Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Theorem 3.2.9. Let \( A \) be a commutative Banach algebra with identity e. Let \( x \in A \) and suppose that \( \sigma \left( x\right) = \mathop{\bigcup }\limits_{{j = 1}}^{m}{C}_{j} \), where the sets \( {C}_{j},1 \leq j \leq m \), are nonempty, pairwise disjoint, and open and closed in \( \sigma \left( x\right) \).... | Proof. Because \( {C}_{1},\ldots ,{C}_{m} \) are compact, there exist pairwise disjoint open subsets \( {V}_{1},\ldots ,{V}_{m} \) of \( \mathbb{C} \) such that \( {C}_{j} \subseteq {V}_{j} \). For each \( j \), choose an open subset \( {W}_{j} \) of \( \mathbb{C} \) such that \( {W}_{j} \cap \sigma \left( x\right) = {... | Yes |
Theorem 3.2.11. Let \( A \) be a commutative Banach algebra with identity \( e \) and let \( x \in A \) . Suppose that \( \Lambda \) is a subset of \( \mathbb{C} \smallsetminus \sigma \left( x\right) \) such that \( \Lambda \cap C \neq \) \( \varnothing \) for every bounded connected component \( C \) of \( \mathbb{C} ... | Proof. By the Hahn-Banach theorem, it suffices to show that if \( l \in {A}^{ * } \) is such that \( {\left. l\right| }_{B} = 0 \), then \( l\left( {\left( \mu e - x\right) }^{-1}\right) = 0 \) for all \( \mu \in \rho \left( x\right) \) . As we have shown in the proof of Theorem 1.2.8, the function \( f \) on \( \rho \... | Yes |
A point \( x \in X \) belongs to the Shilov boundary of \( A \) if and only if given any open neighbourhood \( U \) of \( x \), there exists \( f \in A \) such that\n\n\[ \parallel f{\left| {}_{X \smallsetminus U}{\parallel }_{\infty } < \parallel f\right| }_{U}{\parallel }_{\infty } \] | Proof. First, let \( x \in X \smallsetminus \partial \left( A\right) \) . Then \( U = X \smallsetminus \partial \left( A\right) \) is an open neighbourhood of \( x \) and because \( \partial \left( A\right) \) is a boundary, we have for all \( f \in A \) ,\n\n\[ \parallel f{\left| {}_{U}{\parallel }_{\infty } \leq \par... | Yes |
Theorem 3.3.14. Let \( A \) be a commutative Banach algebra and suppose that \( \partial \left( A\right) \neq \Delta \left( A\right) \) . Then \( \partial \left( A\right) \) contains an infinite number of points. | Proof. Let \( \varphi \) and \( \psi \) be any two distinct elements of \( \Delta \left( A\right) \) . We prove first that there exists \( a \in A \) such that \( \widehat{a}\left( \varphi \right) \neq 0 \) and \( \widehat{a}\left( \psi \right) = 0 \) . This is evident if \( A \) is unital, but requires some argument i... | Yes |
Let \( A \) be a commutative symmetric Banach \( * \) -algebra. Then \( \bar{\partial }\left( A\right) = \Delta \left( A\right) \). | Since \( A \) is symmetric, \( \widetilde{\widehat{x}\left( \varphi \right) } = \widehat{{x}^{ * }}\left( \varphi \right) \) for all \( x \in A \) and \( \varphi \in \Delta \left( A\right) \). Thus \( \Gamma \left( A\right) \) is a subalgebra of \( {C}_{0}\left( {\Delta \left( A\right) }\right) \) which strongly separa... | Yes |
Proposition 3.4.4. Let \( X \) be a locally compact Hausdorff space and let \( A \) be a subalgebra of \( {C}_{0}\left( X\right) \) which strongly separates the points of \( X \) . Then, for any \( f \in A \) , \[ d\left( f\right) = \inf \{ \left| {f\left( x\right) }\right| : x \in \partial \left( A\right) \} . \] | Proof. Let \( g \in A, g \neq 0 \), and choose \( x \in \partial \left( A\right) \) such that \( \left| {g\left( x\right) }\right| = \parallel g{\parallel }_{\infty } \) . Then \[ \inf \{ \left| {f\left( y\right) }\right| : y \in \partial \left( A\right) \} \leq \left| {f\left( x\right) }\right| = \frac{\left| fg\left(... | Yes |
Corollary 3.4.5. Let \( X \) be a locally compact Hausdorff space and let \( A \) be a subalgebra of \( {C}_{0}\left( X\right) \) which strongly separates the points of \( X \) . If \( f \in A \), then \( f \) is a topological divisor of zero if and only if | \[ \inf \{ \left| {f\left( x\right) }\right| : x \in \partial \left( A\right) \} = 0. \] In particular, if \( \partial \left( A\right) \) is compact then \( f \) is a topological divisor of zero if and only \( f\left( x\right) = 0 \) for some \( x \in \partial \left( A\right) \). | Yes |
Corollary 3.4.6. Let \( A \) be a unital commutative Banach algebra and suppose that \( \parallel x{\parallel }^{2} \leq k\begin{Vmatrix}{x}^{2}\end{Vmatrix} \) for some \( k > 0 \) and all \( x \in A \) . Then an element \( x \) of \( A \) is a topological divisor of zero if and only if \( \varphi \left( x\right) = 0 ... | Proof. The hypothesis on \( A \) implies that the Gelfand homomorphism \( \Gamma : A \rightarrow \) \( C\left( {\Delta \left( A\right) }\right) \) is injective and that the two norms \( y \rightarrow \parallel y\parallel \) and \( y \rightarrow \parallel \widehat{y}{\parallel }_{\infty } \) on \( A \) are equivalent (c... | Yes |
Corollary 3.4.8. Let \( A \) be a commutative unital Banach algebra and let \( \varphi \in \) \( \partial \left( A\right) \) . Then every element of \( \ker \varphi \) is a topological divisor of zero. | Proof. We show that every \( x \in \ker \varphi \) satisfies condition (ii) of Theorem 3.4.7. Let \( B \) be any unital commutative Banach algebra such that there exists an isometric isomorphism \( j \) from \( A \) into \( B \) with \( j\left( e\right) = {e}_{B} \) . Since \( \varphi \in \partial \left( A\right) \), b... | Yes |
Theorem 3.4.10. Let \( X \) be a compact Hausdorff space, \( A \) a uniform algebra on \( X \), and \( \varphi \in \Delta \left( A\right) \) . Then \( \varphi \) belongs to \( \partial \left( A\right) \) if and only if \( \ker \varphi \) consists of joint topological zero divisors. | Proof. Let \( \varphi \in \partial \left( A\right) \) and \( {f}_{1},\ldots ,{f}_{n} \in \ker \varphi \) . Since \( \varphi \in \partial \left( A\right) ,\varphi = {\varphi }_{x} \) for some \( x \in X \) . Given \( \epsilon > 0 \), there exists an open neighbourhood \( V \) of \( x \) such that \( \left| {{f}_{j}\left... | Yes |
Theorem 3.4.11. Let \( A \) be a unital commutative Banach algebra and let \( \varphi \in \) \( \partial \left( A\right) \) . Then \( \ker \varphi \) consists of joint topological zero divisors. | Proof. It suffices to show that given \( {a}_{1},\ldots ,{a}_{q} \in A \) such that \( d\left( {{a}_{1},\ldots ,{a}_{q}}\right) > 0 \) , there is no maximal ideal of \( A \) containing all of \( {a}_{1},\ldots ,{a}_{q} \) and corresponding to some point in \( \partial \left( A\right) \) . Of course, we can assume \( d\... | No |
Lemma 3.4.12. Let \( A \) be a commutative normed algebra and let \( M \) be a subset of \( A \) consisting of joint topological divisors of zero. Then the closed ideal of \( A \) generated by \( M \) also consists of joint topological divisors of zero. | Proof. Let \( I \) be the ideal generated by \( M \), so that\n\n\[ I = \left\{ {\mathop{\sum }\limits_{{j = 1}}^{n}{x}_{j}{y}_{j} : {x}_{j} \in M,{y}_{j} \in A, n \in \mathbb{N}}\right\} .\n\]\n\nLet \( {a}_{1},\ldots ,{a}_{m} \in I \) and, for \( i = 1,\ldots, m \), write \( {a}_{i} = \mathop{\sum }\limits_{{j = 1}}^... | Yes |
Theorem 3.4.13. Let \( B \) be a commutative Banach algebra with identity \( e \) and let \( A \) be a closed subalgebra of \( B \) containing e. Then every \( \varphi \in \partial \left( A\right) \) extends to some \( \widetilde{\varphi } \in \partial \left( B\right) \) . | Proof. Let \( \Gamma : B \rightarrow C\left( {\Delta \left( B\right) }\right), x \rightarrow \widehat{x} \) denote the Gelfand homomorphism of \( B \) and let \( {C}_{A} \) and \( {C}_{B} \) be the closure of \( \Gamma \left( A\right) \) and of \( \Gamma \left( B\right) \) in \( C\left( {\Delta \left( B\right) }\right)... | Yes |
Lemma 3.5.2. Let \( A \) be a commutative Banach algebra and let \( {\varphi }_{1} \) and \( {\varphi }_{2} \) be distinct elements of \( \Delta \left( A\right) \) . Then there exists \( x \in A \) such that \( {\varphi }_{1}\left( x\right) = 1 \) and \( {\varphi }_{2}\left( x\right) = 0. | Proof. What follows is a standard argument which is also used to prove the classical Stone-Weierstrass theorem. However, we include the proof for the reader's convenience.\n\nThe set of Gelfand transforms strongly separates the points of \( \Delta \left( A\right) \) . Therefore there exist elements \( {a}_{1},{a}_{2} \... | Yes |
Theorem 3.5.5. Let \( A \) be a semisimple commutative Banach algebra. If \( \Delta \left( A\right) \) is compact, then \( A \) has an identity. | Proof. Since \( \Delta \left( A\right) \) is compact, by Theorem 3.5.1 there exists \( e \in A \) such that \( \widehat{e} = 1 \) on \( \Delta \left( A\right) \) . It follows that \( {xe} - x\left( \varphi \right) = \widehat{x}\left( \varphi \right) \widehat{e}\left( \varphi \right) - \widehat{x}\left( \varphi \right) ... | Yes |
Corollary 3.5.6. Let \( A \) be a commutative Banach algebra and suppose that \( \Delta \left( A\right) \) is totally disconnected. Then \( \widehat{A} = \{ \widehat{a} : a \in A\} \) is dense in \( {C}_{0}\left( {\Delta \left( A\right) }\right) \) . | Proof. Let \( f \in {C}_{0}\left( {\Delta \left( A\right) }\right) \) and \( \epsilon > 0 \) be given. Because \( f \) vanishes at infinity and every point of \( \Delta \left( A\right) \) has a neighbourhood basis of compact open sets, there exists a compact open subset \( K \) of \( \Delta \left( A\right) \) such that... | Yes |
Theorem 3.5.8. Let \( A \) be a nonunital commutative Banach algebra and let \( \left\{ {{I}_{\lambda } : \lambda \in \Lambda }\right\} \) be a family of unital closed ideals of \( A \) satisfying property (ii) of Theorem 3.5.7 and such that the ideal \( \mathop{\sum }\limits_{{\lambda \in \Lambda }}{I}_{\lambda } \) i... | Proof. Of course, each \( \Delta \left( {I}_{\lambda }\right) = \Delta \left( A\right) \smallsetminus h\left( {I}_{\lambda }\right) \) is open in \( \Delta \left( A\right) \) . Let \( \lambda ,\mu \in \Lambda \) such that \( \lambda \neq \mu \), and suppose that there exists \( \varphi \in \Delta \left( {I}_{\lambda }\... | Yes |
Theorem 3.5.9. Let \( A \) be a unital commutative Banach algebra.\n\n(i) If \( \Delta \left( A\right) \) is a disjoint union \( \Delta \left( A\right) = \mathop{\bigcup }\limits_{{j = 1}}^{m}{F}_{j} \) of open (and closed) subsets \( {F}_{j} \), then there exist unital closed ideals \( {I}_{1},\ldots ,{I}_{m} \) of \(... | Proof. A straightforward induction argument shows that for both (i) and (ii), it suffices to consider the case \( m = 2 \) .\n\n(i) Since \( \Delta \left( A\right) \) is compact, \( {F}_{1} \) and \( {F}_{2} \) are compact. By Shilov’s idempotent theorem, there exists an idempotent \( {e}_{1} \in A \) such that \( {\wi... | Yes |
Corollary 3.5.10. Let \( A \) be a commutative Banach algebra.\n\n(i) Suppose that \( \Delta \left( A\right) \) is a disjoint union \( \Delta \left( A\right) = \mathop{\bigcup }\limits_{{j = 1}}^{m}{F}_{j} \), where \( {F}_{1} \) is closed and \( {F}_{2},\ldots ,{F}_{m} \) are compact. Then there exist closed ideals \(... | Proof. To prove (i), in view of Theorem 3.5.9 we can assume that \( A \) does not have an identity. Let \( {A}_{e} \) be the algebra obtained by adjoining an identity \( e \) to \( A \) . Let \( {E}_{1} = {F}_{1} \cup \left\{ {\varphi }_{\infty }\right\} \) and \( {E}_{j} = {F}_{j} \) for \( j = 2,\ldots, m \) . Then\n... | Yes |
Lemma 4.1.2. Let \( B,{B}_{1} \), and \( {B}_{2} \) be subsets of \( A \) and let \( E,{E}_{1} \), and \( {E}_{2} \) be subsets of \( \Delta \left( A\right) \) . Then\n\n(i) \( {B}_{1} \subseteq {B}_{2} \Rightarrow h\left( {B}_{1}\right) \supseteq h\left( {B}_{2}\right) \) .\n\n(ii) \( h\left( \bar{B}\right) = h\left( ... | Proof. (i), (ii), and (iv) are obvious from the definitions. We show the remaining assertions.\n\n(iii) If \( M \in h\left( {k\left( {h\left( B\right) }\right) }\right) \) then \( M \supseteq k\left( {h\left( B\right) }\right) \supseteq B \), so that \( M \in h\left( B\right) \) . Conversely, if \( \varphi \in \Delta \... | Yes |
Lemma 4.1.5. Let \( I \) be a closed ideal of \( A \). (i) Let \( q : A \rightarrow A/I \) denote the quotient homomorphism. The map \( \varphi \rightarrow \varphi \circ q \) is a homeomorphism for the hull-kernel topologies between \( \Delta \left( {A/I}\right) \) and the closed subset \( h\left( I\right) \) of \( \De... | Proof. (i) The map \( \phi : \varphi \rightarrow \varphi \circ q \) clearly is a bijection between \( \Delta \left( {A/I}\right) \) and \( h\left( I\right) \subseteq \Delta \left( A\right) \). Now, for any subset \( E \) of \( \Delta \left( {A/I}\right) \) and \( \varphi \in \Delta \left( {A/I}\right) \) , \( \ker \var... | Yes |
Lemma 4.1.6. Let \( A \) be a commutative Banach algebra without identity and let \( a \in A \) be such that \( \widehat{a} \) is continuous in the hull-kernel topology on \( \Delta \left( A\right) \) . Then \( \widehat{a} \) is also continuous on \( \Delta \left( {A}_{e}\right) \) with respect to the hull-kernel topol... | Proof. Recall that \( \Delta \left( {A}_{e}\right) = \Delta \left( A\right) \cup \left\{ {\varphi }_{\infty }\right\} \), where each \( \varphi \in \Delta \left( A\right) \) is identified with its canonical extension \( x + {\lambda e} \rightarrow \varphi \left( x\right) + \lambda, x \in A,\lambda \in \mathbb{C} \) . I... | Yes |
Lemma 4.1.7. Let \( \alpha \) be an algebra cross-norm on \( A \otimes B \) such that \( \alpha \geq \) \( \epsilon \) . Then the map \( \varphi {\widehat{ \otimes }}_{\alpha }\psi \rightarrow \left( {\varphi ,\psi }\right) \) from \( \Delta \left( {A{\widehat{ \otimes }}_{\alpha }B}\right) \) onto \( \Delta \left( A\r... | Proof. Let \( E \) be a hull-kernel closed subset of \( \Delta \left( A\right) \) . We claim that the set \( F = \left\{ {\varphi {\widehat{ \otimes }}_{\alpha }\psi : \varphi \in E,\psi \in \Delta \left( B\right) }\right\} \) is hull-kernel closed in \( \Delta \left( {A{\widehat{ \otimes }}_{\alpha }B}\right) \) . We ... | Yes |
Example 4.1.8. Let \( \mathbb{D} \) denote the closed unit disc, and let \( A = A\left( \mathbb{D}\right) \) and \( B = C\left( \mathbb{D}\right) \) . Then the map \( \phi : \left( {\varphi ,\psi }\right) \rightarrow \varphi {\widehat{ \otimes }}_{\pi }\psi \) from \( \Delta \left( A\right) \times \Delta \left( B\right... | Assuming that \( \phi \) is hull-kernel continuous, there exist nonempty hull-kernel open subsets \( U \) of \( \Delta \left( A\right) \) and \( V \) of \( \Delta \left( B\right) \), respectively, such that \( U \times V \subseteq {\phi }^{-1}\left( W\right) \) . Because the hull-kernel topology on \( \Delta \left( B\r... | Yes |
Lemma 4.1.9. Let \( I \) be a closed ideal of the commutative Banach algebra \( A \) and let \( E \) be an \( {hk} \) -closed subset of \( \Delta \left( A\right) \) such that \( E \cap h\left( I\right) = \varnothing \) and \( k\left( E\right) \) is modular. Then \( I \) contains an identity modulo \( k\left( E\right) \... | Proof. Because \( A/\left( {I + k\left( E\right) }\right) \) is unital and\n\n\[ h\left( {I + k\left( E\right) }\right) = h\left( I\right) \cap h\left( {k\left( E\right) }\right) = h\left( I\right) \cap E = \varnothing ,\]\n\n it follows that \( I + k\left( E\right) = A \) . Let \( u \in A \) be such that \( {ux} - x \... | Yes |
Lemma 4.1.10. Let \( A \) be a semisimple commutative Banach algebra with bounded approximate identity and regard \( A \) as a closed ideal of its multiplier algebra \( M\left( A\right) \) . Then \( \Delta \left( A\right) \) is hull-kernel dense in \( \Delta \left( {M\left( A\right) }\right) \) . | Proof. We have to show that \( h\left( {k\left( {\Delta \left( A\right) }\right) }\right) = \Delta \left( {M\left( A\right) }\right) \) . For that, consider an arbitrary \( T \in k\left( {\Delta \left( A\right) }\right) \), so \( T \) is a multiplier of \( A \) such that \( \varphi \left( T\right) = 0 \) for all \( \va... | Yes |
Every commutative \( {C}^{ * } \) -algebra \( A \) is regular. | Indeed, \( A \) is isomorphic to \( {C}_{0}\left( {\Delta \left( A\right) }\right) \), and Urysohn’s lemma ensures that for any locally compact Hausdorff space \( T,{C}_{0}\left( T\right) \) is a regular space of functions. | Yes |
Let \( A \) be a commutative Banach algebra.\n\n(i) Let \( I \) be closed ideal of \( A \) . If \( A \) is regular, then so are the algebras \( I \) and \( A/I \) .\n\n(ii) \( A \) is regular if and only if \( {A}_{e} \), the unitisation of \( A \), is regular. | Proof. (i) Because \( A \) is regular, by Theorem 4.2.3 the Gelfand topology coincides with the \( {hk} \) -topology on \( \Delta \left( A\right) \) . By Lemma 4.1.5(ii), the map \( \varphi \rightarrow {\left. \varphi \right| }_{I} \) is a homeomorphism for the \( {hk} \) -topologies on \( \Delta \left( A\right) \small... | Yes |
Lemma 4.2.7. Let \( I \) be an ideal in the regular commutative Banach algebra \( A \) . Given any \( {\varphi }_{0} \in \Delta \left( A\right) \smallsetminus h\left( I\right) \), there exists \( u \in I \) such that \( \widehat{u} = 1 \) in some neighbourhood of \( {\varphi }_{0} \) . | Proof. Because \( A \) is regular, by Theorem 4.2.3 the hull-kernel topology on \( \Delta \left( A\right) \) is Hausdorff and \( {\varphi }_{0} \) possesses a neighbourhood with modular kernel. Therefore we can choose a neighbourhood \( V \) of \( {\varphi }_{0} \) such that \( \bar{V} \cap h\left( I\right) = \varnothi... | Yes |
Theorem 4.2.8. Let \( A \) be a regular commutative Banach algebra, and suppose that \( I \) is an ideal in \( A \) and \( K \) is a compact subset of \( \Delta \left( A\right) \) with \( K \cap h\left( I\right) = \varnothing \) . Then there exists \( x \in I \) such that\n\n\[{\left. \widehat{x}\right| }_{K} = 1\text{... | Proof. We first show the existence of some \( y \in I \) with \( {\left. \widehat{y}\right| }_{K} = 1 \) . As \( K \) is compact, by the preceding lemma there exist open subsets \( {V}_{i} \) of \( \Delta \left( A\right) \) and \( {u}_{i} \in I,1 \leq i \leq r \), such that \( {\left. {\widehat{u}}_{i}\right| }_{{V}_{i... | Yes |
Corollary 4.2.10. Let \( A \) be a regular commutative Banach algebra such that its range under the Gelfand homomorphism \( A \rightarrow {C}_{0}\left( {\Delta \left( A\right) }\right) \) is closed under complex conjugation. Suppose that \( K \) and \( E \) are disjoint closed subsets of \( \Delta \left( A\right) \) wi... | Proof. By Theorem 4.2.8 there exist \( y \in A \) such that \( {\left. \widehat{y}\right| }_{K} = 1 \) and \( \operatorname{supp}\widehat{y} \subseteq \) \( \Delta \left( A\right) \smallsetminus E \) . By hypothesis, there exists \( z \in A \) such that \( \widehat{z} = \overline{\widehat{y}} \) . Let \( f \) be the en... | Yes |
Corollary 4.2.11. Let \( A \) be a semisimple regular commutative Banach algebra. If \( \Delta \left( A\right) \) is compact, then \( A \) has an identity. | Proof. By Theorem 4.2.8 there is \( u \in A \) such that \( \widehat{u} = 1 \) on \( \Delta \left( A\right) \) and hence \( x - {ux} = 0 \) on \( \Delta \left( A\right) \) for all \( x \in A \) . \( A \) being semisimple, this yields \( {ux} = x \) for all \( x \in A \) . | Yes |
Corollary 4.2.12. Let \( A \) be a regular commutative Banach algebra. Suppose that \( K \) is a compact subset of \( \Delta \left( A\right) \) and \( {U}_{1},\ldots ,{U}_{n} \) are open subsets of \( \Delta \left( A\right) \) such that \( K \subseteq \mathop{\bigcup }\limits_{{j = 1}}^{n}{U}_{j} \) . Then there exist ... | Proof. Choose open subsets \( {V}_{j} \) of \( \Delta \left( A\right) ,1 \leq j \leq n \), such that \( {\bar{V}}_{j} \subseteq {U}_{j} \) and \( K \subseteq \mathop{\bigcup }\limits_{{j = 1}}^{n}{V}_{j} \) . Let\n\n\[ \n{I}_{j} = k\left( {\Delta \left( A\right) \smallsetminus {V}_{j}}\right) ,\;1 \leq j \leq n,\text{ ... | Yes |
Lemma 4.2.13. Let \( A \) and \( C \) be commutative Banach algebras and let \( f \) : \( \Delta \left( A\right) \rightarrow \Delta \left( C\right) \) be an injective map with the following properties.\n\n(i) \( f \) is continuous with respect to the hull-kernel topologies.\n\n(ii) \( {f}^{-1} : f\left( {\Delta \left( ... | Proof. We remind the reader that a commutative Banach algebra \( B \) is regular if and only if the Gelfand topology and the \( {hk} \) -topology on \( \Delta \left( B\right) \) coincide (Theorem 4.2.3). Let \( E \) be a subset of \( \Delta \left( A\right) \) which is closed in the Gelfand topology. Then \( f\left( E\r... | Yes |
Theorem 4.2.14. Let \( j : A \rightarrow B \) be an injective algebra homomorphism between commutative Banach algebras. Suppose that \( B \) is regular and that \( j\left( A\right) \) is an ideal in \( B \) . Then \( A \) is regular. | Proof. Let \( I = \overline{j\left( A\right) } \), which is a closed ideal in \( B \) . Since \( B \) is regular, so is \( I \) by Theorem 4.2.6. Consider the dual mapping\n\n\[ \n{j}^{ * } : \Delta \left( I\right) \rightarrow \Delta \left( A\right) ,\;\psi \rightarrow \psi \circ j.\n\]\n\nThen \( {j}^{ * } \) is injec... | Yes |
Lemma 4.2.15. Let \( A \) and \( B \) be commutative Banach algebras, and let \( \phi \) : \( A \rightarrow B \) be a homomorphism with dense range. If \( A \) is regular, then so is \( B \) . | Proof. We have to show that given a closed subset \( F \) of \( \Delta \left( B\right) \) and \( \psi \in \Delta \left( B\right) \smallsetminus \) \( F \), there exists \( b \in B \) such that \( \widehat{b} = 0 \) on \( F \) and \( \widehat{b}\left( \psi \right) \neq 0 \) . \n\nConsider the dual mapping \( {\phi }^{ *... | Yes |
Corollary 4.2.17. Let \( B \) be a commutative Banach algebra and \( A \) a subalgebra of B. Suppose that, for some norm, A is a semisimple regular Banach algebra. Then\n\n(i) Every element of \( \Delta \left( A\right) \) extends to some element of \( \Delta \left( B\right) \) .\n\n(ii) \( {\sigma }_{A}\left( x\right) ... | Proof. (i) is an immediate consequence of Theorem 4.2.16. To show (ii), we apply Theorem 4.2.16 taking for \( \phi \) the inclusion map \( j : A \rightarrow B \) . It follows that\n\n\[ \n{\sigma }_{A}\left( x\right) \cup \{ 0\} = \widehat{x}\left( {\Delta \left( A\right) \cup \{ 0\} }\right) \n\]\n\n\[ \n= \widehat{x}... | Yes |
Corollary 4.2.18. Let \( A \) be a semisimple regular commutative Banach algebra, and let \( \left| \cdot \right| \) be any algebra norm on \( A \) . Then \( {r}_{A}\left( x\right) \leq \left| x\right| \) for all \( x \in A \) . | Proof. Let \( B \) be the completion of \( A \) with respect to \( \left| \cdot \right| \) . Then part (ii) of Corollary 4.2.17 implies that\n\n\[ \n{r}_{A}\left( x\right) = \sup \left\{ {\left| \lambda \right| : \lambda \in {\sigma }_{A}\left( x\right) }\right\} = \sup \left\{ {\left| \lambda \right| : \lambda \in {\s... | Yes |
Lemma 4.2.19. Let \( A \) and \( B \) be commutative Banach algebras and let \( \alpha \) be an algebra cross-norm on \( A \otimes B \) such that \( \alpha \geq \epsilon \) . Then \( A{\widehat{ \otimes }}_{\alpha }B \) is regular whenever both \( A \) and \( B \) are regular. | Proof. We identify \( \Delta \left( A\right) \times \Delta \left( B\right) \) and \( \Delta \left( {A{\widehat{ \otimes }}_{\alpha }B}\right) \) as topological spaces by means of the map \( \left( {\varphi ,\psi }\right) \rightarrow \varphi {\widehat{ \otimes }}_{\alpha }\psi \) (Theorem 2.11.2). Let \( E \) be a close... | Yes |
Theorem 4.2.20. Let \( A \) and \( B \) be commutative Banach algebras and let \( \alpha \) be a cross-norm on \( A \otimes B \) which dominates \( \epsilon \) . Then the tensor product \( A{\widehat{ \otimes }}_{\alpha }B \) is regular if and only if both \( A \) and \( B \) are regular. | Proof. By Lemma 4.2.19, \( A{\widehat{ \otimes }}_{\alpha }B \) is regular whenever \( A \) and \( B \) are regular. So suppose that conversely \( A{\widehat{ \otimes }}_{\alpha }B \) is regular. To see that \( A \) is regular, by Corollary 4.2.4 it suffices to show that, for each \( a \in A \), the function \( \varphi... | Yes |
Lemma 4.3.1. Let \( A \) be a commutative Banach algebra and \( B \) a closed subalgebra of \( A \) . If \( B \) is regular, then for every \( b \in B \) the Gelfand transform \( \widehat{b} \) is continuous on \( \Delta \left( A\right) \) with respect to the hull-kernel topology. | Proof. Suppose first that \( A \) has an identity \( e \) and that \( e \in B \) . Let \( r : \Delta \left( A\right) \rightarrow \) \( \Delta \left( B\right) \) denote the restriction map \( \varphi \rightarrow {\left. \varphi \right| }_{B} \) . Because \( B \) is regular, the Gelfand transform of \( b \in B \) on \( \... | Yes |
Theorem 4.3.2. Let \( A \) be a commutative Banach algebra. Then \( A \) contains a greatest closed regular subalgebra, denoted \( \operatorname{reg}\left( A\right) \) . | Proof. Let \( \operatorname{reg}\left( A\right) \) be the closed subalgebra of \( A \) generated by the collection \( \mathcal{B} \) of all closed regular subalgebras \( B \) of \( A \) . We have to show that \( \operatorname{reg}\left( A\right) \) is regular.\n\nLet \( B \in \mathcal{B} \) and \( b \in B \) . Then, by... | Yes |
Lemma 4.3.4. There exists a largest closed regular ideal \( \operatorname{regid}\left( A\right) \) of \( A \), and for every \( x \in A,\widehat{x} \) is hull-kernel continuous on the open subset \( \Delta \left( {\operatorname{regid}\left( A\right) }\right) \) of \( \Delta \left( A\right) \) . | Proof. It only remains to show the second statement. Let \( J = \operatorname{regid}\left( A\right) \) and \( x \in A \) . Let \( {\varphi }_{0} \) be an arbitrary element of \( \Delta \left( J\right) \) and choose \( y \in J \) such that \( {\varphi }_{0}\left( y\right) \neq 0 \) . Then \( \widehat{y} \neq 0 \) in a n... | Yes |
Lemma 4.3.5. Let \( A \) be a commutative Banach algebra, \( J \) the greatest regular ideal of \( A \) and suppose that \( A/J \) is regular. Then the hull \( h\left( J\right) \) has empty interior in \( \Delta \left( A\right) \) . | Proof. Assume that there exists a nonempty open subset \( U \) of \( \Delta \left( A\right) \) which is contained in \( h\left( J\right) = \Delta \left( {A/J}\right) \) . Because \( A/J \) is regular, \( U \) is \( {hk} \) -open in \( h\left( J\right) \) . Let \( W \) be an \( {hk} \) -open subset of \( \Delta \left( A... | Yes |
Corollary 4.3.6. Let \( A \) and \( J \) be as in Lemma 4.3.5 and let \( E \) be a closed subset of \( \Delta \left( A\right) \) such that \( k\left( E\right) = \{ 0\} \) . Then \( E = \Delta \left( A\right) \) . | Proof. By Lemma 4.3.5, \( \Delta \left( J\right) \) is dense in \( \Delta \left( A\right) \) . It is therefore enough to show that \( \Delta \left( J\right) \subseteq E \) . Assume that \( F = E \cap \Delta \left( J\right) \) is a proper subset of \( \Delta \left( J\right) \) . Then, since \( J \) is regular, there exi... | Yes |
Lemma 4.3.7. Let \( J \) be the greatest closed regular ideal of \( A \) and suppose that \( A/J \) is regular. Let \( I \) be an arbitrary closed ideal of \( A \) . Then there exists a closed ideal \( K \) of \( A/I \) such that both \( K \) and \( \left( {A/I}\right) /K \) are regular. | Proof. Let \( q : A \rightarrow A/I \) denote the quotient homomorphism. Since \( J \) is regular, Lemma 4.2.15 implies that \( K = \overline{q\left( J\right) } \) is a regular ideal of \( A/I \) . Yet, \( \left( {A/I}\right) /K \) is also regular. Indeed, since\n\n\[ \left( {A/I}\right) /K = \left( {A/I}\right) /\left... | Yes |
Theorem 4.3.8. Let \( A \) be a commutative Banach algebra and suppose that \( A \) has a closed ideal \( I \) such that both \( I \) and \( A/I \) are regular. Then \( A \) is regular. | Proof. Let \( J \) be the largest regular closed ideal of \( A \) . Then \( J \supseteq I \) and since \( A/I \) is regular, it follows that \( A/J \) is regular as well. So \( A \) satisfies the hypotheses of Lemma 4.3.7.\n\nLet \( E \) be any closed subset of \( \Delta \left( A\right) \) . We have to show that \( E \... | Yes |
Corollary 4.3.9. Suppose that \( A \) possesses a sequence \( {\left( {I}_{j}\right) }_{j \in \mathbb{N}} \) of closed sub-algebras with the following properties.\n\n(i) \( {I}_{j} \) is an ideal in \( {I}_{j + 1} \) for each \( j \in \mathbb{N} \) and \( \mathop{\bigcup }\limits_{{j = 1}}^{\infty }{I}_{j} \) is dense ... | Proof. Applying Theorem 4.3.8 and induction, it follows from the hypotheses that \( {I}_{j} \) is regular for every \( j \) . Thus \( \mathop{\bigcup }\limits_{{j = 1}}^{\infty }{I}_{j} \subseteq \operatorname{reg}\left( A\right) \), and since \( \mathop{\bigcup }\limits_{{j = 1}}^{\infty }{I}_{j} \) is dense in \( A \... | Yes |
Lemma 4.3.10. Let \( A \) be a commutative Banach algebra and let \( B \) be a Banach algebra consisting of \( A \) -valued functions on a set \( X \) with pointwise operations. Let\n\n\[ R = \{ f \in B : f\left( X\right) \subseteq \operatorname{reg}\left( A\right) \} \]\n\nand suppose that \( R \) is closed in \( B \)... | Proof. For each \( x \in X \), consider the algebra homomorphism\n\n\[ {\phi }_{x} : \operatorname{reg}\left( B\right) \rightarrow A,\;f \rightarrow f\left( x\right) . \]\n\nLemma 4.2.15 yields that \( \overline{{\phi }_{x}\left( {\operatorname{reg}\left( B\right) }\right) } \) is a regular subalgebra of \( A \) and he... | Yes |
Lemma 4.4.2. Given \( f \in {C}^{\infty }\left( G\right) \), the operator \( T \in {C}^{ * }\left( G\right) \) in Definition 4.4.1 is unique and is denoted \( {T}_{f} \) . | Proof. Let \( T, S \in {C}^{ * }\left( G\right) \) and suppose that \( {\left( {f}_{n}\right) }_{n} \) and \( {\left( {g}_{n}\right) }_{n} \) are sequences in \( {L}^{1}\left( G\right) \cap {C}^{b}\left( G\right) \) such that, as \( n \rightarrow \infty \) ,\n\n\[ \n{\lambda }_{{f}_{n}} \rightarrow T,\;{\lambda }_{{g}_... | Yes |
Lemma 4.4.4. The map \( f \rightarrow {T}_{f} \) from \( {C}^{\infty }\left( G\right) \) into \( {C}^{ * }\left( G\right) \) is linear and injective. | Proof. Linearity of the map is obvious. Thus it remains to show that \( f = 0 \) whenever \( {T}_{f} = 0 \) . So suppose there exists a sequence \( {\left( {f}_{n}\right) }_{n} \) in \( {L}^{1}\left( G\right) \cap {C}^{b}\left( G\right) \) such that \( {\lambda }_{{f}_{n}} \rightarrow 0 \) and \( {\begin{Vmatrix}{f}_{n... | Yes |
Corollary 4.4.13. For \( f \in {L}^{2}\left( G\right) \) and \( \alpha \in \widehat{G} \) , \[ \widehat{{f}^{ * }} = \overline{\widehat{f}},\widehat{\bar{f}} = {\left( \widehat{f}\right) }^{ * }\text{and}\widehat{\alpha f} = {L}_{\alpha }\widehat{f}\text{.} \] | Proof. It is sufficient to check all three equations for functions \( f \) in a dense linear subspace of \( {L}^{2}\left( G\right) \) . Now, the first and the third equations hold in \( {C}^{\infty }\left( G\right) \), and hence in \( E \), by (i) and (iii) of Lemma 4.4.6, respectively. As to the second, it is enough t... | No |
Theorem 4.4.14. Let \( G \) be a locally compact Abelian group. Then \( {L}^{1}\left( G\right) \) is regular. | Proof. Let \( E \) be a closed subset of \( \widehat{G} \) and \( \alpha \in \widehat{G} \smallsetminus E \) . We have to find \( f \in {L}^{1}\left( G\right) \) such that \( \widehat{f}\left( \alpha \right) \neq 0 \) and \( {\left. \widehat{f}\right| }_{E} = 0 \) . Choose a neighbourhood \( U \) of \( \alpha \) in \( ... | Yes |
For a commutative Banach algebra \( A \), the following conditions are equivalent.\n\n(i) A has the spectral extension property.\n\n(ii) Every submultiplicative norm \( \left| \cdot \right| \) on \( A \) satisfies \( {r}_{A}\left( a\right) \leq \left| a\right| \) for all \( a \in A \) . | Proof. (i) \( \Rightarrow \) (ii) Let \( \left| \cdot \right| \) be any submultiplicative norm on \( A \) and let \( \left( {B,\parallel \cdot \parallel }\right) \) be the completion of \( \left( {A,\left| \cdot \right| }\right) \) . By (i), for all \( a \in A \),\n\n\[ {r}_{A}\left( a\right) = {r}_{B}\left( a\right) =... | Yes |
Theorem 4.5.3. For a semisimple commutative Banach algebra \( A \) the following are equivalent.\n\n(i) \( A \) has the spectral extension property.\n\n(ii) If \( E \) is a closed subset of \( \Delta \left( A\right) \) that does not contain the Shilov boundary of \( A \), then there exists an element \( a \in A \) such... | Proof. (i) \( \Rightarrow \) (ii) Let \( E \) be a closed subset of \( \Delta \left( A\right) \) that does not contain the Shilov boundary of \( A \) . Towards a contradiction, assume that \( E \) has the property that for any \( a \in A,{\left. \widehat{a}\right| }_{E} = 0 \) implies \( a = 0 \) . Then \( \left| a\rig... | Yes |
Corollary 4.5.4. Let \( A \) be a semisimple commutative Banach algebra and suppose that \( A \) has the spectral extension property. If \( A \) is not one-dimensional, then \( A \) contains zero divisors. | Proof. Notice that if \( E \) is any proper closed subset of \( \partial \left( A\right) \), then, by Theorem 4.5.3, there exists \( a \neq 0 \) in \( A \) such that \( \widehat{a} = 0 \) on \( E \) . Now \( \partial \left( A\right) \) contains at least two elements. Indeed, this follows from Theorem 3.3.14 if \( \part... | Yes |
Lemma 4.5.5. Let \( A \) be a semisimple commutative Banach algebra. Then \( {C}_{0}\left( {\partial \left( A\right) }\right) \) is an extension of \( A \). Furthermore, if \( \varphi \in \Delta \left( A\right) \) extends to some element of \( \Delta \left( {{C}_{0}\left( {\partial \left( A\right) }\right) }\right) \),... | Proof. Because \( A \) is semisimple, the mapping \( a \rightarrow {\left. \widehat{a}\right| }_{\partial \left( A\right) } \) is an injective homomorphism of \( A \) into \( {C}_{0}\left( {\partial \left( A\right) }\right) \). Now, every element of \( \Delta \left( {{C}_{0}\left( {\partial \left( A\right) }\right) }\r... | Yes |
Theorem 4.5.6. Let \( A \) be a semisimple commutative Banach algebra. Then A has the strong spectral extension property if and only if \( A \) has the spectral extension property and the Shilov boundary \( \partial \left( A\right) \) of \( A \) satisfies\n\n\[ \widehat{a}\left( {\partial \left( A\right) }\right) \cup ... | Proof. Suppose first that \( A \) has the strong spectral extension property. Then \( A \) has the spectral extension property. By Lemma 4.5.5, \( B = {C}_{0}\left( {\partial \left( A\right) }\right) \) is an extension of \( A \) . For every \( f \in B \) ,\n\n\[ {\sigma }_{B}\left( f\right) \cup \{ 0\} = f\left( {\par... | Yes |
Theorem 4.5.7. Let \( A \) be a semisimple commutative Banach algebra. Then \( A \) has the multiplicative Hahn-Banach property if and only if \( A \) has the spectral extension property and the Shilov boundary of \( A \) equals \( \Delta \left( A\right) \) . | Proof. Suppose first that \( A \) has the multiplicative Hahn-Banach property. Then \( \partial \left( A\right) = \Delta \left( A\right) \) by Lemma 4.5.5. Let \( B \) be any extension of \( A \) and let \( C \) be the closure of \( A \) in \( B \) . Given \( a \in A \), there exists \( \varphi \in \Delta \left( A\righ... | Yes |
Suppose that \( 0 < r < R \) and let \( X = \{ z \in \mathbb{C} : \left| z\right| \leq R\} \) and \( U = \{ z \in \mathbb{C} : \left| z\right| < r\} \) . Let\n\n\[ A = \{ f \in C\left( X\right) : f\text{ is holomorphic on }U\} ,\]\n\nendowed with the uniform norm. Because a uniform limit of holomorphic functions is hol... | Then the mapping \( f \rightarrow {\left. f\right| }_{X \smallsetminus Y} \) is an isometric isomorphism from \( I \) onto \( {C}_{0}\left( {X \smallsetminus Y}\right) \), and it follows from Tietze’s extension theorem that \( A/I \) is isometrically isomorphic to \( A\left( Y\right) \) (Exercise 4.8.24). Let \( \varph... | Yes |
Let \( X = \mathbb{D} \times \left\lbrack {0,1}\right\rbrack \) and let \( A \) be the algebra of all continuous complex-valued functions \( f \) on \( X \) with the property that \( z \rightarrow f\left( {z,0}\right) \) is holomorphic on \( {\mathbb{D}}^{ \circ } \) . Endowed with the supremum norm, \( A \) is a commu... | In this case, however, the Shilov boundary of \( A \) is all of \( X \) . Indeed, since \( A \) contains every continuous function on \( X \) which is zero on \( \mathbb{D} \times \{ 0\} \), it follows that \( \partial \left( A\right) \supseteq \mathbb{D} \times (0,1\rbrack \) and hence \( \partial \left( A\right) = X ... | Yes |
Example 4.5.10. Let \( X \) denote the closed ball of radius 2 around zero in \( {\mathbb{C}}^{2} \) and \( Y \) the open ball of radius 1 around zero in \( {\mathbb{C}}^{2} \). Let \[ A = \left\{ {f \in C\left( X\right) : {\left. f\right| }_{Y}\text{ is holomorphic }}\right\} . \] Endowed with the supremum norm, \( A ... | We observe next that \( A \) has the spectral extension property. To see this, we once more apply Theorem 4.5.3,(ii) \( \Rightarrow \) (i). Thus, let \( E \) be a closed subset of \( X \) not containing \( \partial \left( A\right) \). Because \( E \) does not contain \( X \smallsetminus \bar{Y} \), we can find an open ... | Yes |
Lemma 4.6.2. Let \( A \) be a commutative Banach algebra and let \( \left| \cdot \right| \) be a uniform norm on \( A \) . Then \( \left| x\right| \leq {r}_{A}\left( x\right) \) for all \( x \in A \) . Let\n\n\[ E = \{ \varphi \in \Delta \left( A\right) : \left| {\varphi \left( x\right) }\right| \leq \left| x\right| \t... | Proof. Let \( \left( {B,\left| \cdot \right| }\right) \) be the completion of \( A \) with respect to \( \left| \cdot \right| \) . Since \( A \) is dense in \( B \) and elements of \( \Delta \left( B\right) \) are continuous, \( {\left. \psi \right| }_{A} \in \Delta \left( A\right) \) for each \( \psi \in \Delta \left(... | Yes |
Corollary 4.6.3. For a commutative Banach algebra \( A \) the following conditions are equivalent.\n\n(i) \( A \) is semisimple.\n\n(ii) The spectral radius is a uniform norm on \( A \) .\n\n(iii) A admits a uniform norm. | Proof. We have already observed in Section 2.1 that \( A \) is semisimple if and only if \( {r}_{A} \) is a (uniform) norm on \( A \) . Thus it suffices to show (iii) \( \Rightarrow \) (i). If \( \parallel \cdot \parallel \) is the original norm on \( A \) and \( \left| \cdot \right| \) is a uniform norm on \( A \), th... | Yes |
Theorem 4.6.5. Let \( A \) be a semisimple commutative Banach algebra. Then the following four conditions are equivalent.\n\n(i) A has the unique uniform norm property.\n\n(ii) The Shilov boundary \( \partial \left( A\right) \) of \( A \) is the smallest set of uniqueness.\n\n(iii) If \( F \) is a closed subset of \( \... | Proof. (i) \( \Rightarrow \) (ii) If \( F \subseteq \Delta \left( A\right) \) is a closed set of uniqueness, then\n\n\[ \left| x\right| = \sup \{ \left| {\varphi \left( x\right) }\right| : \varphi \in F\} \]\n\ndefines a uniform norm on \( A \) . By (i), \( {r}_{A}\left( x\right) = \left| x\right| \) for all \( x \in A... | Yes |
Corollary 4.6.8. Let \( A \) be a semisimple commutative Banach algebra having the unique uniform norm property. Let \( I \) be a spectral synthesis ideal of \( A \) , that is, an ideal with the property that \( I = k\left( {h\left( I\right) }\right) \) . Then \( I \) also has the unique uniform norm property. | Proof. By Theorem 4.6.5,(iii) \( \Rightarrow \) (i), it is sufficient to show that if \( F \) is a closed subset of \( \Delta \left( I\right) \) not containing the Shilov boundary \( \partial \left( I\right) \), then \( F \) is not a set of uniqueness for \( I \) . Let\n\n\[ E = \left( {\Delta \left( A\right) \smallset... | Yes |
Lemma 4.6.12. Let \( A \) be a semisimple commutative Banach algebra and \( B \) be a dense subalgebra of \( A \) . If \( B \) has the unique uniform norm property, then so does \( A \) . | Proof. Let \( \left| \cdot \right| \) be a uniform norm on \( A \) . Then \( \left| \cdot \right| \leq {r}_{A}\left( \cdot \right) \leq \parallel \cdot \parallel \) on \( A \) . Let \( a \in A \) and choose a sequence \( {\left( {b}_{n}\right) }_{n} \) in \( B \) such that \( \begin{Vmatrix}{{b}_{n} - a}\end{Vmatrix} \... | Yes |
Suppose that \( f \in A \) satisfies \( {\varphi }_{z}\left( f\right) = 0 \) for all \( z \in {K}_{r} \). Then \( {\varphi }_{z}\left( {{\delta }_{m} * f}\right) = 0 \) for all \( m \in \mathbb{Z} \) and \( z \in {K}_{r} \), and hence | \[ 0 = {\int }_{0}^{2\pi }{\varphi }_{r{e}^{it}}\left( {{\delta }_{m} * f}\right) {dt} \] \[ = \mathop{\sum }\limits_{{n \in \mathbb{Z}}}\left( {{\delta }_{m} * f}\right) \left( n\right) {r}^{n}{\int }_{0}^{2\pi }{e}^{-{int}}{dt} = \left( {{\delta }_{m} * f}\right) \left( 0\right) \] \[ = f\left( m\right) \text{.} \] T... | Yes |
Theorem 4.7.3. Let \( G \) be a locally compact Abelian group and \( \omega \) a weight on \( G \) . Then \( {L}^{1}\left( {G,\omega }\right) \) has the unique uniform norm property if and only if \( {L}^{1}\left( {G,\omega }\right) \) is regular. | Proof. Since for every semisimple commutative Banach algebra regularity implies the unique uniform norm property, we only have to show the 'only if' part.\n\nLet \( E \) be a proper closed subset of \( \widehat{G}\left( \omega \right) \) and let \( \gamma \in \widehat{G}\left( \omega \right) \smallsetminus E \) . By th... | Yes |
Lemma 4.7.5. Let \( G \) be a nonquasianalytic weight on \( G \) . Then\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\omega {\left( {x}^{n}\right) }^{1/n} = \mathop{\lim }\limits_{{n \rightarrow \infty }}\omega {\left( {x}^{-n}\right) }^{-1/n} = 1 \]\n\nfor all \( x \in G \) . In particular, \( \widehat{G}\left... | Proof. Suppose the statement is false, so that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\omega {\left( {x}^{n}\right) }^{1/n} > 1 \) for some \( x \in G \) . Then there exist some \( \delta > 0 \) and \( N \in \mathbb{N} \) such that \( \omega {\left( {x}^{n}\right) }^{1/n} \geq 1 + \delta \) for all \( n \geq... | Yes |
Lemma 4.7.6. A weight \( \omega \) on \( \mathbb{Z} \) is nonquasianalytic if and only if\n\n\[ \mathop{\sum }\limits_{{n = - \infty }}^{\infty }\frac{\ln \omega \left( n\right) }{1 + {n}^{2}} < \infty \] | Proof. For \( x \in \mathbb{N} \), this condition implies that\n\n\[ \mathop{\sum }\limits_{{n = - \infty }}^{\infty }\frac{\ln \omega \left( {nx}\right) }{1 + {n}^{2}} \leq \mathop{\sum }\limits_{{n = - \infty }}^{\infty }\frac{\ln \left( {\omega {\left( n\right) }^{x}}\right) }{1 + {n}^{2}} = x \cdot \mathop{\sum }\l... | Yes |
Lemma 4.7.10. Let \( {G}_{1} \) and \( {G}_{2} \) be locally compact Abelian groups and let \( G = {G}_{1} \times {G}_{2} \) . Suppose that \( {L}^{1}\left( {{G}_{1},{\omega }_{1}}\right) \) and \( {L}^{1}\left( {{G}_{2},{\omega }_{2}}\right) \) are regular for all nonquasianalytic weights \( {\omega }_{1} \) and \( {\... | Proof. Let \( \omega \) be given and define \( {\omega }_{1} \) and \( {\omega }_{2} \) by \( {\omega }_{1}\left( {x}_{1}\right) = \omega \left( {{x}_{1},{e}_{2}}\right) ,{x}_{1} \in {G}_{1} \) , and \( {\omega }_{2}\left( {x}_{2}\right) = \omega \left( {{e}_{1},{x}_{2}}\right) ,{x}_{2} \in {G}_{2} \), where \( {e}_{1}... | Yes |
Theorem 5.1.2. Let \( A \) be regular and let \( I \) be an ideal of \( A \) and suppose that \( f \) is a function on \( \Delta \left( A\right) \) that belongs locally to \( I \) . Then there exists \( x \in I \) such that \( \widehat{x} = f \) . In particular, if \( A \) is semisimple and \( y \in A \) is such that \... | Proof. Because \( f \) belongs locally to \( I \) at infinity, there exist a compact subset \( C \) of \( \Delta \left( A\right) \) and an element \( {x}_{0} \) in \( I \) such that \( {\widehat{x}}_{0}\left( \psi \right) = f\left( \psi \right) \) for all \( \psi \in \Delta \left( A\right) \smallsetminus \) \( C \) . S... | Yes |
Lemma 5.1.3. Let \( A \) be regular, \( I \) an ideal of \( A \), and \( x \in A \) . Then \( \widehat{x} \) belongs locally to \( I \) at each point of \( h{\left( x\right) }^{0} \), the interior of \( h\left( x\right) \), and at each point of \( \Delta \left( A\right) \smallsetminus h\left( I\right) \) . | Proof. Because \( \widehat{x}\left( \varphi \right) = 0 \) for all \( \varphi \in h\left( x\right) \), the first assertion is clear. If \( \varphi \notin \) \( h\left( I\right) \), then by Lemma 4.1.9 there exists \( y \in I \) such that \( \widehat{y} = 1 \) in some neighbourhood \( V \) of \( \varphi \) . It follows ... | Yes |
Corollary 5.1.4. Suppose that \( A \) is semisimple and regular. Let \( x \in A \) be such that \( \widehat{x} \) has compact support and \( h\left( I\right) \cap \operatorname{supp}\widehat{x} = \varnothing \) . Then \( x \in I \) . | Proof. Since \( \widehat{x} \) has compact support, \( \widehat{x} \) belongs locally to \( I \) at infinity. By Lemma 5.1.3, \( \widehat{x} \) belongs locally to \( I \) at every \( \varphi \in \Delta \left( A\right) \smallsetminus h\left( I\right) \) and also at every \( \varphi \in h\left( I\right) \) since, by hypo... | Yes |
Theorem 5.1.6. Suppose that \( A \) is semisimple and regular and let \( I \) be an ideal of \( A \) and \( E \) a closed subset of \( \Delta \left( A\right) \) . Then \( h\left( I\right) = E \) if and only if \[ j\left( E\right) \subseteq I \subseteq k\left( E\right) \] | Proof. Suppose first that \( j\left( E\right) \subseteq I \subseteq k\left( E\right) \) . Then, since \( A \) is regular, \[ E = h\left( {k\left( E\right) }\right) \subseteq h\left( I\right) \subseteq h\left( {j\left( E\right) }\right) . \] To show that actually \( h\left( I\right) = E \), it therefore suffices to veri... | Yes |
Lemma 5.1.9. Let \( A \) be a regular and semisimple commutative Banach algebra and suppose that \( A \) is Tauberian. Then \( h\left( I\right) \neq \varnothing \) for every proper closed ideal of \( A \) . In particular, if \( a \in A \) is such that \( \widehat{a}\left( \varphi \right) \neq 0 \) for all \( \varphi \i... | Proof. If \( I \) is a proper closed ideal with \( h\left( I\right) = \varnothing \), then \( j\left( \varnothing \right) \subseteq I \) by Theorem 5.1.6. However, \( j\left( E\right) \) is dense in \( A \) since \( A \) is Tauberian.\n\nThe second statement is now obvious. | No |
Lemma 5.1.11. Let \( A \) be a semisimple and regular commutative Banach algebra. Given \( f \in {A}^{ * } \), there exists a largest open subset of \( \Delta \left( A\right) \) on which \( f \) vanishes. | Proof. We first show that if \( f \) vanishes on finitely many open subsets \( {V}_{1},\ldots ,{V}_{n} \) of \( \Delta \left( A\right) \), then \( f \) vanishes on \( \mathop{\bigcup }\limits_{{j = 1}}^{n}{V}_{j} \) . To that end, let \( x \in A \) be such that supp \( \widehat{x} \) is compact and contained in \( \mat... | Yes |
Proposition 5.1.13. Let \( E \) be a closed subset of \( \Delta \left( A\right) \). Then \( E \) is a spectral set if and only if whenever \( f \in {A}^{ * } \) is such that \( \operatorname{supp}f \subseteq E \), then \( f\left( x\right) = 0 \) for all \( x \in k\left( E\right) \). | Proof. Suppose first that \( E \) is a set of synthesis and let \( f \in {A}^{ * } \) such that \( \operatorname{supp}f \subseteq E \). Then \( f \) vanishes on \( \Delta \left( A\right) \smallsetminus E \) and hence \( f\left( x\right) = 0 \) for all \( x \in j\left( E\right) \). Thus \( f\left( x\right) = 0 \) for al... | Yes |
Lemma 5.1.14. Let \( A \) be semisimple and regular and let \( I \) be a closed ideal of \( A \) . Let \( x \in A \) and let \( \varphi \) be an isolated point of \( \Delta \left( {x, I}\right) \) . In addition, suppose that \( \overline{j\left( \varphi \right) } \) possesses an approximate identity. Then \( \widehat{x... | Proof. Towards a contradiction, assume that \( \widehat{x} \) belongs locally to \( \overline{j\left( \varphi \right) } \) at \( \varphi \) , and let \( U \) be a neighbourhood of \( \varphi \) and \( y \in \overline{j\left( \varphi \right) } \) such that \( \widehat{x} = \widehat{y} \) on \( U \) . Then, because \( \v... | Yes |
Proposition 5.1.15. Let \( A \) be a regular and semisimple commutative Banach algebra. Let \( I \) be a closed ideal of \( A \) and let \( x \in A \) be such that \( h\left( I\right) \subseteq h\left( x\right) \). Then\n\n(i) \( \Delta \left( {x, I}\right) \) is contained in \( h\left( I\right) \cap \partial \left( {h... | Proof. (i) By Lemma 5.1.3, \( x \) belongs locally to \( I \) at each point of \( h{\left( x\right) }^{0} \) and at each point of \( \Delta \left( A\right) \smallsetminus h\left( I\right) \). Thus\n\n\[ \Delta \left( {x, I}\right) \subseteq h\left( I\right) \cap \left( {\Delta \left( A\right) \smallsetminus h{\left( x\... | Yes |
Lemma 5.2.1. The union of two Ditkin sets is a Ditkin set. | Proof. Let \( {E}_{1} \) and \( {E}_{2} \) be Ditkin sets in \( \Delta \left( A\right) \) and let \( E = {E}_{1} \cup {E}_{2} \) . We have to show that given \( x \in k\left( E\right) \) and \( \varepsilon > 0 \), there exists \( y \in j\left( E\right) \) such that \( \parallel x - {xy}\parallel \leq \epsilon \) . Now,... | Yes |
Lemma 5.2.3. Let \( {E}_{1} \) and \( {E}_{2} \) be closed subsets of \( \Delta \left( A\right) \) . Let \( E = {E}_{1} \cup {E}_{2} \) and \( F = {E}_{1} \cap {E}_{2} \), and suppose that \( F \) is a Ditkin set. Let \( I \) be any closed ideal of \( A \) with \( h\left( I\right) = E \) and let \[ {I}_{k} = \overline{... | Proof. We only have to show that \( {I}_{1} \cap {I}_{2} \subseteq I \) . Because \( I \) is closed in \( A \) it suffices to prove that given \( a \in {I}_{1} \cap {I}_{2} \) and \( \epsilon > 0 \), there exists \( u \in A \) such that \( {ua} \in I \) and \( \parallel {ua} - a\parallel \leq \epsilon \) . Note that \[... | Yes |
Lemma 5.2.4. Let \( {E}_{1},{E}_{2} \), and \( I \) be as in Lemma 5.2.3. If \( {J}_{1} \) and \( {J}_{2} \) are closed ideals of \( A \) with \( h\left( {J}_{k}\right) = {E}_{k}, k = 1,2 \), and \( {J}_{1} \cap {J}_{2} = I \), then \[ {J}_{k} = \overline{I + j\left( {E}_{k}\right) },\;k = 1,2. \] | Proof. As in Lemma 5.2.3, let \( {I}_{i} = \overline{I + j\left( {E}_{i}\right) } \) for \( i = 1,2 \) . We prove that \( {J}_{1} \subseteq {I}_{1} \), the converse inclusion, \( {I}_{1} \subseteq {J}_{1} \), being obvious since \( I \subseteq {J}_{1} \) and \( j\left( {E}_{1}\right) \) is the smallest ideal with hull ... | Yes |
Theorem 5.2.5. Let \( A \) be a semisimple and regular commutative Banach algebra. Suppose that \( {E}_{1} \) and \( {E}_{2} \) are closed subsets of \( \Delta \left( A\right) \) such that \( {E}_{1} \cap {E}_{2} \) is a Ditkin set. Then \( {E}_{1} \cup {E}_{2} \) is a spectral set if and only if both \( {E}_{1} \) and... | Proof. First, let \( {E}_{1} \) and \( {E}_{2} \) be spectral sets. Let \( E = {E}_{1} \cup {E}_{2} \) and apply Lemma 5.2.3 with \( I = \overline{j\left( E\right) } \) . It follows that\n\n\[ \overline{j\left( E\right) } = \overline{I + j\left( {E}_{1}\right) } \cap \overline{I + j\left( {E}_{2}\right) } \]\n\n\[ = \o... | Yes |
Theorem 5.2.6. Let \( A \) be a regular and semisimple commutative Banach algebra satisfying Ditkin’s condition at infinity and let \( E \) be a closed subset of \( \Delta \left( A\right) \). (i) Suppose that each point of \( E \) has a closed relative neighbourhood in \( E \) which is a set of synthesis for \( A \). T... | Proof. (i) We have to show that every \( x \in k\left( E\right) \) belongs to \( \overline{j\left( E\right) } \). Because \( \varnothing \) is a Ditkin set, we can assume that \( \widehat{x} \) has compact support. By Theorem 5.1.2 and Lemma 5.1.3 it then suffices to show that \( \widehat{x} \) belongs locally to \( \o... | Yes |
Theorem 5.2.7. Let \( A \) be a regular and semisimple commutative Banach algebra, \( I \) a closed ideal of \( A \), and \( E \) a closed subset of \( \Delta \left( {A/I}\right) \) . (i) If \( i\left( E\right) \) is a spectral set (respectively, Ditkin set) for \( A \), then \( E \) is a spectral set (respectively, Di... | Proof. (i) Suppose that \( i\left( E\right) \) is a spectral set (respectively, Ditkin set) for \( A \) and let \( x \in A \) be such that \( q\left( x\right) \in k\left( E\right) \) . Then, given \( \epsilon > 0 \), there exists \( y \in j\left( {i\left( E\right) }\right) \) such that \( \parallel x - y\parallel < \ep... | Yes |
Lemma 5.2.8. Let \( A \) be a semisimple and regular commutative Banach algebra and let \( E \subseteq \Delta \left( A\right) \) be a set of synthesis. Suppose that there exists a constant \( c > 0 \) such that for every compact subset \( K \) of \( \Delta \left( A\right) \) which is disjoint from \( E \), there exists... | Proof. Let \( x \in k\left( E\right) \) and \( \epsilon > 0 \) be given. Since \( E \) is a set of synthesis, there exists \( u \in j\left( E\right) \) such that \( \parallel u - x\parallel \leq \epsilon \) . By hypothesis, there exists \( y \in j\left( E\right) \) satisfying \( \parallel y\parallel \leq c \) and \( \w... | Yes |
Lemma 5.2.10. Suppose that \( A \) has an identity \( e \) and let \( E \) be a closed subset of \( \Delta \left( A\right) \) . Then the following are equivalent.\n\n(i) \( E \) satisfies condition (D).\n\n(ii) There exists a constant \( c > 0 \) such that for every compact subset \( K \) of \( \Delta \left( A\right) \... | Proof. (ii) \( \Rightarrow \) (i) Let \( U \) be an open set containing \( E \) . Choose an open set \( V \) so that \( E \subseteq V \) and \( \bar{V} \subseteq U \), and let \( K = \Delta \left( A\right) \smallsetminus V \) . By (ii), there exists \( a \in j\left( E\right) \) such that \( \widehat{a} = 1 \) on \( K \... | Yes |
Proposition 5.2.14. Let \( A \) be a regular and semisimple commutative Banach algebra and let \( E \) be an open and closed subset of \( \Delta \left( A\right) \). (i) If \( A \) is Tauberian and \( x \in \overline{Ax} \) for every \( x \in k\left( E\right) \), then \( E \) is a set of synthesis. (ii) If \( A \) satis... | Proof. (i) Let \( x \in k\left( E\right) \) and \( I = \overline{j\left( E\right) } \). By Lemma 5.1.3, \( \widehat{x} \) belongs locally to \( I \) at every point of \( \Delta \left( A\right) \smallsetminus E \) and, since \( E \) is open, at every point of \( E \subseteq h{\left( x\right) }^{ \circ } \). Because \( x... | Yes |
For each \( a \in \left\lbrack {0,1}\right\rbrack \), the quotient algebra \( {C}^{n}\left\lbrack {0,1}\right\rbrack /\overline{j\left( a\right) } \) is isomorphic to the \( \left( {n + 1}\right) \) -dimensional algebra \( \mathbb{C}\left\lbrack X\right\rbrack /J \), where \( J \) denotes the ideal in \( \mathbb{C}\lef... | Proof. Define \( \phi : {C}^{n}\left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{C}\left\lbrack X\right\rbrack /J \) by\n\n\[ \phi \left( f\right) = \mathop{\sum }\limits_{{i = 0}}^{n}\frac{1}{i!}{f}^{\left( i\right) }\left( a\right) {X}^{i} + J. \]\n\n\( \phi \) is a homomorphism because, for \( f, g \in {C}^{n}\lef... | Yes |
Lemma 5.3.3. Let \( J \) be as in Lemma 5.3.2. There are exactly \( n + 1 \) proper ideals in \( \mathbb{C}\left\lbrack X\right\rbrack /J \) . | Proof. For \( 0 \leq k \leq n + 1 \), let \( {I}_{k} \) denote the ideal in \( \mathbb{C}\left\lbrack X\right\rbrack \) generated by \( {X}^{k} \) . Then\n\n\[ J = {I}_{n + 1} \varsubsetneq {I}_{n} \varsubsetneq \ldots \varsubsetneq {I}_{0} = \mathbb{C}\left\lbrack X\right\rbrack \]\n\nsince \( {X}^{k} \in {I}_{k} \sma... | Yes |
Theorem 5.3.4. Given \( a \in \left\lbrack {0,1}\right\rbrack \), there are exactly \( n + 1 \) closed primary ideals in \( {C}^{n}\left\lbrack {0,1}\right\rbrack \) with hull equal to \( \{ a\} \), namely the ideals\n\n\[ \n{P}_{a, m} = \left\{ {f \in {C}^{n}\left\lbrack {0,1}\right\rbrack : {f}^{\left( i\right) }\lef... | Proof. It is clear that \( h\left( {P}_{a, m}\right) = \{ a\} \) and that \( {P}_{a, m} \) is properly contained in \( {P}_{a, k} \) for \( m > k \) . The statement now follows from Lemmas 5.3.2 and 5.3.3. | No |
Lemma 5.3.7. Let \( f \in {C}^{n}\left\lbrack {0,1}\right\rbrack \) and suppose that \( a \) is an accumulation point of \( h\left( f\right) \), the zero set of \( f \) . Then \( f \in \overline{j\left( a\right) } \) . | Proof. According to Theorem 5.3.1, it suffices to show that \( {f}^{\left( m\right) }\left( a\right) = 0 \) for \( 0 \leq m \leq n \) . This is done by induction on \( m \), the case \( m = 0 \) being clear. Thus, let \( m > 1 \) and assume that \( {f}^{\left( i\right) }\left( a\right) = 0 \) for \( i < m \) . By hypot... | Yes |
Lemma 5.3.8. Let \( I \) be a closed ideal in \( {C}^{n}\left\lbrack {0,1}\right\rbrack \) and a an isolated point of \( h\left( I\right) \) . If \( I \subseteq \overline{j\left( a\right) } \), then every \( f \in \overline{j\left( a\right) } \) belongs locally to \( I \) at \( a \) . | Proof. Assume that some \( f \in \overline{j\left( a\right) } \) does not belong locally to \( I \) at \( a \) . By hypothesis, there is an open set in \( U \) in \( \left\lbrack {0,1}\right\rbrack \) such that \( U \cap h\left( I\right) = \{ a\} \) . Because \( {C}^{n}\left\lbrack {0,1}\right\rbrack \) is regular, \( ... | Yes |
For \( f, g \in M \) define \( f * g \) on \( \mathbb{T} \) by\n\n\[ \left( {f * g}\right) \left( x\right) = \frac{1}{2\pi }{\int }_{\mathbb{T}}f\left( {x - t}\right) g\left( t\right) {dt}. \]\n\nThen \( f * g \in C\left( \mathbb{T}\right) \), and with this convolution product \( M \) becomes a commutative Banach algeb... | Proof. Observe first that, for \( x, y \in G \) ,\n\n\[ \left| {\left( {f * g}\right) \left( {x}^{-1}\right) - \left( {f * g}\right) \left( {y}^{-1}\right) }\right| \leq \frac{1}{2\pi }\parallel g{\parallel }_{2}{\begin{Vmatrix}{L}_{x}f - {L}_{y}f\end{Vmatrix}}_{2}. \]\n\nThe map \( x \rightarrow {L}_{x}f \) from \( \m... | Yes |
Lemma 5.4.2. The linear span of the functions \( {e}_{n}, n \in \mathbb{Z} \), is dense in \( M \) . | Proof. By the Stone-Weierstrass theorem, the trigonometric functions are uniformly dense in \( C\left( \mathbb{T}\right) \), and hence also dense in the \( \parallel \cdot {\parallel }_{2} \) -norm. Thus the trigonometric functions are dense in \( \left( {C\left( \mathbb{T}\right) ,\parallel \cdot \parallel }\right) \)... | Yes |
Theorem 5.4.3. For \( n \in \mathbb{Z} \), define \( {\varphi }_{n} : M \rightarrow \mathbb{C} \) by\n\n\[ \n{\varphi }_{n}\left( f\right) = \frac{1}{2\pi }{\int }_{\mathbb{T}}f\left( t\right) {e}_{-n}\left( t\right) {dt}, f \in M.\n\]\n\nThen \( {\varphi }_{n} \in \Delta \left( M\right) \) and the map \( n \rightarrow... | Proof. It is easy to check that \( {\varphi }_{n} \in \Delta \left( M\right) \) . Moreover, since \( {\varphi }_{n}\left( {e}_{m}\right) \neq 0 \) if and only if \( m = n \), the map \( n \rightarrow {\varphi }_{n} \) is injective. Observe next that given \( \varphi \in \Delta \left( M\right) \), there exists \( n \in ... | Yes |
Corollary 5.4.4. \( M \) is semisimple and regular. | Proof. If \( f \in M \) is such that \( \widehat{f} = 0 \), then \( \left\langle {f,{e}_{n}}\right\rangle = 0 \) for all \( n \in \mathbb{Z} \). Because the functions \( {e}_{n}, n \in \mathbb{Z} \), form an orthonormal basis of \( {L}^{2}\left( \mathbb{T}\right), f = 0 \) in \( {L}^{2}\left( \mathbb{T}\right) \) and t... | Yes |
Lemma 5.4.5. Let \( E \) be any subset of \( \mathbb{Z} \). Then \( j\left( E\right) \) equals the linear span of all functions \( {e}_{n} \), where \( n \in \mathbb{Z} \smallsetminus E \). | Proof. Clearly, if \( n \notin E \), then \( {e}_{n} \in j\left( E\right) \) since \( {\varphi }_{m}\left( {e}_{n}\right) = 0 \) for all \( m \neq n \), so for all \( m \in E \). Conversely, let \( f \in j\left( E\right) \) and let \( F = \left\{ {n \in \mathbb{Z} : {\varphi }_{n}\left( f\right) \neq 0}\right\} \). The... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.