, we find \( s \in \left\lbrack {a, b}\right\rbrack \) such that \( M = \{ f \in A : f\left( s\right) = 0\} \). It follows that \( M = \ker {\varphi }_{s} \). Hence... | Yes |
We determine the structure space of \( {l}^{1}\left( \mathbb{Z}\right) \) . For \( z \in \mathbb{T} \) , define \( {\varphi }_{z} : {l}^{1}\left( \mathbb{Z}\right) \rightarrow \mathbb{C} \) by\n\n\[ \n{\varphi }_{z}\left( f\right) = \mathop{\sum }\limits_{{n \in \mathbb{Z}}}f\left( n\right) {z}^{-n} \n\]\n\nThen, for \... | \[ \n{\varphi }_{z}\left( {f * g}\right) = \mathop{\sum }\limits_{{n \in \mathbb{Z}}}\left( {\mathop{\sum }\limits_{{m \in \mathbb{Z}}}f\left( {n - m}\right) g\left( m\right) }\right) {z}^{-n} \n\]\n\n\[ \n= \mathop{\sum }\limits_{{n, m \in \mathbb{Z}}}f\left( n\right) g\left( m\right) {z}^{-\left( {n + m}\right) } \n\... | Yes |
Theorem 2.2.11. If \( f \in {AC}\left( \mathbb{T}\right) \) is such that \( f\left( z\right) \neq 0 \) for all \( z \in \mathbb{T} \), then \( 1/f \in {AC}\left( \mathbb{T}\right) \) ; that is, \( 1/f \) has an absolutely convergent Fourier series. | Proof. With the previous identification of \( \Delta \left( {{AC}\left( \mathbb{T}\right) }\right) \) with \( \mathbb{T} \), the assumption on \( f \) means that \( f \) belongs to no maximal ideal of \( {AC}\left( \mathbb{T}\right) \) . Thus \( f \) is invertible in \( {AC}\left( \mathbb{T}\right) \) and so \( 1/f \in... | Yes |
Lemma 2.2.12. Let \( A \) and \( B \) be commutative Banach algebras. If \( A \) and \( B \) are algebraically isomorphic, then \( \Delta \left( A\right) \) and \( \Delta \left( B\right) \) are homeomorphic. | Proof. Suppose \( \phi : A \rightarrow B \) is an algebra isomorphism. Let \( {\phi }^{ * } : \Delta \left( B\right) \rightarrow \Delta \left( A\right) \) be the dual mapping; that is,\n\n\[{\phi }^{ * }\left( \varphi \right) \left( a\right) = \varphi \left( {\phi \left( a\right) }\right) ,\;a \in A,\;\varphi \in \Delt... | Yes |
Corollary 2.2.13. For locally compact Hausdorff spaces \( X \) and \( Y \) the following conditions are equivalent.\n\n(i) \( {C}_{0}\left( X\right) \) and \( {C}_{0}\left( Y\right) \) are isometrically isomorphic.\n\n(ii) \( {C}_{0}\left( X\right) \) and \( {C}_{0}\left( Y\right) \) are algebraically isomorphic.\n\n(i... | Proof. (i) \( \Rightarrow \) (ii) is trivial, and (ii) \( \Rightarrow \) (iii) is a consequence of the preceding lemma and Example 2.2.8. Finally, if \( \phi : X \rightarrow Y \) is a homeomorphism, then \( f \rightarrow f \circ \phi \) is an isometric algebra isomorphism from \( {C}_{0}\left( Y\right) \) to \( {C}_{0}... | Yes |
Proposition 2.2.14. Let \( X \) be a locally compact Hausdorff space and let \( A \) be a family of functions in \( {C}_{0}\left( X\right) \) which strongly separates the points of \( X \) . Then the topology of \( X \) equals the weak topology with respect to the functions \( x \rightarrow f\left( x\right), f \in A \)... | Proof. The given topology on \( X \) is stronger than the weak topology. Thus it suffices to show that given \( x \in X \) and an open neighbourhood \( U \) of \( x \) in \( X \), there exists a set \( V \) such that \( x \in V \subseteq U \) and \( V \) is open in the weak topology. Let \( \widetilde{X} \) be \( X \) ... | Yes |
Lemma 2.2.15. Let \( I \) be a closed ideal of \( A \) and \( q : A \rightarrow A/I \) the quotient homomorphism.\n\n(i) The map \( \varphi \rightarrow \varphi \circ q \) is a homeomorphism from \( \Delta \left( {A/I}\right) \) onto \( h\left( I\right) \) .\n\n(ii) The map \( \varphi \rightarrow {\left. \varphi \right|... | Proof. (i) It is obvious that the map is a bijection. It is a homeomorphism since\n\n\[ U\left( {\varphi, x + I,\epsilon }\right) \circ q = \{ \psi \circ q : \psi \in \Delta \left( {A/I}\right) ,\left| {\psi \left( {x + I}\right) - \varphi \left( {x + I}\right) }\right| < \epsilon \} \]\n\n\[ = \{ \rho \in h\left( I\ri... | Yes |
Proposition 2.2.16. Let \( A \) be a commutative Banach algebra and let \( T \in \) \( M\left( A\right) \) . Then there exists a unique continuous function \( f \) on \( \Delta \left( A\right) \) such that \( \widehat{Tx}\left( \varphi \right) = f\left( \varphi \right) \widehat{x}\left( \varphi \right) \) for all \( \v... | Proof. If \( \varphi \in \Delta \left( A\right) \) and \( x, y \in A \) are such that \( \widehat{x}\left( \varphi \right) \neq 0 \) and \( \widehat{y}\left( \varphi \right) \neq 0 \), then it follows from \( \left( {Tx}\right) y = x\left( {Ty}\right) \) that\n\n\[ \frac{\widehat{Tx}\left( \varphi \right) }{\widehat{x}... | Yes |
Lemma 2.3.3. Let \( A \) be a unital commutative Banach algebra, and suppose that \( E \subseteq A \) generates \( A \) . Then the mapping\n\n\[ \phi : \Delta \left( A\right) \rightarrow \mathop{\prod }\limits_{{x \in E}}{\sigma }_{A}\left( x\right) ,\;\varphi \rightarrow {\left( \varphi \left( x\right) \right) }_{x \i... | Proof. Assume first that \( {\varphi }_{1},{\varphi }_{2} \in \Delta \left( A\right) \) are such that \( {\varphi }_{1}\left( x\right) = {\varphi }_{2}\left( x\right) \) for all \( x \in E \) . Let \( B \) denote the smallest subalgebra of \( A \) containing \( E \) and the identity. Then \( B \) is dense in \( A \), a... | Yes |
Lemma 2.3.5. Every compact convex subset \( K \) of \( {\mathbb{C}}^{n} \) is polynomially convex. | Proof. We view \( {\mathbb{C}}^{n} \) as a \( {2n} \) -dimensional real vector space. Then, given \( w \in \) \( {\mathbb{C}}^{n} \smallsetminus K \), there exist a real linear functional \( \psi \) on \( {\mathbb{C}}^{n} = {\mathbb{R}}^{2n} \) and \( \alpha \in \mathbb{R} \) such that\n\n\[ \psi \left( w\right) > \alp... | Yes |
Example 2.3.9. Let \( n \geq 2 \) and\n\n\[ K = \left\{ {z = \left( {{z}_{1},\ldots ,{z}_{n}}\right) \in {\mathbb{C}}^{n} : \left| {z}_{j}\right| = 1,1 \leq j \leq n}\right\} .\n\]\n\nAssuming that \( K \) is polynomially convex we find a polynomial \( p \) in \( n \) variables such that \( \left| {p\left( z\right) }\r... | Define a polynomial \( q \) in one variable by\n\n\[ q\left( w\right) = p\left( {w,1,\ldots ,1}\right) ,\;w \in \mathbb{C}.\n\]\n\nThen \( \left| {q\left( w\right) }\right| < 1 \) for all \( w \in \mathbb{C} \) with \( \left| w\right| = 1 \) and \( q\left( 0\right) = 1 \) . This contradicts the maximum modulus principl... | No |
Lemma 2.4.4. Let \( A \) be a commutative \( {C}^{ * } \) -algebra. Then the Gelfand homomorphism is a \( * \) -homomorphism; that is, \( \widehat{{x}^{ * }} = \overline{\widehat{x}} \) for all \( x \in A \) . | Proof. We have to show that \( \varphi \left( {x}^{ * }\right) = \overline{\varphi \left( x\right) } \) for \( \varphi \in \Delta \left( A\right) \) and \( x \in A \) . Of course, we can assume that \( A \) has an identity \( e \) . Let\n\n\[ \varphi \left( x\right) = \alpha + {i\beta }\text{ and }\varphi \left( {x}^{ ... | Yes |
For a commutative \( {C}^{ * } \) -algebra \( A \) the Gelfand homomorphism is an isometric \( * \) -isomorphism from \( A \) onto \( {C}_{0}\left( {\Delta \left( A\right) }\right) \) . | To prove that \( x \rightarrow \widehat{x} \) is isometric, note first that if \( y = {y}^{ * } \in A \), then \( \parallel y{\parallel }^{2} = \begin{Vmatrix}{{y}^{ * }y}\end{Vmatrix} = \begin{Vmatrix}{y}^{2}\end{Vmatrix} \) and hence by induction \( \parallel y{\parallel }^{{2}^{n}} = \begin{Vmatrix}{y}^{{2}^{n}}\end... | Yes |
Corollary 2.4.6. For two commutative \( {C}^{ * } \) - algebras \( A \) and \( B \) the following are equivalent.\n\n(i) \( \Delta \left( A\right) \) and \( \Delta \left( B\right) \) are homeomorphic.\n\n(ii) There exists an isometric \( * \) -isomorphism between \( A \) and \( B \) .\n\n(iii) There exists an algebra i... | Proof. The implication (ii) \( \Rightarrow \) (iii) is trivial and, as we have seen earlier (Lemma 2.2.10), the implication (iii) \( \Rightarrow \) (i) holds even for general commutative Banach algebras \( A \) and \( B \) . To prove (i) \( \Rightarrow \) (ii), note first that if \( \phi : \Delta \left( A\right) \right... | Yes |
Corollary 2.4.7. Let \( A \) be a commutative \( {C}^{ * } \) -algebra. For \( x \in A \) consider the following conditions.\n\n(i) \( x = {x}^{ * } \) .\n\n(ii) \( {\sigma }_{A}\left( x\right) \subseteq \mathbb{R} \) .\n\n(iii) \( \widehat{x} \) is real-valued.\n\n(iv) \( x = {y}^{ * }y \) for some \( y \in A \) .\n\n... | Proof. The equivalence of (ii) and (iii) and of (v) and (vi) follows immediately from\n\n\[ \widehat{x}\left( {\Delta \left( A\right) }\right) \cup \{ 0\} = {\sigma }_{A}\left( x\right) \cup \{ 0\} . \]\n\nThe Gelfand homomorphism is injective and satisfies \( \widehat{{x}^{ * }} = \overline{\widehat{x}} \) . Therefore... | Yes |
Lemma 2.4.8. Let \( A \) be a commutative \( {C}^{ * } \) -algebra with identity \( e \) and \( B \) a \( {C}^{ * } \) -subalgebra of \( A \) containing e. Then \( {\sigma }_{A}\left( x\right) = {\sigma }_{B}\left( x\right) \) for each \( x \in B \) . | Proof. It suffices to show that if \( y \in B \) is invertible in \( A \), then \( y \) is already invertible in \( B \) . Let \( y \in B \cap G\left( A\right) \) and note first that \( {y}^{ * } \in G\left( A\right) \) since\n\n\[ \n{\left( {y}^{-1}\right) }^{ * }{y}^{ * } = {\left( y{y}^{-1}\right) }^{ * } = {e}^{ * ... | Yes |
Example 2.5.3. Let \( \mathbb{D} = \{ z \in \mathbb{C} : \left| z\right| \leq 1\} \) and \( \mathbb{T} = \{ z \in \mathbb{C} : \left| z\right| = 1\} \), the boundary of \( \mathbb{D} \). (1) The algebra \( P\left( \mathbb{D}\right) \) is generated by the function \( f\left( z\right) = z, z \in \mathbb{D} \) . Now, \( {... | In fact, if \( \left| \lambda \right| > 1 \), then the function\n\n\[ z \rightarrow \frac{1}{\lambda - f\left( z\right) } = \frac{1}{\lambda }\frac{1}{1 - \frac{z}{\lambda }} = \frac{1}{\lambda }\mathop{\sum }\limits_{{n = 0}}^{\infty }{\left( \frac{z}{\lambda }\right) }^{n} \]\n\nis a uniform limit of polynomials on \... | Yes |
Lemma 2.5.6. For any compact subset \( X \) of \( {\mathbb{C}}^{n} \) , \[ {\widehat{X}}_{r} = \left\{ {z \in {\mathbb{C}}^{n} : p\left( z\right) \in p\left( X\right) \text{ for every polynomial }p}\right\} . \] | Proof. Let \( z \in {\mathbb{C}}^{n} \) and suppose that there is a polynomial \( p \) such that \( p\left( z\right) \notin \) \( p\left( X\right) \) . Then \( q\left( w\right) = p\left( w\right) - p\left( z\right) \) is non-zero on \( X \) and \[ 1 > 0 = \left| {q\left( z\right) }\right| \cdot {\begin{Vmatrix}{\left. ... | Yes |
Theorem 2.5.7. Let \( X \) be a compact subset of \( {\mathbb{C}}^{n} \). (i) The restriction map \( \phi : f \rightarrow {\left. f\right| }_{X} \) is an isometric isomorphism from \( P\left( {\widehat{X}}_{p}\right) \) onto \( P\left( X\right) \). Moreover, for \( x \in {\widehat{X}}_{p} \), define \( {\varphi }_{x} :... | Proof. (i) The map \( {\left. q\right| }_{{\widehat{X}}_{p}} \rightarrow {\left. q\right| }_{X} \) takes the dense subalgebra of \( P\left( {\widehat{X}}_{p}\right) \) consisting of all polynomial functions on \( {\widehat{X}}_{p} \) homomorphically onto the corresponding subalgebra of \( P\left( X\right) \). This map ... | Yes |
Theorem 2.5.8. If \( X \) is a compact subset of \( {\mathbb{C}}^{n} \), then \( P\left( X\right) = R\left( X\right) \) if and only if \( {\widehat{X}}_{p} = {\widehat{X}}_{r} \). In particular, for a compact subset \( X \) of \( \mathbb{C}, P\left( X\right) = R\left( X\right) \) if and only if \( \mathbb{C} \smallsetm... | Proof. Suppose first that \( P\left( X\right) = R\left( X\right) \), and let \( x \in {\widehat{X}}_{p} \). Then the function \( {\varphi }_{x} : f \rightarrow \left( {{\phi }^{-1}f}\right) \left( x\right) \), where \( \phi \) is as in part (i) of Theorem 2.5.7, defines an element of \( \Delta \left( {P\left( X\right) ... | Yes |
Theorem 2.5.9. If \( X \) is a compact subset of \( {\mathbb{C}}^{n} \), then \( R\left( X\right) \) is generated by \( n + 1 \) elements. | Proof. The set of \( n + 1 \) generators we produce consists of the coordinate functions \( {p}_{j}\left( z\right) = {z}_{j}, z \in X,1 \leq j \leq n \), and an additional function \( f \) which has to be constructed. Notice first that since \( P\left( X\right) \) contains a countable dense subset, there exists a seque... | Yes |
Corollary 2.5.10. Every rationally convex compact subset of \( {\mathbb{C}}^{n} \) is homeomorphic to some polynomially convex subset of \( {\mathbb{C}}^{n + 1} \) . | Proof. If \( X \) is a compact subset of \( {\mathbb{C}}^{n} \) and rationally convex, then \( X = \) \( \Delta \left( {R\left( X\right) }\right) \) by Theorem 2.5.7. On the other hand, \( R\left( X\right) \) is generated by \( n + 1 \) elements \( {f}_{1},\ldots ,{f}_{n + 1} \), and hence, by Lemma 2.3.3, \( \Delta \l... | Yes |
Example 2.5.11. As before, let \( \mathbb{D} \) denote the closed unit disc. We are going to show the existence of a sequence of closed discs \( {\Delta }_{j}, j \in \mathbb{N} \), of radii \( {r}_{j} > 0 \) with the following properties.\n\n(1) \( {\Delta }_{j} \subseteq {\mathbb{D}}^{ \circ } = \{ z \in \mathbb{C} : ... | Let \( {y}_{1},{y}_{2},\ldots \) be a numbering of the countable set of complex numbers \( \alpha + {i\beta } \in \) \( {\mathbb{D}}^{ \circ } \) with \( \alpha ,\beta \) rational. We construct by induction on \( n \) a sequence \( {\left( {\Delta }_{n}\right) }_{n} \) of closed discs such that (1) holds for \( 1 \leq ... | Yes |
Theorem 2.5.12. Let \( X \) be a countable compact subset of \( \mathbb{C} \). Then \( P\left( X\right) = C\left( X\right) \). | Proof. We first observe that \( \mathbb{C} \smallsetminus X \) is connected. To see this, let \( {z}_{1},{z}_{2} \in \mathbb{C} \smallsetminus X \). Since \( X \) is countable, there is a ray \( L \) emanating from \( {z}_{1} \) which does not intersect \( X \). For any point \( z \in L \), let \( \overline{z,{z}_{2}} ... | Yes |
Corollary 2.5.13. Let \( X \) be a countable compact Hausdorff space and let \( A \) be a closed subalgebra of \( C\left( X\right) \) . Then \( A \) is self-adjoint. | Proof. Let \( f \in A \) . Then \( f\left( X\right) \cup \{ 0\} \) is a countable compact subset of \( \mathbb{C} \) . By the preceding theorem there exists a sequence of polynomials \( {p}_{n}, n \in \mathbb{N} \), such that \( {p}_{n}\left( z\right) \rightarrow \bar{z} \) uniformly on \( f\left( X\right) \cup \{ 0\} ... | Yes |
Lemma 2.5.14. Let \( X \) be a compact Hausdorff space and \( A \) a uniform algebra on \( X \) . If \( f \) and \( g \) are functions in \( A \) such that \( \parallel 1 + f + \bar{g}{\parallel }_{\infty } < 1 \), then \( f + g \) is invertible in \( A \) . | Proof. Let \( h = f + g \) and \( c = \parallel 1 + \operatorname{Re}h{\parallel }_{\infty } \) . Since \( \parallel 1 + f + \bar{g}{\parallel }_{\infty } < 1 \) and hence \( \parallel 1 + \bar{f} + g{\parallel }_{\infty } < 1 \), we have\n\n\[ \parallel 1 + \operatorname{Re}h{\parallel }_{\infty } = \frac{1}{2}\parall... | Yes |
Theorem 2.5.15. Let \( A \) be a uniform algebra on the unit circle \( \mathbb{T} \) such that \( P\left( \mathbb{T}\right) \subseteq A \) . Then either \( A = P\left( \mathbb{T}\right) \) or \( A = C\left( \mathbb{T}\right) \) . | Proof. For \( h \in C\left( \mathbb{T}\right) \) and \( k \in \mathbb{Z} \), let\n\n\[ \n{c}_{k}\left( h\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }h\left( {e}^{it}\right) {e}^{-{ikt}}{dt} \n\]\n\nthe \( k \) -th Fourier coefficient of \( h \) . Then \( h \in P\left( \mathbb{T}\right) \) if and only if \( {c}_{k}\left(... | Yes |
Lemma 2.6.2. Let \( X \) be a Borel subset of \( \mathbb{C} \). Then, for any \( z \in \mathbb{C} \), \[ {\int }_{X}\frac{1}{\left| x - z\right| }{dx} \leq 2{\left( \pi \lambda \left( X\right) \right) }^{1/2} \] In particular, the functions \( x \rightarrow 1/\left( {x - z}\right), z \in \mathbb{C} \), are integrable o... | Proof. Nothing has to be shown if \( \lambda \left( X\right) = 0 \) or \( \lambda \left( X\right) = \infty \). Thus we can assume that \( 0 < \lambda \left( X\right) < \infty \). Let \( R = {\pi }^{-1/2}\lambda {\left( X\right) }^{1/2}, S = \{ x \in \mathbb{C} : \left| {x - z}\right| \leq R\} \) and, for any \( \vareps... | Yes |
Lemma 2.6.4. Let \( X \) and \( K \) be compact subsets of \( \mathbb{C} \) and let \( f \in A\left( X\right) \) . Extend \( f \) to all of \( \mathbb{C} \) by setting \( f\left( x\right) = 0 \) for all \( x \in \mathbb{C} \smallsetminus X \), and define \( h \) on \( \mathbb{C} \) by \[ h\left( z\right) = {\int }_{K}\... | Proof. Since \( x \rightarrow f\left( x\right) - f\left( z\right) \) is a bounded Borel measurable function on \( \mathbb{C} \) , \( h\left( z\right) \) is defined for all \( z \in \mathbb{C} \) and \( h \) is a continuous function (Lemma 2.6.3). Therefore, to show that \( h \) is holomorphic on \( {X}^{ \circ } \), by... | Yes |
Theorem 2.6.6. Let \( X \) be a compact subset of \( \mathbb{C} \). Then the mapping \( x \rightarrow \) \( {\varphi }_{x} \), where \( {\varphi }_{x}\left( f\right) = f\left( x\right) \) for all \( f \in A\left( X\right) \), is a homeomorphism between \( X \) and \( \Delta \left( {A\left( X\right) }\right) \). With th... | Proof. We only have to show that given \( \varphi \in \Delta \left( {A\left( X\right) }\right) \), there exists \( x \in X \) such that \( \varphi \left( f\right) = f\left( x\right) \) for all \( f \in A\left( X\right) \).\n\nLet \( x = \varphi \left( {\operatorname{id}}_{X}\right) \). Then \( x \in X \) since, for eve... | Yes |
Lemma 2.7.3. Let \( f \in {L}^{1}\left( G\right) \) and \( \alpha \in \widehat{G} \) . (i) For all \( x \in G,\left( {f * \alpha }\right) \left( x\right) = \alpha \left( x\right) \widehat{f}\left( \alpha \right) = \widehat{{L}_{{x}^{-1}}f}\left( \alpha \right) \) . In particular, \( \widehat{{L}^{1}\left( G\right) } \)... | Proof. (i) \( f * \alpha \) is a continuous function and \[ \left( {f * \alpha }\right) \left( x\right) = {\int }_{G}f\left( y\right) \alpha \left( {{y}^{-1}x}\right) {dy} = \alpha \left( x\right) \widehat{f}\left( \alpha \right) \] for all \( x \in G \) . On the other hand, \[ \left( {f * \alpha }\right) \left( x\righ... | Yes |
Lemma 2.7.4. Let \( f \in {L}^{1}\left( G\right) \) and \( \epsilon > 0 \) and let \( \sigma \) denote the Gelfand topology on \( \widehat{G} \) . Then there exists a neighbourhood \( W \) of \( e \) in \( G \) with the following property. If \( y, x \in G \) and \( \beta ,\alpha \in \widehat{G} \) are such that \( y \... | Proof. For arbitrary \( y, x \in G \) and \( \beta ,\alpha \in \widehat{G} \) such that \( \widehat{f}\left( \alpha \right) = 1 \) we obtain from Lemma 2.7.3,\n\n\[ \left| {\beta \left( y\right) - \alpha \left( x\right) }\right| \leq \left| {\overline{\beta \left( y\right) } - \overline{\beta \left( y\right) }\widehat{... | Yes |
(1) The dual group of the real line \( \mathbb{R} \) is topologically isomorphic to \( \mathbb{R} \) . | In fact, for each \( y \in \mathbb{R} \), define a character \( {\alpha }_{y} \) of \( \mathbb{R} \) by \( {\alpha }_{y}\left( x\right) = \exp \left( {2\pi ixy}\right), x \in \mathbb{R} \) . Then the map \( y \rightarrow {\alpha }_{y} \) from \( \mathbb{R} \) into \( \widehat{\mathbb{R}} \) is injective and every chara... | No |
Theorem 2.7.7. The mapping \( \lambda : f \rightarrow {\lambda }_{f} \) from \( {L}^{1}\left( G\right) \) into \( \mathcal{B}\left( {{L}^{2}\left( G\right) }\right) \) is an injective \( * \) -homomorphism. | Proof. It is clear that \( \lambda \) is linear. For \( {f}_{1},{f}_{2} \in {L}^{1}\left( G\right) \) and \( g \in {C}_{c}\left( G\right) \) ,\n\n\[{\lambda }_{{f}_{1} * {f}_{2}}\left( g\right) = {f}_{1} * \left( {{f}_{2} * g}\right) = {\lambda }_{{f}_{1}}\left( {{\lambda }_{{f}_{2}}\left( g\right) }\right) .\]\n\nThus... | Yes |
Lemma 2.7.10. Let \( G \) be a locally compact Abelian group, and suppose that the Gelfand homomorphism \( \Gamma : f \rightarrow \widehat{f} \) from \( {L}^{1}\left( G\right) \) into \( {C}_{0}\left( \widehat{G}\right) \) is surjective. Then \( G \) has to be discrete. | Proof. Let \( {\Gamma }^{ * } : M\left( \widehat{G}\right) = {\left( {C}_{0}\left( \widehat{G}\right) \right) }^{ * } \rightarrow {L}^{1}{\left( G\right) }^{ * } = {L}^{\infty }\left( G\right) \) denote the dual mapping of \( \Gamma \) . Since \( \Gamma \) is surjective, it is an isomorphism of Banach spaces and hence ... | Yes |
Theorem 2.7.12. Let \( G \) be a locally compact Abelian group. Then the Gelfand homomorphism \( \Gamma : {L}^{1}\left( G\right) \rightarrow {C}_{0}\left( \widehat{G}\right) \) is surjective if and only if \( G \) is finite. | Proof. Suppose that \( \Gamma \) is surjective. Then \( G \) is discrete by Lemma 2.7.10. Towards a contradiction, assume that \( G \) is infinite. For each \( x \in G \), define a character \( {\chi }_{x} \) of \( \widehat{G} \) by \( {\chi }_{x}\left( \alpha \right) = \alpha \left( x\right) \) . Then \( x \rightarrow... | Yes |
Theorem 2.8.2. Let \( G \) be a locally compact Abelian group and \( \omega \) a weight on G. For \( \alpha \in \widehat{G}\left( \omega \right) \), define \( {\varphi }_{\alpha } : {L}^{1}\left( {G,\omega }\right) \rightarrow \mathbb{C} \) by\n\n\[ \n{\varphi }_{\alpha }\left( f\right) = {\int }_{G}f\left( x\right) \o... | Proof. It is straightforward to show that \( {\varphi }_{\alpha } \) is a nonzero homomorphism and that, since \( {C}_{c}\left( G\right) \subseteq {L}^{1}\left( {G,\omega }\right) \), the map \( \alpha \rightarrow {\varphi }_{\alpha } \) is injective (compare the proof of Theorem 2.7.2).\n\nTo show that every \( \varph... | Yes |
Lemma 2.8.4. Let \( f \in {L}^{1}\left( {G,\omega }\right), x \in G \), and \( \epsilon > 0 \) . Then there exist a neighbourhood \( W \) of \( e \) in \( G \) and \( \delta > 0 \) with the following property. If \( y \in G \) and \( \beta ,\alpha \in \widehat{G}\left( \omega \right) \) are such that \( y \in {Wx},{\va... | Proof. Note first that \( \widehat{{L}_{z}f}\left( \gamma \right) = \overline{\gamma \left( z\right) }\widehat{f}\left( \gamma \right) \) for all \( z \in G \) and \( \gamma \in \widehat{G}\left( \omega \right) \) since \( \gamma \) is multiplicative. For arbitrary \( y, x \in G \) and \( \beta ,\alpha \in \widehat{G}\... | Yes |
Proposition 2.8.7. Let \( \omega \) be any weight on \( \mathbb{R} \) and let \( {R}_{ + } \) and \( {R}_{ - } \) be as in Lemma 2.8.6. Let \( {S}_{\omega } \) be the vertical strip in the complex plane defined by \[ {S}_{\omega } = \left\{ {z \in \mathbb{C} : - \ln {R}_{ + } \leq \operatorname{Re}z \leq - \ln {R}_{ - ... | Proof. It is clear that the map \( z \rightarrow {\varphi }_{z} \) from \( {S}_{\omega } \) into \( \Delta \left( {{L}^{1}\left( {\mathbb{R},\omega }\right) }\right) \) is injective. We show that every \( \varphi \in \Delta \left( {{L}^{1}\left( {\mathbb{R},\omega }\right) }\right) \) arises in this manner. To see this... | Yes |
Lemma 2.8.9. \( {L}^{1}\left( {G,\omega }\right) \) is either semisimple or radical. | Proof. Assume that \( {L}^{1}\left( {G,\omega }\right) \) is not radical, and fix any \( \varphi \in \Delta \left( {{L}^{1}\left( {G,\omega }\right) }\right) \) . By Theorem 2.8.2, there exists a continuous function \( \gamma : G \rightarrow \mathbb{C} \) satisfying \( \gamma \left( {xy}\right) = \gamma \left( x\right)... | Yes |
Theorem 2.8.10. Let \( G \) be a locally compact Abelian group and \( \omega \) a weight on \( G \) . Then the Beurling algebra \( {L}^{1}\left( {G,\omega }\right) \) is semisimple. | Proof. By virtue of Lemma 2.8.9, it suffices to show that \( {L}^{1}\left( {G,\omega }\right) \) is not radical. We construct a function \( f \in {L}^{1}\left( {G,\omega }\right) \) such that\n\n\[ \n{r}_{{L}^{1}\left( {G,\omega }\right) }\left( f\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}... | Yes |
Lemma 2.9.3. Let \( a \in G \) and \( f \in A\left( G\right) \) such that \( f\left( a\right) = 0 \) . Then, given \( \epsilon > 0 \), there exists \( h \in A\left( G\right) \cap {C}_{c}\left( G\right) \) vanishing in a neighbourhood of a such that \( \parallel h - f{\parallel }_{A\left( G\right) } \leq \epsilon \) . | Proof. Notice first that, since \( A\left( G\right) \cap {C}_{c}\left( G\right) \) is dense in \( A\left( G\right) \), without loss of generality we can assume that \( f \neq 0, f \) has compact support and \( \epsilon \leq \parallel f{\parallel }_{\infty } \) and \( \epsilon < 1 \) . Let\n\n\[ W = \left\{ {y \in G : {... | Yes |
Theorem 2.9.4. Let \( G \) be a locally compact group. For \( x \in G \), let \( {\varphi }_{x} \) : \( A\left( G\right) \rightarrow \mathbb{C} \) denote the evaluation at \( x \) . Then the map \( x \rightarrow {\varphi }_{x} \) is a homeomorphism from \( G \) onto \( \Delta \left( {A\left( G\right) }\right) \) . | Proof. It is obvious that \( {\varphi }_{x} \in \Delta \left( {A\left( G\right) }\right) \) and that the map \( x \rightarrow {\varphi }_{x} \) is injective. Now let \( \varphi \in \Delta \left( {A\left( G\right) }\right) \) be given and suppose that \( \varphi \neq {\varphi }_{x} \) for all \( x \in G \) . Then, for e... | Yes |
Lemma 2.9.5. Let \( G \) be a locally compact group, \( K \) a compact subset of \( G \) and \( U \) an open subset of \( G \) such that \( U \supseteq K \) . Then there exists \( u \in A\left( G\right) \cap {C}_{c}\left( G\right) \) with the following properties: \( 0 \leq u \leq 1, u\left( x\right) = 1 \) for all \( ... | Proof. Since \( K \) is compact, there exists a compact symmetric neighbourhood \( V \) of the identity such that \( K{V}^{2} \subseteq U \) . Let\n\n\[ u\left( x\right) = {\left| V\right| }^{-1}\left( {{1}_{KV} * {1}_{V}}\right) \left( x\right) = {\left| V\right| }^{-1} \cdot \left| {{xV} \cap {KV}}\right| . \]\n\nThe... | Yes |
Let \( G \) be a compact group. Then \( {AP}\left( G\right) = C\left( G\right) \) . | In fact, for \( f \in C\left( G\right) \) the map \( x \rightarrow {L}_{x}f \) from \( G \) into \( C\left( G\right) \) is continuous because \( f \) is uniformly continuous and\n\n\[ \n{\begin{Vmatrix}{L}_{x}f - {L}_{y}f\end{Vmatrix}}_{\infty } = \mathop{\sup }\limits_{{t \in G}}\left| {f\left( {{x}^{-1}t}\right) - f\... | No |
Lemma 2.10.2. \( {AP}\left( G\right) \) is a closed \( * \) -subalgebra of \( {C}^{b}\left( G\right) \) . | Proof. It is clear that \( {C}_{f + g} \subseteq {C}_{f} + {C}_{g},{C}_{\alpha f} = \alpha {C}_{f} \) and \( {C}_{fg} \subseteq {C}_{f}{C}_{g} \) for \( f, g \in \) \( {C}^{b}\left( G\right) \) and \( \alpha \in \mathbb{C} \) . Thus \( {AP}\left( G\right) \) is a subalgebra of \( {C}^{b}\left( G\right) \) . Also \( f \... | Yes |
Corollary 2.10.3. Let \( \Delta \left( {{AP}\left( G\right) }\right) \) denote the structure space of \( {AP}\left( G\right) \) . Then the Gelfand homomorphism is an isometric \( * \) -isomorphism from \( {AP}\left( G\right) \) onto \( C\left( {\Delta \left( {{AP}\left( G\right) }\right) }\right) \) . | Each \( x \in G \) defines an element \( {\varphi }_{x} \in \Delta \left( {{AP}\left( G\right) }\right) \) by \( {\varphi }_{x}\left( f\right) = f\left( x\right), f \in \) \( {AP}\left( G\right) \) . | No |
Lemma 2.10.4. The mapping \( \phi : x \rightarrow {\varphi }_{x} \) from \( G \) into \( \Delta \left( {{AP}\left( G\right) }\right) \) is continuous and has dense range. | Proof. Because \( \Delta \left( {{AP}\left( G\right) }\right) \) carries the \( {w}^{ * } \) -topology and the functions \( x \rightarrow \) \( {\varphi }_{x}\left( f\right) = f\left( x\right), f \in {AP}\left( G\right) \), are continuous on \( G \), it follows that \( \phi \) is continuous.\n\nSuppose that there exist... | Yes |
Lemma 2.10.5. Let \( f \in {AP}\left( G\right) \) and \( \epsilon > 0 \) . Then there exist finitely many \( {a}_{1},\ldots ,{a}_{n} \in G \) with the following property. For every \( a \in G \) there exists some \( j \in \{ 1,\ldots, n\} \) such that\n\n\[ \left| {f\left( {xay}\right) - f\left( {x{a}_{j}y}\right) }\ri... | Proof. There exist \( {b}_{1},\ldots ,{b}_{m} \in G \) such that the set \( \left\{ {{L}_{{b}_{j}}f : 1 \leq j \leq m}\right\} \) forms an \( \epsilon /4 \) -net for \( {C}_{f} \) . Let \( \Gamma \) be the finite set of all mappings \( \gamma \) from \( \{ 1,\ldots, m\} \) to itself with the property that there exists ... | Yes |
Corollary 2.10.6. Let \( f \in {AP}\left( G\right) \) and \( \epsilon > 0 \) . Then there exist \( {a}_{1},\ldots ,{a}_{n} \in G \) such that the functions \( {L}_{{a}_{i}}{R}_{{a}_{j}}f,1 \leq i, j \leq n \), form an \( \epsilon \) -net for the set of all two-sided translates \( {L}_{a}{R}_{b}f, a, b \in G \) . | Proof. Choose \( 0 < \delta < \epsilon /2 \) . By Lemma 2.10.5, there exist \( {a}_{1},\ldots ,{a}_{n} \in G \) with the property that for any \( a \in G \) there is \( j \in \{ 1,\ldots, n\} \) such that, for all \( x, y \in G,\left| {f\left( {xay}\right) - f\left( {x{a}_{j}y}\right) }\right| < \delta \) . Thus, given... | Yes |
Corollary 2.10.7. Retain the notation of Corollary 2.10.6. If \( x \) and \( y \) are elements of \( G \) such that\n\n\[ \left| {f\left( {{a}_{i}x{a}_{j}}\right) - f\left( {{a}_{i}y{a}_{j}}\right) }\right| < \epsilon \]\n\nfor all \( 1 \leq i, j \leq n \), then\n\n\[ \left| {f\left( {axb}\right) - f\left( {ayb}\right)... | Proof. Given \( a, b \in G \), by Corollary 2.10.6 there exist \( i \) and \( j \) such that\n\n\[ {\begin{Vmatrix}{L}_{a}{R}_{b}f - {L}_{{a}_{i}}{R}_{{a}_{j}}f\end{Vmatrix}}_{\infty } < \epsilon . \]\n\nCombining with the presumed inequality, we get\n\n\[ \left| {f\left( {axb}\right) - f\left( {ayb}\right) }\right| \l... | Yes |
Lemma 2.10.8. Let \( \alpha ,\beta \in \Delta \left( {{AP}\left( G\right) }\right) \) and \( f \in {AP}\left( G\right) \) . Let \( \epsilon > 0 \) and let \( \left\{ {{L}_{{x}_{1}}f,\ldots ,{L}_{{x}_{n}}f}\right\} \) be an \( \epsilon \) -net for \( {C}_{f} \) and \( \left\{ {{R}_{{y}_{1}}f,\ldots ,{R}_{{y}_{m}}f}\righ... | Proof. Choose \( j \in \{ 1,\ldots, n\} \) and \( k \in \{ 1,\ldots, m\} \) such that\n\n\[ \n{\begin{Vmatrix}{L}_{{x}^{-1}}f - {L}_{{x}_{j}}f\end{Vmatrix}}_{\infty } < \epsilon \text{ and }{\begin{Vmatrix}{R}_{b}f - {R}_{{y}_{k}}f\end{Vmatrix}}_{\infty } < \epsilon .\n\]\n\nThen we have\n\n\[ \n\left| {{\varphi }_{xy}... | Yes |
Corollary 2.10.9. For each pair of elements \( \varphi ,\psi \) of \( \Delta \left( {{AP}\left( G\right) }\right) ,{\Delta }_{\varphi ,\psi } \) is a singleton. | Proof. Let \( \alpha ,\beta \in {\Delta }_{\varphi ,\psi } \) and \( f \in {AP}\left( G\right) \) . We show that \( \left| {\alpha \left( f\right) - \beta \left( f\right) }\right| < \delta \) for each \( \delta > 0 \) . Fix \( \delta \) and let \( \epsilon = \delta /{24} \) . Let \( U \) and \( V \) be defined as in Le... | Yes |
Theorem 2.10.13. Let \( G \) be a locally compact Abelian group. The Gelfand isomorphism \( f \rightarrow \widehat{f} \) from \( {AP}\left( G\right) \) onto \( C\left( {b\left( G\right) }\right) \) maps \( \widehat{G} \) onto \( \widehat{b\left( G\right) } \) and hence \( T\left( G\right) \) onto \( T\left( {b\left( G\... | Proof. It suffices to show that if \( \gamma \in \widehat{G} \), then \( \widehat{\gamma } \in \widehat{b\left( G\right) } \), and that every character of \( b\left( G\right) \) arises in this way. For \( x, y \in G \), we have\n\n\[ \widehat{\gamma }\left( {{\varphi }_{x}{\varphi }_{y}}\right) = \widehat{\gamma }\left... | Yes |
Corollary 2.10.14. Let \( G \) be a locally compact Abelian group, and let \( {\widehat{G}}_{d} \) denote the algebraic group \( \widehat{G} \) endowed with the discrete topology. Then the discrete dual group \( \widehat{b\left( G\right) } \) of \( b\left( G\right) \) is isomorphic to \( {\widehat{G}}_{d} \) . | Proof. Being the dual group of the compact group \( b\left( G\right) ,\widehat{b\left( G\right) } \) is discrete. By Theorem 2.10.13, the Gelfand homomorphism of \( {AP}\left( G\right) \) maps \( \widehat{G} \) onto \( b\left( \widehat{G}\right) \) and this map is obviously a group isomorphism. Thus \( {\widehat{G}}_{d... | Yes |
Theorem 2.11.2. Let \( A \) and \( B \) be commutative Banach algebras and let \( \gamma \) be an algebra cross-norm on \( A \otimes B \) such that \( \gamma \geq \epsilon \) . Then the mapping \[ \Delta \left( A\right) \times \Delta \left( B\right) \rightarrow \Delta \left( {A{\widehat{ \otimes }}_{\gamma }B}\right) ,... | Proof. As to continuity, it suffices to show that for each \( c \in A{\widehat{ \otimes }}_{\gamma }B \), the function \( \left( {\varphi ,\psi }\right) \rightarrow \left( {\varphi {\widehat{ \otimes }}_{\gamma }\psi }\right) \left( c\right) \) is continuous on \( \Delta \left( A\right) \times \Delta \left( B\right) \)... | Yes |
Corollary 2.11.3. Let \( A, B \) and \( \gamma \) be as before. If \( A{\widehat{ \otimes }}_{\gamma }B \) is semisimple, then so are \( A \) and \( B \) . | Proof. Let \( a \in A \) such that \( \widehat{a} = 0 \) . Fix any nonzero \( b \in B \) . Then\n\n\[ \widehat{a \otimes b}\left( {\varphi {\widehat{ \otimes }}_{\gamma }\psi }\right) = \varphi \left( a\right) \psi \left( b\right) = 0 \]\n\nfor all \( \varphi \in \Delta \left( A\right) \) and \( \psi \in \Delta \left( ... | Yes |
Lemma 2.11.5. Let \( A \) and \( B \) be commutative Banach algebras and let \( \gamma \) be an algebra cross-norm on \( A \otimes B \) such that \( \gamma \geq \epsilon \) . Let \( \psi \in \Delta \left( B\right) \) . Then there is a unique continuous homomorphism \( {\phi }_{\psi } : A{\widehat{ \otimes }}_{\gamma }B... | Proof. The map \( A \times B \rightarrow A,\left( {a, b}\right) \rightarrow \psi \left( b\right) a \) is bilinear. Hence there is a unique linear map \( {\phi }_{\psi } : A \otimes B \rightarrow A \) satisfying \( {\phi }_{\psi }\left( {a \otimes b}\right) = \psi \left( b\right) a \) for all \( a \in A \) and \( b \in ... | Yes |
Theorem 2.11.6. Let \( A \) and \( B \) be commutative Banach algebras. Then the projective tensor product \( A{\widehat{ \otimes }}_{\pi }B \) is semisimple if and only if the following two conditions are satisfied.\n\n(i) \( A \) and \( B \) are semisimple.\n\n(ii) The natural homomorphism \( A{\widehat{ \otimes }}_{... | Proof. Suppose first that \( A{\widehat{ \otimes }}_{\pi }B \) is semisimple. Then \( A \) and \( B \) are semisimple by Corollary 2.11.3. Let \( \phi \) be the natural homomorphism from \( A{\widehat{ \otimes }}_{\pi }B \) into \( A{\widehat{ \otimes }}_{\epsilon }B \) and let \( c = \mathop{\sum }\limits_{{j = 1}}^{\... | Yes |
Proposition 2.11.7. Let \( A \) and \( B \) be commutative Banach algebras. Then \( A{\widehat{ \otimes }}_{\pi }B \) is unital if and only if both \( A \) and \( B \) are unital. | Proof. It is apparent that if \( {e}_{A} \) and \( {e}_{B} \) are identities of \( A \) and \( B \), respectively, then \( {e}_{A} \otimes {e}_{B} \) is an identity of \( A{\widehat{ \otimes }}_{\pi }B \) . Conversely, let \( \mathop{\sum }\limits_{{j = 1}}^{\infty }{a}_{j} \otimes {b}_{j} \), where \( {a}_{j} \in A \)... | Yes |
Theorem 2.11.8. Let \( G \) be a locally compact \( A \) belian group and \( A \) a semisimple commutative Banach algebra. Then \( {L}^{1}\left( {G, A}\right) = {L}^{1}\left( G\right) {\widehat{ \otimes }}_{\pi }A \) is semisimple. | Proof. Let \( \phi : {L}^{1}\left( G\right) {\widehat{ \otimes }}_{\pi }A \rightarrow {L}^{1}\left( {G, A}\right) \) be the isometric isomorphism satisfying \( \phi \left( {f \otimes a}\right) \left( x\right) = f\left( x\right) a \) for all \( f \in {L}^{1}\left( G\right) \) and \( a \in A \) and almost all \( x \in G ... | Yes |
Lemma 3.1.6. Let \( U \) be an open neighbourhood of \( {\sigma }_{A}\left( x\right) \) and let \( f \) be a holomorphic function on \( U \) . Moreover, let \( {\gamma }_{1},\ldots ,{\gamma }_{n} \) and \( {\delta }_{1},\ldots ,{\delta }_{m} \) be systems of closed, piecewise smooth curves in \( U \smallsetminus {\sigm... | Proof. Let \( {C}_{k} \) and \( {D}_{j} \) denote the trace of \( {\gamma }_{k} \) and \( {\delta }_{j} \), respectively, and set\n\n\[ K = \left( {\mathop{\bigcup }\limits_{{k = 1}}^{n}{C}_{k}}\right) \cup \left( {\mathop{\bigcup }\limits_{{j = 1}}^{m}{D}_{j}}\right) \]\n\nNotice that, because the function \( z \right... | Yes |
Theorem 3.1.8. Let \( A \) be a commutative unital Banach algebra. For \( x \in A \) the following assertions hold.\n\n(i) \( f \rightarrow f\\left( x\\right) \) is a homomorphism from \( H\\left( x\\right) \) into \( A \) . | Proof. (i) Recall first that the definition of \( f\\left( x\\right) \) for \( f \in H\\left( x\\right) \) does not depend on the choice of curves \( \\gamma _{1},\\ldots ,\\gamma _{n} \) in \( U \\smallsetminus \\sigma _{A}\\left( x\\right) \) as long as these have the properties in Lemma 3.1.2. Moreover, if \( \\gamm... | Yes |
Theorem 3.1.9. Let \( A \) be a commutative Banach algebra without identity. For \( x \in A \) let\n\n\[ \n{H}_{0}\left( x\right) = \{ f \in H\left( x\right) : f\left( 0\right) = 0\} .\n\]\n\nThen the functional calculus \( H\left( x\right) \rightarrow {A}_{e} \) (note that \( {\sigma }_{{A}_{e}}\left( x\right) = {\sig... | Proof. Recall that \( \Delta \left( {A}_{e}\right) = \widetilde{\Delta \left( A\right) } \cup \left\{ {\varphi }_{\infty }\right\} \), where \( \widetilde{\varphi }\left( {y + {\lambda e}}\right) = \varphi \left( y\right) + \lambda \) for \( \varphi \in \Delta \left( A\right), y \in A,\lambda \in \mathbb{C} \) . By The... | Yes |
Theorem 3.1.10. Let \( A \) be a unital commutative Banach algebra and let \( {x}_{1},\ldots ,{x}_{n} \in A \) . Let \( f \) be a complex-valued function of \( n \) variables which is defined and holomorphic on some open set containing the joint spectrum \( {\sigma }_{A}\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) of \( ... | To prove Theorem 3.1.10, the following result, which is due to Oka and usually referred to as Oka's extension theorem, is employed (see Section 3.7 for references).\n\nLet \( n, m \in \mathbb{N} \) . Let \( {p}_{1},\ldots ,{p}_{m} \) be polynomials in \( n \) complex variables and let \( \pi : {\mathbb{C}}^{n} \rightar... | No |
Proposition 3.1.11. Let \( n, m \in \mathbb{N} \) and \( {c}_{j} > 0 \) for \( 1 \leq j \leq n + m \), and let \( {p}_{1},\ldots ,{p}_{m} \) be polynomials in \( n \) variables. Let\n\n\[ D = \\left\\{ {z \\in {\\mathbb{C}}^{n + m} : \\left| {z}_{j}\\right| \\leq {c}_{j}\\text{ for }j = 1,\\ldots, n + m}\\right\\} \]\n... | Proof. Define three mappings \( \\rho : {\\mathbb{C}}^{n} \\rightarrow {\\mathbb{C}}^{n},\\sigma : {\\mathbb{C}}^{n + m} \\rightarrow {\\mathbb{C}}^{n + m} \), and \( \\tau : {\\mathbb{C}}^{n} \\rightarrow \) \( {\\mathbb{C}}^{n + m} \), respectively, by\n\n\[ \\rho \\left( {{w}_{1},\\ldots ,{w}_{n}}\\right) = \\left( ... | Yes |
Lemma 3.1.12. Let \( x = \left( {{x}_{1},\ldots ,{x}_{n}}\right) \in {A}^{n} \) and let \( U \) be an open neighbourhood of \( {\sigma }_{A}\left( x\right) \) in \( {\mathbb{C}}^{n} \) . Then there exists a finitely generated closed subalgebra \( B \) of \( A \) containing \( e,{x}_{1},\ldots ,{x}_{n} \) such that \( {... | Proof. Let \( z = \left( {{z}_{1},\ldots ,{z}_{n}}\right) \in {\mathbb{C}}^{n} \smallsetminus {\sigma }_{A}\left( x\right) \) . Then the ideal generated by the elements \( {z}_{j}e - {x}_{j},1 \leq j \leq n \), is not contained in any maximal ideal of \( A \), and hence there exists \( y = \left( {{y}_{1},\ldots ,{y}_{... | Yes |
Lemma 3.1.13. Let \( \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) be a set of generators for \( A \) and \( \left( {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right) \) \( \in {\mathbb{C}}^{n} \smallsetminus {\sigma }_{A}\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) . Then there exists a polynomial \( p \) such that\n\n\[ \le... | Proof. Because \( \left( {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right) \notin {\sigma }_{A}\left( {{x}_{1},\ldots ,{x}_{n}}\right) \), there exist \( {y}_{1},\ldots ,{y}_{n} \in A \) so that \( \mathop{\sum }\limits_{{j = 1}}^{n}\left( {{\lambda }_{j}e - {x}_{j}}\right) {y}_{j} = e \) . Choose \( \delta > 0 \) such th... | Yes |
Theorem 3.2.1. Let \( A \) be a commutative Banach algebra and suppose that \( \Delta \left( A\right) \) is compact. Let \( x \in A \) be such that \( \widehat{x}\left( \varphi \right) \neq 0 \) for all \( \varphi \in \Delta \left( A\right) \), and let \( f \) be a holomorphic function on some open neighbourhood of \( ... | Proof. By hypothesis, \( \widehat{x}\left( {\Delta \left( A\right) }\right) \) is a compact subset of \( \mathbb{C} \smallsetminus \{ 0\} \) . Choose disjoint open sets \( U \) and \( V \) in \( \mathbb{C} \) such that \( \widehat{x}\left( {\Delta \left( A\right) }\right) \subseteq U,0 \in V \), and \( f \) is holomorp... | Yes |
Corollary 3.2.2. Let \( A \) be a semisimple commutative Banach algebra. Suppose that \( \Delta \left( A\right) \) is compact and that there exists \( x \in A \) such that \( \widehat{x}\left( \varphi \right) \neq 0 \) for all \( \varphi \in \Delta \left( A\right) \) . Then \( A \) has an identity. | Proof. Let \( f \) be the function \( f\left( z\right) = {z}^{-1} \) on \( \mathbb{C} \smallsetminus \{ 0\} \) . Since \( \widehat{x}\left( {\Delta \left( A\right) }\right) \subseteq \mathbb{C} \smallsetminus \{ 0\} \) , by Theorem 3.2.1 there exists \( y \in A \) such that\n\n\[ \varphi \left( y\right) = f\left( {\var... | Yes |
Corollary 3.2.3. Let \( A \) be a semisimple commutative Banach \( * \) -algebra. If \( \Delta \left( A\right) \) is compact and the Gelfand homomorphism of \( A \) is a \( * \) -homomorphism, then \( A \) is unital. | Proof. According to Corollary 3.2.2 it suffices to show the existence of some \( x \in A \) such that \( \widehat{x}\left( \varphi \right) \neq 0 \) for all \( \varphi \in \Delta \left( A\right) \) . For each \( \varphi \in \Delta \left( A\right) \), there exists \( {y}_{\varphi } \in A \) with \( \varphi \left( {y}_{\... | Yes |
Lemma 3.2.4. Let \( x \in A \) be such that \( {\sigma }_{A}\left( x\right) \subseteq \mathbb{C} \smallsetminus ( - \infty ,0\rbrack \) . Then\n\n\[ \exp \left( {\log x}\right) = x \] | Proof. Let \( g \in H\left( x\right) \) and let \( p\left( z\right) = \mathop{\sum }\limits_{{k = 0}}^{m}{\alpha }_{k}{z}^{k} \) be any polynomial. Then, because the mapping from \( H\left( x\right) \) into \( A \) is a homomorphism and the definition of \( f\left( x\right) \) for \( f \in H\left( x\right) \) does not ... | Yes |
Corollary 3.2.5. If \( x \in A \) is such that \( \parallel e - x\parallel < 1 \), then \( x = \exp y \) for some \( y \in A \) . | Proof. If \( \parallel e - x\parallel < 1 \), then \( {\sigma }_{A}\left( x\right) \subseteq \{ z \in \mathbb{C} : \left| {z - 1}\right| < 1\} \) and hence \( x = \exp \left( {\log x}\right) \in \exp A \) by Lemma 3.2.4. | Yes |
Theorem 3.2.6. Let \( A \) be a commutative Banach algebra with identity \( e \) . Then \( \exp A \) equals the connected component of \( e \) in \( G\left( A\right) \) . | Proof. Note first that Theorem 3.1.8(ii) and the functional equation of the exponential function imply that\n\n\[ \exp x \cdot \exp y = \exp \left( {x + y}\right) \]\n\nfor all \( x, y \in A \) . It follows that \( \exp A \) is a group and \( \exp A \subseteq G\left( A\right) \) . For \( x \in A \) , the map \( \gamma ... | Yes |
Lemma 3.2.7. Let a be an element of finite order in \( G\left( A\right) \). Then a belongs to the connected component of \( e \) in \( G\left( A\right) \). | Proof. Choose \( n \in \mathbb{N} \) such that \( {a}^{n} = e \), and for each \( \lambda \in \mathbb{C} \) define an element \( a\left( \lambda \right) \in A \) by\n\n\[ a\left( \lambda \right) = \mathop{\sum }\limits_{{j = 0}}^{{n - 1}}{\left( \lambda - 1\right) }^{j}{\left( \lambda a\right) }^{n - 1 - j}. \]\n\nThen... | Yes |
Theorem 3.2.8. Let \( A \) be a unital commutative Banach algebra. Then either \( G\left( A\right) \) is connected or \( G\left( A\right) \) has infinitely many connected components. | Proof. Let \( {C}_{e} \) denote the connected component of the identity \( e \) in \( G\left( A\right) \) . Since elements of finite order of \( G\left( A\right) \) lie in \( {C}_{e} \), it suffices to show that if \( x \in G\left( A\right) \) is such that \( x \notin {C}_{e} \), then no two of the elements \( {x}^{n},... | Yes |
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