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Theorem 4.2 (Existence of liftings). Let \( \varphi : \mathcal{S} \rightarrow {\mathcal{S}}^{ \star } \) be a sheaf epimorphism, \( {\varepsilon }^{ \star } : q\mathcal{O} \rightarrow {\mathcal{S}}^{ \star } \) an arbitrary sheaf homomorphism. Then for every \( {\mathfrak{z}}_{0} \in B \) there is a neighborhood \( U\l... | Proof. Let \( {s}_{i}^{ * } \mathrel{\text{:=}} {\varepsilon }^{ * } \circ {\mathrm{e}}_{i} \in \Gamma \left( {B,{\mathcal{S}}^{ * }}\right) \) for \( i = 1,\ldots, q \) . Then for \( {\mathfrak{z}}_{0} \in B \) there are elements \( {\sigma }_{i} \in {\mathcal{S}}_{\text{30 }} \) with \( \varphi \left( {\sigma }_{i}\r... | Yes |
Theorem 4.4. Let \( {\mathcal{S}}^{ \star }\overset{j}{ \rightarrow }\mathcal{S}\overset{p}{ \rightarrow }{\mathcal{S}}^{\star \star } \rightarrow \mathrm{O} \) be an exact sequence of sheaves over B. If \( {\mathcal{S}}^{ \star } \) and \( \mathcal{S} \) are coherent, then \( {\mathcal{S}}^{\star \star } \) is also co... | Proof\n\n1. Since \( p \) is surjective, it follows immediately that \( {\mathcal{S}}^{\star \star } \) is finitely generated.\n\n2. Let \( {\varepsilon }^{\star \star } : {q}^{\star \star }\mathcal{O} \rightarrow {\mathcal{S}}^{\star \star } \) be an arbitrary sheaf homomorphism on an open set \( W \subset B,\varepsil... | Yes |
Theorem 4.5. Let \( \mathrm{O} \rightarrow {\mathcal{S}}^{ \star }\overset{j}{ \rightarrow }\mathcal{S}\overset{p}{ \rightarrow }{\mathcal{S}}^{\star \star } \) be an exact sequence of analytic sheaves over \( B \) . If \( \mathcal{S},{\mathcal{S}}^{\star \star } \) are coherent, then \( {\mathcal{S}}^{ \star } \) is a... | Proof. We may regard \( {\mathcal{S}}^{ \star } \) as an analytic subsheaf of \( \mathcal{S} \), so it suffices to show that \( {\mathcal{S}}^{ * } \) is finitely generated. Let \( {\mathfrak{z}}_{0} \in B \) be chosen arbitrarily. Since \( \mathcal{S} \) and \( {\mathcal{S}}^{\star \star } \) are coherent there is a n... | Yes |
Theorem 4.7. Let \( \varphi : {\mathcal{S}}_{1} \rightarrow {\mathcal{S}}_{2} \) be a homomorphism of coherent sheaves over B. Then \( \operatorname{Ker}\varphi \) and \( \operatorname{Im}\varphi \) are also coherent. | Proof. The sequence \( \mathrm{O} \rightarrow \operatorname{Ker}\varphi \rightarrow {\mathcal{S}}_{1} \rightarrow {\mathcal{S}}_{2} \) is exact, so \( \operatorname{Ker}\varphi \) is coherent. Since Im \( \varphi \simeq {\mathcal{S}}_{1}/\mathrm{{Ker}}\;\varphi \) , the coherence of Im \( \varphi \) follows from Theore... | Yes |
Theorem 4.8 (Serre’s five lemma). Let \( {\mathcal{S}}^{\prime }\overset{{j}_{1}}{ \rightarrow }{\mathcal{S}}^{\prime \prime }\overset{{j}_{2}}{ \rightarrow }\mathcal{S}\overset{{p}_{1}}{ \rightarrow }{\mathcal{S}}^{ * }\overset{{p}_{2}}{ \rightarrow }{\mathcal{S}}^{* * } \) be an exact sequence of sheaves. If \( {\mat... | Proof. The sequence \( \mathrm{O} \rightarrow {\mathcal{S}}^{\prime \prime }/\operatorname{Im}{j}_{1} \rightarrow \mathcal{S} \rightarrow \operatorname{Ker}{p}_{2} \rightarrow \mathrm{O} \) is exact and the sheaves \( {\mathcal{S}}^{\prime \prime }/\operatorname{Im}{j}_{1} \) and \( \operatorname{Ker}{p}_{2} \) are coh... | Yes |
Theorem 4.9. Let \( \mathcal{R} \) be a coherent sheaf of \( \mathbb{C} \) -algebras over \( X,\mathcal{I} \subset \mathcal{R}a \) coherent ideal sheaf. Then as a sheaf of \( \mathbb{C} \) -algebras \( \mathcal{R}/\mathcal{I} \) is coherent. | Proof. We already know that as a sheaf of \( \mathcal{R} \) -modules \( \mathcal{R}/\mathcal{I} \) is coherent. Now let \( \pi : \mathcal{R} \rightarrow \mathcal{R}/\mathcal{I} \) the canonical projection, \( U \subset X \) open, \( \varphi : q\left( {\mathcal{R}/\mathcal{I}}\right) \mid U \rightarrow \) \( \left( {\ma... | Yes |
Theorem 1.1. Let \( \varphi = \left( {\widetilde{\varphi },{\varphi }_{ \star }}\right) : \left( {{X}_{1},{\mathcal{H}}_{1}}\right) \rightarrow \left( {{X}_{2},{\mathcal{H}}_{2}}\right) \) be an isomorphism of complex ringed spaces. Then for every open set \( V \subset {X}_{2} \) there is a \( \mathbb{C} \) -algebra is... | Proof. \( \widehat{\varphi }\left( s\right) : {\widetilde{\varphi }}^{-1}\left( V\right) \rightarrow {\mathcal{H}}_{1} \) is clearly continuous, and\n\n\[ {\pi }_{1} \circ \left( {{\varphi }_{ \star }^{-1} \circ s \circ \widetilde{\varphi }}\right) = \left( {{\pi }_{1} \circ {\varphi }_{ \star }^{-1}}\right) \circ \lef... | Yes |
Lemma 1. If \( R \) is a local \( \mathbb{C} \) -algebra with the maximal ideal \( \mathfrak{m} \), then \( a \in \mathfrak{m} \) if and only if \( a - c \cdot 1 \notin \mathfrak{m} \) for all \( c \in \mathbb{C} - \{ 0\} \) . | Proof\n\n1. Let \( a \in \mathfrak{m}, a - c \cdot 1 \in \mathfrak{m} \) . Then also \( c \cdot 1 = a - \left( {a - c \cdot 1}\right) \in \mathfrak{m} \) . That is, \( c \) cannot lie in \( \mathbb{C} - \{ 0\} \) .\n\n2. For all \( c \in \mathbb{C} - \{ 0\} \) let \( a - c \cdot 1 \notin \mathfrak{m} \) . We set \( c \... | No |
Lemma 2. Let \( \rho : \left( {{R}_{1},{m}_{1}}\right) \rightarrow \left( {{R}_{2},{m}_{2}}\right) \) be a \( \mathbb{C} \) -algebra homomorphism between local \( \mathbb{C} \) -algebras. Then \( \rho \left( {\mathfrak{m}}_{1}\right) \subset {\mathfrak{m}}_{2} \) . If in particular \( \rho \) is an isomorphism, then \(... | Proof\n\n1. If \( \sigma \in {m}_{1} \), then \( \sigma - c \cdot 1 \notin {m}_{1} \) for all \( c \in \mathbb{C} - \{ 0\} \) . Therefore for every \( c \in \mathbb{C} - \{ 0\} \) there is a \( {\sigma }_{c} \) with \( {\sigma }_{c} \cdot \left( {\sigma - \mathrm{c} \cdot 1}\right) = 1 \), and then\n\n\[ \rho \left( {\... | Yes |
Theorem 1.2. Let \( \varphi = \left( {\widetilde{\varphi },{\varphi }_{ \star }}\right) : \left( {{X}_{1},{\mathcal{H}}_{1}}\right) \rightarrow \left( {{X}_{2},{\mathcal{H}}_{2}}\right) \) be an isomorphism of complex ringed spaces. For an open set \( V \subset {X}_{2} \) let \( {\varphi }^{ * } : \mathcal{F}\left( {V,... | Proof. Let \( V \subset {X}_{2} \) be open, \( y \in V, x \mathrel{\text{:=}} {\widetilde{\varphi }}^{-1}\left( y\right) \) and \( s \in \Gamma \left( {V,{\mathcal{H}}_{2}}\right) \) . Then \( s\left( y\right) = \left( {\left\lbrack s\right\rbrack \left( y\right) }\right) \cdot 1 + {\sigma }^{ * } \) with \( {\sigma }^... | Yes |
Theorem 1.3. Let \( \left( {X,\mathcal{H}}\right) \) be a complex manifold, \( W \subset X \) an open subset. Then the mapping \( \Gamma \left( {W,\mathcal{H}}\right) \rightarrow \mathcal{F}\left( {W,\mathbb{C}}\right) \) given by \( f \mapsto \left\lbrack f\right\rbrack \) is injective, and for every \( f \in \Gamma \... | Proof. There is an open covering \( {\left( {U}_{\imath }\right) }_{\imath \in I} \) of \( W \) and a system \( {\left( {B}_{\imath }\right) }_{\imath \in I} \) of open sets such that \( \left( {{U}_{\imath },\mathcal{H} \mid {U}_{\imath }}\right) \simeq \left( {{B}_{\imath },\mathcal{O} \mid {B}_{\imath }}\right) \) .... | Yes |
Theorem 2.1. Let \( \left( {X,\mathcal{H}}\right) \) be an n-dimensional complex manifold, \( W \subset X \) open. Then \( \left( {W,\mathcal{H} \mid W}\right) \) is also an n-dimensional complex manifold. | Proof. It is clear that \( W \) is a Hausdorff space and \( \mathcal{H} \mid W \) a sheaf of local \( \mathbb{C} \) -algebras. For every point \( {x}_{0} \in W \) there is a neighborhood \( U\left( {x}_{0}\right) \subset X \) and an isomorphism \( \varphi : \left( {U,\mathcal{H} \mid U}\right) \rightarrow \left( {B,\ma... | Yes |
Theorem 2.2 (Identity theorem). Let \( \left( {X,\mathcal{H}}\right) \) be a connected complex manifold, \( {f}_{1},{f}_{2} \) holomorphic functions on \( X \) and \( V \subset X \) a non-empty open subset with \( {f}_{1}\left| {V = {f}_{2}}\right| V \) . Then \( {f}_{1} = {f}_{2} \) . | Proof. Let \( {W}_{1} \mathrel{\text{:=}} \left\{ {x \in X : r{f}_{1}\left( x\right) = r{f}_{2}\left( x\right) }\right\} ,{W}_{2} \mathrel{\text{:=}} X - {W}_{1}.{W}_{1} \) is not empty since \( V \) is contained in \( {W}_{1} \), and \( {W}_{1} \) is open since the set where two sections coincide is always open. Let \... | Yes |
Theorem 2.3 (Maximum principle). Let \( \left( {X,\mathcal{H}}\right) \) be a connected complex manifold, \( f \) holomorphic on \( X,{x}_{0} \in X \) a point. If \( \left| f\right| \) has a local maximum at \( {x}_{0} \), then \( f \) is constant. | Proof. There is a neighborhood \( U\left( {x}_{0}\right) \subset X \) and an isomorphism \( \varphi \) : \( \left( {U,\mathcal{H}}\right) \rightarrow \left( {B,\mathcal{O}}\right) \) . Without loss of generality we may assume that \( \widetilde{\varphi }\left( {x}_{0}\right) = 0 \) and \( B \) is a polycylinder about t... | Yes |
Theorem 2.4. Let \( \left( {X,\mathcal{H}}\right) \) be a connected compact complex manifold. Then every function holomorphic on \( X \) is constant. | Proof. If \( f \) is holomorphic on \( X \), then \( \left| f\right| \) is continuous on \( X \) and therefore attains a maximum on the compact manifold \( X \) . By the maximum principle, \( f \) is constant. | Yes |
Theorem 2.5. A mapping \( \psi : {X}_{1} \rightarrow {X}_{2} \) is biholomorphic if and only if there exists an isomorphism \( \varphi : \left( {{X}_{1},{\mathcal{H}}_{1}}\right) \rightarrow \left( {{X}_{2},{\mathcal{H}}_{2}}\right) \) with \( \widetilde{\varphi } = \psi \) . | ## Proof\n\n1. If \( \psi : {X}_{1} \rightarrow {X}_{2} \) is a biholomorphic mapping, then \( \psi \) and \( {\psi }^{-1} \) carry holomorphic functions into holomorphic functions and hence induce an isomorphism between the canonical pre-sheaves. Naturally \( \widetilde{\varphi } = \psi \) for the corresponding isomor... | Yes |
Theorem 2.6. Let \( X \) be a connected complex manifold and \( M \neq X \) an analytic subset of \( X \) . Then \( \mathring{M} = \varnothing \) . | Proof. If \( \mathring{M} \neq \varnothing \), there exists a point \( {x}_{0} \in \partial \mathring{M} \) such that for every open neighborhood \( U\left( {x}_{0}\right) \subset X \) the set \( U \cap \mathring{M} \) is open and non-empty. We could choose a connected \( U \) such that there exist holomorphic function... | Yes |
Theorem 2.8. For a domain \( \left( {X,\psi }\right) \) over \( {\mathbb{C}}^{n} \) the following properties are equivalent:\n\n1. \( X \) is a Stein manifold.\n\n2. \( X \) is holomorphically convex.\n\n3. \( X \) is a domain of holomorphy. | Proof. If \( K \subset X \) is compact, then \( \widehat{K} \subset X \) is always closed. If \( X \) is compact it follows that \( \widehat{K} \) is also compact. Therefore \( X \) is holomorphically convex.\n\nSince \( X \) is compact there exists a decomposition of \( X \) into finitely many connected components, wh... | No |
Theorem 3.2. An analytic set \( A \subset X \) is free of singularities of codimension \( d \) if and only if for every point \( {x}_{0} \in A \) there exists a neighborhood \( U\left( {x}_{0}\right) \subset X \) and an isomorphism \( \varphi = \left( {\widetilde{\varphi },{\varphi }_{ \star }}\right) : \left( {U,\math... | Proof. Let \( {x}_{0} \in A \) be given, \( U\left( {x}_{0}\right) \subset X \) a neighborhood and \( \varphi : \left( {U,\mathcal{H}}\right) \rightarrow \) \( \left( {\widehat{B},\mathcal{O}}\right) \) an isomorphism. \( A \cap U \) is singularity free of codimension \( d \) if and only if \( \widetilde{\varphi }\left... | Yes |
Theorem 3.3. Let \( A \subset X \) be an analytic set, free of singularities of codimension d. Then \( X \) induces a canonical \( \left( {n - d}\right) \) -dimensional manifold structure on \( A \) , and the natural imbedding \( {j}_{A} : A \hookrightarrow X \) is holomorphic. | Proof. \( A \), with the relative topology induced by \( X \), is clearly a Hausdorff space. A function \( f \) defined on an open set \( W \subset A \) will be called holomorphic if for every \( x \in W \) there is an open neighborhood \( U\left( x\right) \subset X \) and a holomorphic function \( \widehat{f} \) on \(... | Yes |
Theorem 3.5. Let \( X \) be an n-dimensional complex manifold. Then the diagonal \( D \mathrel{\text{:=}} \{ \left( {x, x}\right) : x \in X\} \subset X \times X \) is an analytic subset free of singularities of codimension \( n \) . | ## Proof\n\n1. Since \( X \) is a Hausdorff space, the diagonal \( D \subset X \times X \) is closed. Therefore \( D \) is analytic at each point \( \left( {x, y}\right) \in X \times X - D \) .\n\n2. Let \( \left( {{x}_{0},{x}_{0}}\right) \in D \) . Then there is a neighborhood \( U\left( {x}_{0}\right) \subset X \) an... | Yes |
Theorem 3.6. Let \( X \) be a complex manifold, \( D \subset X \times X \) the diagonal. Then the diagonal mapping \( d : X \rightarrow D \) by \( d\left( x\right) \mathrel{\text{:=}} \left( {x, x}\right) \) is biholomorphic. | Proof. \( d \) is bijective, and the inverse mapping \( {d}^{-1} = {p}_{1} \mid D \) is holomorphic. It remains to be shown that \( d \) is holomorphic. Let \( W \subset D \) be open, \( g \) holomorphic on \( W,\left( {{x}_{0},{x}_{0}}\right) \in W \) . Then there exists a neighborhood \( U\left( {x}_{0}\right) \subse... | Yes |
Theorem 3.7. The n-dimensional complex projective space is an n-dimensional complex manifold, and the natural projection \( \pi : {\mathbb{C}}^{n + 1} - \{ 0\} \rightarrow {\mathbb{P}}^{n} \) is holomorphic. | Proof. In order to complete the proof we have to demonstrate the holomorphy of \( \pi \) . Let \( W \subset X \) be open, \( g \) holomorphic in \( W \) . Without loss of generality we may assume that \( W \subset {U}_{1} \) . Then \( g \circ {\varphi }_{1}^{-1} = g \circ \pi \circ {\left( {\alpha }_{1} \mid {W}_{1}\ri... | No |
Theorem 3.8. \( {\mathbb{P}}^{n} \) is compact. | Proof. Let \( S \mathrel{\text{:=}} \left\{ {3 \in {\mathbb{C}}^{n + 1} : \parallel 3\parallel = 1}\right\} = {S}^{{2n} + 1} \) . For \( 3 \in {\mathbb{C}}^{n + 1} - \{ 0\} ,\widehat{3} \mathrel{\text{:=}} \) \( \left( {1/\begin{Vmatrix}3\end{Vmatrix}}\right) \cdot 3 \) lies in \( S \) and \( \pi \left( 3\right) = \pi ... | Yes |
Theorem 3.9. Let \( X = \mathbb{C} \cup \{ \infty \} \) be the Riemann sphere. A biholomorphic mapping \( \varphi : X \rightarrow {\mathbb{P}}^{1} \) is defined by \( \varphi \left( \infty \right) \mathrel{\text{:=}} G\left( {0,1}\right) \) and \( \varphi \left( z\right) \mathrel{\text{:=}} {\varphi }_{1}^{-1}\left( z\... | Proof. It is clear that \( \varphi \) is bijective. On \( X \) one has two coordinate systems \( {\psi }_{1} : X - \{ \infty \} \rightarrow \mathbb{C} \), and \( {\psi }_{2} : X - \{ 0\} \rightarrow \mathbb{C} \) . Let \( {X}_{1} \mathrel{\text{:=}} X - \{ \infty \} ,{X}_{2} \mathrel{\text{:=}} \) \( X - \{ 0\} \) . Th... | Yes |
Theorem 3.11. \( H \) is a compact \( n \) -dimensional complex manifold (the so-called Hopf manifold), and \( {\pi }_{H} : {\mathbb{C}}^{n} - \{ 0\} \rightarrow H \) is holomorphic. If, for \( {\mathfrak{z}}_{1},{\mathfrak{z}}_{2} \in {\mathbb{C}}^{n} - \{ 0\} ,{\pi }_{H}\left( {\mathfrak{z}}_{1}\right) = {\pi }_{H}\l... | But then \( G\left( {\mathfrak{z}}_{2}\right) = G\left( {\mathfrak{z}}_{1}\right) \) . Therefore there is a mapping \( h : H \rightarrow {\mathbb{P}}^{n} \) defined by \( h\left( {{\pi }_{H}\left( 3\right) }\right) \mathrel{\text{:=}} G\left( 3\right) \) . We obtain the following commutative diagram. ![cd020fea-4b30-4e... | No |
Theorem 3.12. Let \( U \subset X \) be open, \( {x}_{0} \in U \) . Let \( g \) , \( h \) be holomorphic functions on \( U \) with \( g\left( {x}_{0}\right) = h\left( {x}_{0}\right) = 0 \) . If the germs \( {g}_{{x}_{0}},{h}_{{x}_{0}} \) are relatively prime, then for every complex number \( c \) there exists a point \(... | Proof. Without loss of generality we can assume that \( U \) is a polycylinder in \( {\mathbb{C}}^{n} \) and \( {x}_{0} = 0 \) . By the Weierstrass preparation theorem one can further assume that \( {g}_{{x}_{0}},{h}_{{x}_{0}} \) are elements of \( {\mathcal{O}}_{{x}_{0}}^{\prime }\left\lbrack {z}_{1}\right\rbrack \) .... | Yes |
Theorem 3.13. Let \( Y \subset X \) be an open dense subset, \( f \) a holomorphic function on \( Y \) . For every point \( {x}_{0} \in X - Y \) let there be a neighborhood \( U\left( {x}_{0}\right) \subset X \) and holomorphic functions \( h, g \) on \( U \) such that \( {g}_{{x}_{0}} \) and \( {h}_{{x}_{0}} \) are re... | Proof. Let \( {x}_{0} \in X - Y \) . By assumption there exists a neighborhood \( U\left( {x}_{0}\right) \subset X \) and holomorphic functions \( g, h \) on \( U \) which are relatively prime at \( {x}_{0} \), such that \( g\left( x\right) = f\left( x\right) \cdot h\left( x\right) \) for \( x \in U \cap Y \) . If \( h... | Yes |
Theorem 3.15. If \( X \) is a projective-algebraic manifold, then the Betti numbers satisfy\n\n\[ \n{B}_{{2i} + 1}\left( X\right) \in 2\mathbb{Z} \]\n\n\[ \n{B}_{2i}\left( X\right) \neq 0\text{. } \]\n | This theorem is proved within the framework of the theory of \ | No |
If \( Y \) is a regular closure of \( {\mathbb{C}}^{n} \), then \( Y - {\mathbb{C}}^{n} \) is an analytic set of codimension 1. | Let \( {z}_{1},\ldots ,{z}_{n} \) be the coordinates of \( {\mathbb{C}}^{n} \) . By hypothesis they can be continued to meromorphic functions \( {f}_{1},\ldots ,{f}_{n} \) on \( Y \) .\n\nThe set \( {P}_{i} \) of poles of \( {f}_{i} \) is an analytic set of codimension 1, and so is\n\n\( P : = \mathop{\bigcup }\limits_... | Yes |
Theorem 4.4. Let \( \left( {X, M,\pi, N, Y}\right) \) be a proper modification, \( \widehat{\pi } : X \rightarrow Y \) a continuation of \( \pi \) in the sense of Def. 4.5. Then \( M \) and \( N \) are analytic sets, and \( \widehat{\pi }\left( M\right) = N \) . | Proof. By Theorem 4.2, \( M = E\left( \widehat{\pi }\right) \) is analytic, and by Theorem 4.3, \( {N}^{ \star } \mathrel{\text{:=}} \) \( \widehat{\pi }\left( M\right) \) is analytic. It remains to show that \( N = {N}^{ \star } \) :\n\n1. Suppose there is a \( {y}_{0} \in {N}^{ \star } - N \) . We set \( {x}_{0} \mat... | Yes |
Theorem 4.5. Let \( G \subset {\mathbb{C}}^{n} \) be a domain with \( 0 \in G,\pi : {\mathbb{C}}^{n} - \{ 0\} \rightarrow {\mathbb{P}}^{n - 1} \) the natural projection. Then \( X \mathrel{\text{:=}} \left\{ {\left( {\mathfrak{z}, x}\right) \in \left( {G-\{ 0\} }\right) \times {\mathbb{P}}^{n - 1} : x = \pi \left( \mat... | Proof. Let \( {\varphi }_{i} : {U}_{i} \rightarrow {\mathbb{C}}^{n - 1} \) be the canonical coordinate system of \( {\mathbb{P}}^{n - 1} \) . If \( \mathfrak{z} = \left( {{z}_{1},\ldots ,{z}_{n}}\right) \in G - \{ 0\} \) and \( x = \pi \left( \mathfrak{z}\right) \in {U}_{1} \), then\n\n\[ x = \pi \left( {1,\frac{{z}_{2... | Yes |
Theorem 1.1. \( \left( {{\widehat{\Gamma }}_{U} : {\mathcal{S}}_{N} \rightarrow \widehat{\Gamma }\left( {U,\mathcal{S}}\right) ,\varphi \mapsto {\varphi }_{ \star }}\right) \) is an exact covariant functor from the category of \( R \) -module sheaves over \( X \) to the category of \( R \) -modules. Therefore:\n\n1. if... | The proof is completely trivial. | No |
Theorem 1.3. If \( \mathcal{S} \) is a sheaf of \( R \) -modules over \( X \), then \( W\left( \mathcal{S}\right) \) is a flabby sheaf. | Proof. We can identify \( \Gamma \left( {U, W\left( \mathcal{S}\right) }\right) \) with \( \widehat{\Gamma }\left( {U,\mathcal{S}}\right) \) . If \( s \in \widehat{\Gamma }\left( {U,\mathcal{S}}\right) \) then we define \( {s}^{ \star } \in \widehat{\Gamma }\left( {X,\mathcal{S}}\right) \) by\n\n\[ \n{s}^{ \star }\left... | Yes |
Theorem 1.4. \( \left( {W : \mathcal{S} \mapsto W\left( \mathcal{S}\right) ,\varphi \mapsto {W\varphi }}\right) \) is an exact covariant functor from the category of \( R \) -module sheaves over \( X \) to itself. | 1. Let \( \psi : {\mathcal{S}}_{1} \rightarrow \mathcal{S},\varphi : \mathcal{S} \rightarrow {\mathcal{S}}_{2} \) be sheaf homomorphisms and \( s \in \Gamma \left( {U,{\mathcal{S}}_{1}}\right) \). Then \( W\left( {\varphi \circ \psi }\right) \circ {rs} = r\left( {{\left( \varphi \circ \psi \right) }_{ \star }s}\right) ... | Yes |
Theorem 1.5. Let \( \varphi : {\mathcal{S}}_{1} \rightarrow {\mathcal{S}}_{2} \) be a homomorphism of sheaves of \( R \) -modules over \( X \) . Then there are canonical homomorphisms \( {W}_{i}\varphi : {W}_{i}\left( {\mathcal{S}}_{1}\right) \rightarrow {W}_{i}\left( {\mathcal{S}}_{2}\right) \) with \( \left( {{W}_{i ... | Proof. We proceed by induction. Let \( {W}_{0}\varphi \mathrel{\text{:=}} {W\varphi } \) . If \( {W}_{0}\varphi ,{W}_{1}\varphi ,\ldots ,{W}_{t}\varphi \) have been constructed, then we have the following commutative diagram.\n\n\[ \n{W}_{\ell - 1}\left( {\mathcal{S}}_{1}\right) \overset{{d}_{\ell - 1}}{ \rightarrow }{... | Yes |
Lemma 1. Let the following diagram of sheaves of \( R \) -modules be commutative, have exact rows and columns, and moreover let the mapping \( {\varphi }_{0} \) be surjective:\n\n\n\nIf \( \sigma \in {\mathcal{S}}_{6... | Proof. Let \( {\sigma }_{1} \mathrel{\text{:=}} {\varphi }_{3}\left( \sigma \right) \in {\mathcal{S}}_{7} \) .\n\n1. Because \( {\psi }_{6}\left( {\sigma }_{1}\right) = \mathbf{O} \) there exists a \( {\sigma }_{2} \in {\mathcal{S}}_{4} \) with \( {\psi }_{4}\left( {\sigma }_{2}\right) = {\sigma }_{1} \).\n\n2. \( {\ps... | Yes |
Lemma 2. In the sequence\n\n\[ \n{\mathcal{S}}_{1}\overset{{\varphi }_{1}}{ \rightarrow }{\mathcal{S}}_{2}\overset{{\varphi }_{2}}{ \rightarrow }{\mathcal{S}}_{3}\overset{{\varphi }_{3}}{ \rightarrow }{\mathcal{S}}_{4}\overset{{\varphi }_{4}}{ \hookrightarrow }{\mathcal{S}}_{5} \]\n\nlet \( {\varphi }_{1} \) be surject... | Proof. \( \operatorname{Im}\left( {{\varphi }_{2} \circ {\varphi }_{1}}\right) = \operatorname{Im}{\varphi }_{2} = \operatorname{Ker}{\varphi }_{3} = \operatorname{Ker}\left( {{\varphi }_{4} \circ {\varphi }_{3}}\right) \) . | Yes |
Theorem 1.7. \( \left( {\Gamma : {\mathcal{S}}_{N} \rightarrow \Gamma \left( {X,\mathcal{S}}\right) ,\varphi \mapsto {\varphi }_{ \star }}\right) \) is a left-exact functor; that is, if\n\n\[ 0 \rightarrow {\mathcal{S}}^{\prime }\overset{\varphi }{ \rightarrow }\mathcal{S}\overset{\psi }{ \rightarrow }{\mathcal{S}}^{\p... | Proof. Since \( \widehat{\Gamma } \) is exact (see Theorem 1.1), it is clear that \( {\varphi }_{ \star } \) is injective and \( {\psi }_{ \star } \circ {\varphi }_{ \star } = 0 \) . Now let \( s \in \Gamma \left( {X,\mathcal{S}}\right) \) with \( \mathbf{O} = {\psi }_{ \star }\left( s\right) = \psi \circ s \) . Then t... | Yes |
Theorem 1.8. Let \( \\mathcal{S} \) be a sheaf of \( R \) -modules over \( X \) , \n\n\[ \n{W}^{ \\bullet }\\left( \\mathcal{S}\\right) : \\Gamma \\left( {X,{W}_{0}\\left( \\mathcal{S}\\right) }\\right) \\rightarrow \\Gamma \\left( {X,{W}_{1}\\left( \\mathcal{S}\\right) }\\right) \\rightarrow \\Gamma \\left( {X,{W}_{2}... | Proof. Clearly \( {W}^{ \\bullet }\\left( \\mathcal{S}\\right) \) is a complex, \( {\\varepsilon }_{ \\star } : \\Gamma \\left( {X,\\mathcal{S}}\\right) \\rightarrow \\Gamma \\left( {X,{W}_{0}\\left( \\mathcal{S}\\right) }\\right) \) an \( R \) - module monomorphism, and \( {\\left( {d}_{0}\\right) }_{ \\star } \\circ ... | Yes |
Theorem 1.10. Let \( \mathcal{S} \) be a flabby sheaf of \( R \) -modules over \( X \) and \( \mathrm{O} \rightarrow \mathcal{S} \rightarrow \) \( {\mathcal{S}}_{0} \rightarrow {\mathcal{S}}_{1} \rightarrow \cdots \) a flabby resolution of \( \mathcal{S} \) . Then the sequence\n\n\[ 0 \rightarrow \Gamma \left( {X,\math... | Proof. Let \( {\mathcal{B}}_{\lambda } \mathrel{\text{:=}} \operatorname{Im}\left( {{\varphi }_{\lambda } : {\mathcal{S}}_{\lambda - 1} \rightarrow {\mathcal{S}}_{\lambda }}\right) \) for \( \lambda = 0,1,2,\ldots \) and \( {\mathcal{S}}_{-1} \mathrel{\text{:=}} \mathcal{S} \).\n\n1. We show by induction that all \( {\... | Yes |
Theorem 2.1. \( {\delta }^{\ell } : {C}^{\ell }\left( {\mathfrak{U},\mathcal{S}}\right) \rightarrow {C}^{\ell + 1}\left( {\mathfrak{U},\mathcal{S}}\right) \) with\n\n\[ \n\left( {{\delta }^{\ell }\xi }\right) \left( {{\iota }_{0},\ldots ,{\iota }_{\ell }}\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{\lambda = 0}}... | Proof\n\n1. First we show that \( {\delta }^{\ell }\xi \) is alternating. It suffices to consider transpositions.\n\n\[ \n\left( {{\delta }^{\ell }\xi }\right) \left( {{\iota }_{0},\ldots ,{\iota }_{v},{\iota }_{v + 1},\ldots ,{\iota }_{\ell + 1}}\right) \n\]\n\n\[ \n= \mathop{\sum }\limits_{{\lambda \neq v, v + 1}}{\l... | Yes |
Theorem 2.2. \( \left( {\Gamma \left( {X,\mathcal{S}}\right) ,\varepsilon ,{C}^{ \bullet }\left( {\mathfrak{U},\mathcal{S}}\right) }\right) \) is an augmented cochain complex. | Proof. Clearly \( \varepsilon \) is an \( R \) -module homomorphism. If \( {\varepsilon s} = 0 \), then \( s \mid {U}_{\iota } = \mathbf{O} \) for every \( \iota \in I \) ; therefore \( s = \mathbf{O} \) . Hence \( \varepsilon \) is injective.\n\nLet \( \xi \in {C}^{0}\left( {\mathfrak{U},\mathcal{S}}\right) \) and \( ... | Yes |
Theorem 2.3. If \( X \) itself belongs to the elements of the covering \( \mathfrak{U} \), then \( {H}^{\ell }\left( {\mathfrak{U},\mathcal{S}}\right) = 0 \) for \( \ell \geq 1 \) . | Proof. If \( \mathfrak{U} = {\left( {U}_{\imath }\right) }_{\imath \in I} \), then there is a \( \rho \in I \) with \( X = {U}_{\rho } \) . Let \( \xi \in {Z}^{\ell }\left( {\mathfrak{U},\mathcal{S}}\right) \) , \( \ell \geq 1 \) . There is an element \( \eta \in {C}^{\ell - 1}\left( {\mathfrak{U},\mathcal{S}}\right) \... | Yes |
Theorem 2.5. Let \( {\left( {U}_{\iota },{f}_{\iota }\right) }_{\iota \in I} \) be a Cousin I distribution on \( X \) , \( \mathcal{H} \) the structure sheaf of \( X,\mathfrak{U} \mathrel{\text{:=}} {\left( {U}_{\imath }\right) }_{\imath \in I} \) . Then\n\n1. \( \gamma \left( {{\iota }_{0},{\iota }_{1}}\right) \mathre... | Proof\n\n1. Clearly\n\n\[ \gamma \left( {{\iota }_{0},{\iota }_{1}}\right) + \gamma \left( {{\iota }_{1},{\iota }_{2}}\right) = \left( {{f}_{{\iota }_{0}} - {f}_{{\iota }_{1}}}\right) + \left( {{f}_{{\iota }_{1}} - {f}_{{\iota }_{2}}}\right) = {f}_{{\iota }_{0}} - {f}_{{\iota }_{2}} = \gamma \left( {{\iota }_{0},{\iota... | Yes |
Theorem 3.1. Let \( \left( {M,{\varepsilon }_{1},{A}^{ \bullet }}\right) ,\left( {M,{\varepsilon }_{2},{B}^{ \bullet }}\right) \) be two augmented cochain complexes. Let there be given a double complex \( \left( {{C}_{\mathrm{v}\mu },{d}^{\prime },{d}^{\prime \prime }}\right) \) and homomorphisms \( {d}_{j}^{\prime } :... | ## Proof\n\n1. \( {Z}_{0j} = \left\{ {\xi \in {C}_{0j} : {d}^{\prime }\xi = 0}\right. \) and \( \left. {{d}^{\prime \prime }\xi = 0}\right\} = \left\{ {\xi \in {C}_{0j}}\right. \) : There is an \( \eta \in {A}^{j} \) with \( {d}_{j}^{\prime }\eta = \xi ,{d}^{\prime \prime }\xi = 0\} = \left\{ {\xi \in {C}_{0j}}\right. ... | Yes |
Let the \( {d}^{\prime } \) -sequences be exact at the locations \( \left( {i, j}\right) \) and \( \left( {i - 1, j}\right) \) . Then there are homomorphisms\n\n\[ \n{\varphi }_{ij} : {H}_{ij} \rightarrow {H}_{i - 1, j + 1}\;\text{ for }\;i \geq 1,\;\mathrm{{with}}\;{\varphi }_{ij} \circ {q}_{ij} \circ {d}^{\prime } = ... | 1. If \( {\xi }_{ij} \in {Z}_{ij} \), then \( {d}^{\prime }{\xi }_{ij} = 0 \) . Therefore there is an \( {\eta }_{i - 1, j} \in {C}_{i - 1, j} \) with \( {d}^{\prime }{\eta }_{i - 1, j} = {\xi }_{ij} \) . We set \( {\varphi }_{ij}\left( {{q}_{ij}\left( {\xi }_{ij}\right) }\right) \mathrel{\text{:=}} {q}_{i - 1, j + 1}\... | Yes |
Theorem 3.3. Let \( X \) be a complex manifold, \( \mathcal{S} \) a sheaf of \( R \) -modules over \( X \) , and \( \mathfrak{U} \) an open covering of \( X \) . Then there is a (canonical) \( R \) -module homomorphism \[ {\varphi }_{\ell } : {H}^{\ell }\left( {\mathfrak{U},\mathcal{S}}\right) \rightarrow {H}^{\ell }\l... | ## Proof 1. Let \( \left( {{C}_{ij},{d}^{\prime },{d}^{\prime \prime }}\right) \) be the canonical double complex of \( \left( {X,\mathcal{S},\mathfrak{U}}\right) \) . Then \( {H}^{j}\left( {X,\mathcal{S}}\right) \simeq {H}_{0j},{H}^{i}\left( {\mathfrak{U},\mathcal{S}}\right) \simeq {H}_{i0} \), and we can define \[ {\... | Yes |
Theorem 3.4. If \( \mathfrak{U} \) is a Leray covering of \( \mathcal{S} \), then \( {\varphi }_{\ell } : {H}^{\ell }\left( {\mathfrak{U},\mathcal{S}}\right) \rightarrow {H}^{\ell }\left( {X,\mathcal{S}}\right) \) is an isomorphism for every \( \ell \geq 1 \) . | Proof. If \( {H}^{\ell }\left( {{U}_{{\iota }_{0}}\ldots {}_{{\iota }_{i}},\mathcal{S}}\right) = 0 \), then by the definition of flabby cohomology the following sequence is exact:\n\n\[ \Gamma \left( {{U}_{{\iota }_{0}\cdots {\iota }_{i}},{\mathcal{S}}_{j - 1}}\right) \overset{{d}_{ * }}{ \rightarrow }\Gamma \left( {{U... | Yes |
1. If \( \xi \in {Z}^{i}\left( {X,{\mathcal{S}}^{ \star }}\right) \), then \( {\left( {W}_{i}\varphi \right) }_{ \star }\xi \in {Z}^{i}\left( {X,\mathcal{S}}\right) \) . 2. If \( \xi \in {B}^{i}\left( {X,{\mathcal{S}}^{ \star }}\right) \), then \( {\left( {W}_{i}\varphi \right) }_{ \star }\xi \in {B}^{i}\left( {X,\math... | Proof. The following diagram is commutative: \[ \Gamma \left( {X,{\mathcal{S}}_{i - 1}^{ * }}\right) \overset{d}{ \rightarrow }\Gamma \left( {X,{\mathcal{S}}_{i}^{ * }}\right) \overset{d}{ \rightarrow }\Gamma \left( {X,{\mathcal{S}}_{i + 1}^{ * }}\right) \] \[ \Gamma \left( {X,{\mathcal{S}}_{i - 1}}\right) \overset{d}{... | Yes |
Theorem 4.2. \( \left( {{H}^{i} : \mathcal{S} \rightsquigarrow {H}^{i}\left( {X,\mathcal{S}}\right) ,\varphi \rightsquigarrow \bar{\varphi }}\right) \) is a covariant functor, that is:\n\n1. \( {\overline{\mathrm{{id}}}}_{\mathcal{S}} = {\mathrm{{id}}}_{{H}^{i}\left( {X,\mathcal{S}}\right) } \) .\n\n2. \( \overline{\ps... | The proof is trivial. | No |
Theorem 4.5. \( \mathrm{O} \rightarrow \mathbb{Z} \rightarrow \mathcal{O}\overset{\exp }{ \rightarrow }{\mathcal{O}}^{ \star } \rightarrow \mathrm{O} \) is an exact sequence of sheaves of \( \mathbb{Z} \) - modules (where \( \mathbb{Z} \) also denotes the sheaf of germs of continuous \( \mathbb{Z} \) -valued functions)... | Proof. Continuous \( \mathbb{Z} \) -valued functions are locally constant, in particular, locally holomorphic. Hence we can regard \( \mathbb{Z} \) as a subsheaf of \( \mathcal{O} \), and we need only show that \( \operatorname{Ker}\left( \exp \right) = \mathbb{Z} \) and \( \operatorname{Im}\left( \exp \right) = {\math... | Yes |
Let \( f \in \Gamma \left( {X,{\mathcal{O}}^{ \star }}\right) \) . Then there is an \( h \in \Gamma \left( {X,\mathcal{O}}\right) \) with \( f = {e}^{2\pi ih} \) if and and only if \( \partial \left( f\right) = 0 \) . | Look at the long exact cohomology sequence of the short exact sequence \( \mathrm{O} \rightarrow \mathbb{Z} \rightarrow \mathcal{O} \rightarrow {\mathcal{O}}^{ \star } \rightarrow \mathrm{O} \) . | No |
Theorem 4.7. Let \( {\left( {U}_{\imath },{f}_{\imath }\right) }_{\imath \in I} \) be a Cousin II distribution on \( X,\mathfrak{U} = {\left( {U}_{\imath }\right) }_{\imath \in I} \). Then:\n\n1. \( h\left( {{\iota }_{0},{\iota }_{1}}\right) \mathrel{\text{:=}} r{h}_{{\iota }_{0}{\iota }_{1}} \) defines an element \( h... | Proof.\n\n1a. Because\n\n\[ \n{f}_{{\imath }_{1}} = {h}_{{\imath }_{0}{\imath }_{1}}^{-1} \cdot {f}_{{\imath }_{0}} = {h}_{{\imath }_{1}{\imath }_{0}} \cdot {f}_{{\imath }_{0}} \n\]\n\nit follows that \( h\left( {{\iota }_{1},{\iota }_{0}}\right) = - h\left( {{\iota }_{0},{\iota }_{1}}\right) \).\n\nb. Because\n\n\[ \n... | Yes |
Theorem 4.8. Let \( X \) be an \( n \) -dimensional complex manifold with structure sheaf 0 . 1. If \( {H}^{1}\left( {X,\mathcal{O}}\right) = 0 \), then every Cousin I distribution on \( X \) is solvable. 2. If \( {H}^{1}\left( {X,{\mathcal{O}}^{ \star }}\right) = 0 \), then every Cousin II distribution on \( X \) is s... | Proof. The canonical homomorphisms \( {H}^{1}\left( {\mathfrak{U},\mathcal{O}}\right) \rightarrow {H}^{1}\left( {X,\mathcal{O}}\right) \) and \( {H}^{1}(\mathfrak{U} \) , \( \left. {\mathcal{O}}^{ \star }\right) \rightarrow {H}^{1}\left( {X,{\mathcal{O}}^{ \star }}\right) \) are injective for every covering \( \mathfra... | No |
Theorem 4.9. If \( {H}^{\ell }\left( {X,\mathcal{O}}\right) = 0 \) for \( \ell \geq 1 \), then the Cousin II distribution \( {\left( {U}_{\imath },{f}_{\imath }\right) }_{\imath \in I} \) (with the corresponding cocycle \( h \) ) is solvable if and only if \( c\left( h\right) = 0 \) (and that is a purely topological co... | Proof. By Theorem 4.6 \( {H}^{1}\left( {X,{\mathcal{O}}^{ \star }}\right) \simeq {H}^{2}\left( {X,\mathbb{Z}}\right) \), under \( \partial .h \) is thus solvable if and only if \( \underline{h} = 0 \), and that is the case if and only if \( c\left( h\right) = \partial \left( \underline{h}\right) = 0 \) . | No |
Theorem 5.3. Let \( X \) be a Stein manifold, and \( \mathrm{O} \rightarrow {\mathcal{S}}^{ \star } \rightarrow \mathcal{S} \rightarrow {\mathcal{S}}^{\star \star } \rightarrow \mathrm{O} \) an exact sequence of coherent analytic sheaves over \( X \) . Then\n\n\[ 0 \rightarrow \Gamma \left( {X,{\mathcal{S}}^{ \star }}\... | Proof. The cohomology sequence\n\n\[ 0 \rightarrow \Gamma \left( {X,{\mathcal{S}}^{ \star }}\right) \rightarrow \Gamma \left( {X,\mathcal{S}}\right) \rightarrow \Gamma \left( {X,{\mathcal{S}}^{\star \star }}\right) \rightarrow {H}^{1}\left( {X,{\mathcal{S}}^{ \star }}\right) \rightarrow \cdots \]\n\nis exact and by The... | Yes |
Theorem 5.4. Let \( X \) be a complex manifold and \( {U}_{1},{U}_{2} \subset X \) open Stein manifolds. Then \( U \mathrel{\text{:=}} {U}_{1} \cap {U}_{2} \) is also a Stein manifold. | ## Proof\n\n1. If \( {x}_{0} \in U \), then there are holomorphic functions \( {f}_{1},\ldots ,{f}_{\ell } \) on \( {U}_{1} \) such that \( {x}_{0} \) is an isolated point in\n\n\[ \left\{ {x \in {U}_{1} : {f}_{1}\left( x\right) = \cdots = {f}_{\ell }\left( x\right) = 0}\right\} .\n\]\n\nThen the functions \( {f}_{1}\l... | Yes |
Theorem 5.5 (Leray). Let \( X \) be a complex manifold, \( \mathcal{S} \) a coherent analytic sheaf on \( X,\mathfrak{U} \) a Stein covering of \( X \). Then \( \mathfrak{U} \) is a Leray covering of \( X \) and for all \( \ell ,{H}^{\ell }\left( {\mathfrak{U},\mathcal{S}}\right) \simeq {H}^{\ell }\left( {X,\mathcal{S}... | Proof. If \( \mathfrak{U} = {\left( {U}_{\imath }\right) }_{\imath \in I} \) is Stein, then by Theorem 5.4 all sets \( {U}_{{\imath }_{0}\cdots {\imath }_{i}} \) are Stein, and by Theorem \( \mathrm{B},{H}^{\ell }\left( {{U}_{{\iota }_{0}}\ldots {}_{{\iota }_{i}},\mathcal{S}}\right) = 0 \) for \( \ell \geq 1 \). Theref... | Yes |
Theorem 5.6. If \( X \) is a complex manifold, then there are arbitrarily fine Stein coverings of \( X \) . If \( \mathcal{S} \) is coherent analytic on \( X \), then for every open covering \( \mathfrak{U} \) of \( X \) there exists a refinement \( \mathfrak{B} \) such that \( {H}^{\ell }\left( {\mathfrak{V},\mathcal{... | Proof. Let \( \mathcal{H} \) be the structure sheaf of \( X \) . If \( {x}_{0} \in X \), then there is an open neighborhood \( U\left( {x}_{0}\right) \subset X \), a domain \( G \subset {\mathbb{C}}^{n} \) and an isomorphism \( \varphi : \left( {U,\mathcal{H}}\right) \rightarrow \left( {G,\mathcal{O}}\right) \) . If \(... | No |
Theorem 5.7. Let \( X \) be a Stein manifold, \( \mathcal{S} \) a coherent analytic sheaf over \( X \) , \( \mathfrak{U} \) an arbitrary open covering of \( X \) . Then \( {H}^{1}\left( {\mathfrak{U},\mathcal{S}}\right) = 0 \) . In particular, every Cousin I distribution over \( X \) is solvable. | Proof. \( {H}^{1}\left( {X,\mathcal{S}}\right) = 0 \) and \( {\varphi }_{1} : {H}^{1}\left( {\mathfrak{U},\mathcal{S}}\right) \rightarrow {H}^{1}\left( {X,\mathcal{S}}\right) \) is injective. | Yes |
Theorem 5.9. Let \( X \) be Stein, \( {\left( {U}_{\imath },{f}_{\imath }\right) }_{\imath \in I} \) a Cousin II distribution on \( X \) , \( h \in {Z}^{1}\left( {\mathfrak{U},{\mathcal{O}}^{ \star }}\right) \) the corresponding cocycle. Then \( {\left( {U}_{t},{f}_{t}\right) }_{t \in I} \) is solvable if and only if \... | Proof. Theorem B and Theorem 4.9. | No |
Theorem 5.10. If \( X \) is a Stein manifold and \( {H}^{2}\left( {X,\mathbb{Z}}\right) = 0 \), then every Cousin II problem on \( X \) is solvable. | Proof. Immediate corollary of Theorem 5.9. | No |
Theorem 5.11. Let \( \\left( {X,\\mathcal{O}}\\right) \) be a Stein manifold, \( A \\subset X \) an analytic subset and \( f \) a function holomorphic on \( A \) . Then there is a holomorphic function \( \\widehat{f} \) on \( X \) with \( \\widehat{f} \\mid A = f \) . (Global continuation!) | Proof. We assign to every point \( x \\in A \) a neighborhood \( {U}_{x} \\subset X \) and a holomorphic function \( {\\widetilde{f}}_{x} \) such that \( {\\widetilde{f}}_{x}\\left| {A \\cap {U}_{x} = f}\\right| A \\cap {U}_{x} \) . To every point \( x \\in X - A \) let there be assigned a neighborhood \( {U}_{x} \\sub... | Yes |
Theorem 5.12. Let \( \left( {X,\mathcal{O}}\right) \) be Stein, \( {X}^{\prime } \subset \subset X \) open, \( \mathcal{S} \) a coherent analytic sheaf over \( X \) . Then there are sections \( {s}_{1},\ldots ,{s}_{\ell } \in \Gamma \left( {X,\mathcal{S}}\right) \) which at each point \( x \in {X}^{\prime } \) generate... | Proof\n\n1. Let \( {x}_{0} \in {\bar{X}}^{\prime } \) . Then there exists an open neighborhood \( U\left( {x}_{0}\right) \subset X \) and sections \( {t}_{1},\ldots ,{t}_{q} \in \Gamma \left( {U,\mathcal{S}}\right) \) such that for every point \( x \in U \) the stalk \( {\mathcal{S}}_{x} \) over \( {\mathcal{O}}_{x} \)... | Yes |
Theorem 5.13. Let \( \left( {X,\mathcal{O}}\right) \) be Stein, \( {X}^{\prime } \subset \subset X \) open, \( A \subset X \) analytic. Then there are holomorphic functions \( {f}_{1},\ldots ,{f}_{\ell } \) on \( X \) such that \[ A \cap {X}^{\prime } = \left\{ {x \in {X}^{\prime } : {f}_{1}\left( x\right) = \cdots = {... | Proof. Since \( \mathcal{I}\left( A\right) \) is a coherent analytic sheaf on \( X \), by Theorem 5.12 there exist global sections \( {f}_{1},\ldots ,{f}_{\ell } \in \Gamma \left( {X,\mathcal{I}\left( A\right) }\right) \subset \Gamma \left( {X,\mathcal{O}}\right) \) which generate each stalk of \( \mathcal{I}\left( A\r... | Yes |
Theorem 1.1. If \( {c}_{1},\ldots ,{c}_{n} \) are arbitrary complex numbers then there exists exactly one tangent vector \( D \) with \( D\left( f\right) = \mathop{\sum }\limits_{{v = 1}}^{n}{c}_{v}\frac{\partial }{\partial {x}_{v}}\left( f\right) \) for each function \( f \) holomorphic at \( {x}_{0} \) . In particula... | Proof. If \( {c}_{v} = {a}_{v} + i{b}_{v} \) for \( v = 1,\ldots, n \), we set \[ D : = \mathop{\sum }\limits_{{v = 1}}^{n}{a}_{v}\frac{\partial }{\partial {x}_{v}} + \mathop{\sum }\limits_{{v = 1}}^{n}{b}_{v}\frac{\partial }{\partial {y}_{v}}. \] Then for each function \( f \) holomorphic at \( {x}_{0} \) (because \( ... | Yes |
Theorem 1.2. If \( c \in \mathbb{C} \) and \( D \in {T}_{{x}_{0}} \), then there exists exactly one tangent vector \( c \cdot D \in {T}_{{x}_{0}} \) such that \( \left( {c \cdot D}\right) \left( f\right) = c \cdot \left( {D\left( f\right) }\right) \) for every function \( f \) holomorphic at \( {x}_{0} \) . | Proof. There exist complex numbers \( {c}_{1},\ldots ,{c}_{n} \) such that\n\n\[ D\left( f\right) = \mathop{\sum }\limits_{{v = 1}}^{n}{c}_{v}\frac{\partial }{\partial {x}_{v}}\left( f\right) \]\n\nfor every function \( f \) holomorphic at \( {x}_{0} \) ; and by Theorem 1.1 there is exactly one tangent vector \( {D}^{ ... | Yes |
Theorem 1.3. Let\n\n\[ \ni \cdot \frac{\partial }{\partial {x}_{v}} = \frac{\partial }{\partial {y}_{v}}\;\text{ and }\;i \cdot \frac{\partial }{\partial {y}_{v}} = - \frac{\partial }{\partial {x}_{v}}\text{ for }v = 1,\ldots, n.\]\n\nThen \( {T}_{{x}_{0}} \) is an n-dimensional complex vector space with basis \( \left... | Proof. If \( f \) is holomorphic at \( {x}_{0} \), then\n\n\[ \left( {i \cdot \frac{\partial }{\partial {x}_{v}}}\right) \left( f\right) = i\left( {\frac{\partial }{\partial {x}_{v}}\left( f\right) }\right) = i \cdot {f}_{{z}_{v}} = \frac{\partial }{\partial {y}_{v}}\left( f\right) .\n\nThe axioms of a \( \mathbb{C} \)... | Yes |
Theorem 1.5. If \( \varphi \in {F}_{{x}_{0}}^{\left( r\right) },\varphi \neq 0 \) and \( \varphi \) is of type \( \left( {p, q}\right) \), then \( p \) and \( q \) are uniquely determined. | Proof. Suppose \( \varphi \) is of type \( \left( {p, q}\right) \) and of type \( \left( {{p}^{\prime },{q}^{\prime }}\right) \) . Since \( \varphi \neq 0 \) there exist tangent vectors \( {\xi }_{1},\ldots ,{\xi }_{r} \) such that \( \varphi \left( {{\xi }_{1},\ldots ,{\xi }_{r}}\right) \neq 0 \) . Then\n\n\[ \varphi ... | Yes |
If \( \varphi \) is of type \( \left( {p, q}\right) \), then \( \bar{\varphi } \) is of type \( \left( {q, p}\right) \). | (1) \( \begin{aligned} \overline{\varphi }\left( {c{\xi }_{1},\ldots, c{\xi }_{r}}\right) & = \overline{\varphi \left( {c{\xi }_{1},\ldots, c{\xi }_{r}}\right) } \\ & = \overline{{c}^{p}{\overline{c}}^{q}\varphi \left( {{\xi }_{1},\ldots ,{\xi }_{r}}\right) } \\ & = {\overline{c}}^{p}{c}^{q}\overline{\varphi }\left( {{... | Yes |
Theorem 1.7. If \( \varphi \in {F}_{{x}_{0}}^{\left( r\right) } \), then \( \varphi \) has a uniquely determined representation\n\n\[ \varphi = \mathop{\sum }\limits_{{p + q = r}}{\varphi }^{\left( p, q\right) } \]\n\nwhere \( {\varphi }^{\left( p, q\right) } \in {F}_{{x}_{0}}^{\left( r\right) } \) are forms of the typ... | Proof. Clearly \( d{z}_{v} \) is of type \( \left( {1,0}\right), d{\bar{z}}_{v} \) of type \( \left( {0,1}\right) \) . Hence it follows that monomials \( d{z}_{{i}_{1}} \land \cdots \land d{z}_{{i}_{p}} \land d{\bar{z}}_{{j}_{1}} \land \cdots \land d{\bar{z}}_{{j}_{q}} \) (with \( 1 \leq {i}_{1} < \cdots < {i}_{p} \leq... | Yes |
Theorem 2.1. If \( \varphi \in {A}^{\left( p, q\right) } \), then \( {d\varphi } = {d}^{\prime }\varphi + {d}^{\prime \prime }\varphi \) with \( {d}^{\prime }\varphi \in {A}^{\left( p + 1, q\right) } \) and \( {d}^{\prime \prime }\varphi \in {A}^{\left( p, q + 1\right) } \) . | Proof. One usually abbreviates the normal form of \( {\varphi }^{\left( p, q\right) } \) as\n\n\[ \n{\varphi }^{\left( p, q\right) } = \mathop{\sum }\limits_{{I, J}}{a}_{I, J}{d}_{3I} \land {d}_{3J}^{ - }\n\]\n\nThen\n\n\[ \nd{\varphi }^{\left( p, q\right) } = \mathop{\sum }\limits_{{I, J}}d{a}_{I, J} \land {d}_{3I} \l... | Yes |
Theorem 3.2. Let \( f \) be continuously differentiable on \( \mathbb{C},\operatorname{Supp}\left( f\right) \subset \subset \mathbb{C} \) , \( P \subset \mathbb{C} \) a circular disk with \( \operatorname{Supp}\left( f\right) \subset P \) . Then \( g \mathrel{\text{:=}} {\mathrm{{Ch}}}_{f}^{\left( P\right) } \) is cont... | Proof. Let \[ {P}_{c} \mathrel{\text{:=}} \{ z \in \mathbb{C} : z + c \in P\} , \] \[ \gamma \left( {w, c}\right) \mathrel{\text{:=}} {\operatorname{Ch}}_{f}^{\left( P\right) }\left( {w + c}\right) = \frac{1}{2\pi i}{\int }_{P}\frac{f\left( z\right) }{z - w - c}{dz} \land d\bar{z} \] \[ = \frac{1}{2\pi i}{\int }_{{P}_{... | Yes |
Theorem 3.3. Let \( B \subset \subset \mathbb{C} \) be a region, \( f \) continuously differentiable and bounded on B.\n\nThen \( g \mathrel{\text{:=}} {\mathrm{{Ch}}}_{f}^{\left( B\right) } \) is continuously differentiable on \( B \) and \( {g}_{\bar{z}} = f \) . | Proof. Let \( {w}_{0} \in B \) be given, \( H \) an open circular disk about \( {w}_{0} \) with \( H \subset \subset B \) . We can then find an arbitrarily often differentiable function \( \rho : \mathbb{C} \rightarrow \mathbb{R} \) for which\n\n1. \( 0 \leq \rho \leq 1 \) ,\n\n2. \( \rho \mid H = 1 \) ,\n\n3. \( \oper... | Yes |
Theorem 5.1. Let \( \mathcal{S},{\mathcal{S}}^{\prime } \) be fine sheaves over \( X,\varphi : \mathcal{S} \rightarrow {\mathcal{S}}^{\prime } \) an epimorphism of sheaves of \( T \) -modules. Then \( {\varphi }_{ \star } : \Gamma \left( {X,\mathcal{S}}\right) \rightarrow \Gamma \left( {X,{\mathcal{S}}^{\prime }}\right... | Proof\n\n1. Let \( {s}^{\prime } \in \Gamma \left( {X,{\mathcal{S}}^{\prime }}\right), x \in X \) . Then there exist a \( \sigma \in {\mathcal{S}}_{x} \) with \( \varphi \left( \sigma \right) = {s}^{\prime }\left( x\right) \) , a neighborhood \( W\left( x\right) \subset X \) and a section \( {s}^{ \star } \in \Gamma \l... | Yes |
Theorem 5.2. If \( \mathcal{S} \) is fine, then \( {H}^{\ell }\left( {X,\mathcal{S}}\right) = 0 \) for \( \ell \geq 1 \) . | Proof. Let \( \mathrm{O} \rightarrow \mathcal{S} \rightarrow {\mathcal{S}}_{0} \rightarrow {\mathcal{S}}_{1} \rightarrow {\mathcal{S}}_{2} \rightarrow \cdots \) be the canonical flabby resolution of \( \mathcal{S} \) . \( \mathcal{S} \) and all the \( {\mathcal{S}}_{v} \) are sheaves of \( T \) -modules. By induction o... | Yes |
Theorem 5.5. Let \( X \) be a Stein manifold, \( q \geq 1 \) . If \( \varphi \) is a form of the type \( \left( {p, q}\right) \) on \( X \) with \( {d}^{\prime \prime }\varphi = 0 \), then on \( X \) there exists a form \( \psi \) of the type \( \left( {p, q - 1}\right) \) , with \( {d}^{\prime \prime }\psi = \varphi \... | Proof. By Theorem B \( {H}^{q}\left( {X,{\Omega }^{p}}\right) = 0 \) for \( q \geq 1 \) ; therefore \( {H}^{p, q}\left( X\right) = 0 \) for \( q \geq 1 \) . | No |
Theorem 5.6. \( {H}^{r}\left( X\right) \simeq {H}^{r}\left( {X,\mathbb{C}}\right) \) for \( r \geq 0 \) . | Since\n\n\[ \n{\mathcal{A}}^{\prime } = {\bigoplus }_{p + q = \ell }{\mathcal{A}}^{p, q} \n\]\n\nwe would expect that a connection between the topological cohomology groups \( {H}^{r}\left( {X,\mathbb{C}}\right) \) and the analytically defined cohomology groups \( {H}^{q}\left( {X,{\Omega }^{p}}\right) \) exists. That ... | No |
Theorem 1.1.2. Let \( S \) and \( T \) be normal operators. Then \( S \) and \( T \) are algebraically equivalent if, and only if, they have the same spectrum. | Proof. Assume first that \( \operatorname{sp}\left( S\right) = \operatorname{sp}\left( T\right) \) . Then by the above we have \( \parallel f\left( S\right) \parallel = \sup \{ \left| {f\left( z\right) }\right| : z \in \operatorname{sp}\left( S\right) \} = \parallel f\left( T\right) \parallel \), for every continuous f... | Yes |
Proposition 1.3.1. Let \( A \) be a \( {C}^{ \star } \) -algebra and let \( J \) be an ideal in \( A \) . Then for every \( x \) in \( J \) there is a sequence \( {e}_{1},{e}_{2},\ldots \) of self-adjoint elements of \( J \) satisfying\n\n(i) \( \operatorname{sp}\left( {e}_{n}\right) \subseteq \left\lbrack {0,1}\right\... | Proof. Consider first the case where \( A \) has an identity \( e \) and \( x = {x}^{ * } \) . Define \( {e}_{n} \in A \) (via the functional calculus) by\n\n\[ \n{e}_{n} = n{x}^{2}{\left( e + n{x}^{2}\right) }^{-1}.\n\]\n\nThe function \( {f}_{n} : \mathbb{R} \rightarrow \mathbb{R} \) defined by \( {f}_{n}\left( t\rig... | Yes |
Corollary 1. Every ideal in a \( {C}^{ \star } \) -algebra is self-adjoint (i.e., is closed under the *-operation). | Proof. Let \( J \) be an ideal in a \( {C}^{ * } \) -algebra \( A \), and let \( x \) be an element of \( J \) . By 1.3.1 we can find a sequence \( {e}_{n} = {e}_{n}^{ \star } \) in \( J \) so that \( x = \mathop{\lim }\limits_{n}x{e}_{n} \) . By taking adjoints we have \( {x}^{ \star } = \lim {e}_{n}{x}^{ \star } \), ... | Yes |
Corollary 2. \( A/J \) is a \( {C}^{ \star } \) -algebra. | Proof. By Exercise 1.1.B, it suffices to show that \( \parallel \dot{x}{\parallel }^{2} \leq \begin{Vmatrix}{{\dot{x}}^{ \star }\dot{x}}\end{Vmatrix} \), for every \( x \) in \( A \) .\n\nFor that, fix \( x \), and let \( E \) denote the set of all self-adjoint elements \( u \) of \( J \) satisfying \( \operatorname{sp... | Yes |
Theorem 1.3.2. Let \( A \) and \( B \) be \( {C}^{ * } \) -algebras and let \( \pi \) be a \( * \) -homomorphism of \( A \) into \( B \) . Then \( \pi \) is continuous and \( \pi \left( A\right) \) is a \( {C}^{ \star } \) -subalgebra of \( B.\pi \) induces an isometric *-isomorphism of the quotient \( A/\ker \pi \) on... | Proof. We shall merely indicate how the general case can be reduced to the above situation where both \( A \) and \( B \) have units and \( \pi \left( {e}_{A}\right) = {e}_{B} \) . Assume first that \( A \) has a unit \( {e}_{A} \) . By passing from \( B \) to the closure of \( \pi \left( A\right) \) if necessary, we m... | Yes |
Lemma 1.4.1. Let \( E \) be a nonzero projection in \( \mathcal{A} \). Then \( E \) is minimal iff \( E\mathcal{A}E = \{ {\lambda E} : \lambda \in \mathbb{C}\} \). | Proof. If \( E\mathcal{A}E \) consists of scalar multiples of \( E \), then \( E \) is minimal. Conversely, assume \( E \) is minimal. It suffices to show that \( {ETE} \) is a scalar multiple of \( E \), for every self-adjoint \( T \in \mathcal{A} \). Considering the spectral formula for \( {ETE} \) we have \( {ETE} =... | Yes |
Theorem 1.4.2. If \( \mathcal{A} \) is irreducible then \( \mathcal{A} = \mathcal{C}\left( \mathcal{H}\right) \) . | Proof. Note first that \( \mathcal{A} \) contains a projection of rank 1 . For if \( F \) is a nonzero spectral projection of any self-adjoint operator in \( \mathcal{A} \) then by the lemma \( F \) contains a minimal projection \( E \), and it suffices to show that \( E \) has rank 1 . Choose \( \xi ,\eta \in E\mathca... | Yes |
Corollary 1. \( \mathcal{C}\left( \mathcal{H}\right) \) contains no ideals other than 0 and \( \mathcal{C}\left( \mathcal{H}\right) \) . | Proof. Let \( \mathcal{A} \neq 0 \) be an ideal in \( \mathcal{C}\left( \mathcal{H}\right) \) . Applying 1.3.4 to the identity representation of \( \mathcal{C}\left( \mathcal{H}\right) \) we see that \( \mathcal{A} \) is irreducible, and 1.4.2 shows that \( \mathcal{A} = \mathcal{C}\left( \mathcal{H}\right) \) | Yes |
Corollary 2. Let \( \mathcal{B} \) be an irreducible \( {C}^{ \star } \) -algebra of operators on \( \mathcal{H} \) which contains a nonzero compact operator. The \( \mathcal{B} \) contains \( \mathcal{C}\left( \mathcal{H}\right) \) . | Proof. \( \mathcal{B} \cap \mathcal{C}\left( \mathcal{H}\right) \) is a nonzero ideal in \( \mathcal{B} \) . Arguing as in Corollary 1 we see that \( \mathcal{B} \cap \mathcal{C}\left( \mathcal{H}\right) \) is irreducible, and by \( {1.4.2}\mathcal{B} \cap \mathcal{C}\left( \mathcal{H}\right) = \mathcal{C}\left( \mathc... | Yes |
Proposition 1.4.3. Let \( E \) be a minimal projection in \( \mathcal{A} \), let \( \xi \) be a unit vector in \( E\mathcal{H} \), and let \( {\mathcal{H}}_{0} = \left\lbrack {\mathcal{A}\xi }\right\rbrack \) . Then \( {\left. \mathcal{A}\right| }_{{\mathcal{H}}_{0}} \) is irreducible, and in fact \( {\left. \mathcal{A... | Proof. The map \( T \rightarrow {\left. T\right| }_{{\mathcal{H}}_{0}} \) is a \( \star \) -homomorphism of \( \mathcal{A} \) into \( \mathcal{C}\left( {\mathcal{H}}_{0}\right) \) whose range is \( {\left. \mathcal{A}\right| }_{{\mathcal{H}}_{0}} \) . By \( {1.3.2}{\left. \mathcal{A}\right| }_{{\mathcal{H}}_{0}} \) is ... | Yes |
Theorem 1.4.4. Let \( \pi \) be any nondegenerate representation of \( \mathcal{A} \). Then there is an orthogonal family \( \left\{ {\pi }_{i}\right\} \) of irreducible subrepresentations of \( \pi \) such that \( \pi = \mathop{\sum }\limits_{i}{\pi }_{i} \), and each \( {\pi }_{i} \) is equivalent to a subrepresentat... | Proof. First note that there is a minimal projection \( E \in \mathcal{A} \) such that \( \pi \left( E\right) \neq \) 0 . To see that, choose \( T = {T}^{ \star } \in \mathcal{A} \) such that \( \pi \left( T\right) \neq 0 \) . By our initial remarks about the spectral theorem there is a spectral projection \( F \) of \... | Yes |
Corollary 1. Every representation of \( \mathcal{C}\left( \mathcal{H}\right) \) is equivalent to a multiple of the identity representation. | Proof. Let \( \pi \) be a representation of \( \mathcal{A} \) on \( \mathcal{K} \) . Since the identity representation \( {id} \) of \( \mathcal{C}\left( \mathcal{H}\right) \) is itself irreducible, it follows that \( \pi \) has a decomposition \( \mathop{\sum }\limits_{i}{\pi }_{i} \) into orthogonal subrepresentation... | Yes |
Corollary 3. Let \( \mathcal{H} \) and \( \mathcal{K} \) be Hilbert spaces. Then every \( \star \) -isomorphism \( \alpha \) of \( \mathcal{C}\left( \mathcal{H}\right) \) (resp. \( \mathcal{L}\left( \mathcal{H}\right) \) ) onto \( \mathcal{C}\left( \mathcal{K}\right) \) (resp. \( \mathcal{L}\left( \mathcal{K}\right) \)... | Proof. Consider first the case \( \alpha : \mathcal{C}\left( \mathcal{H}\right) \rightarrow \mathcal{C}\left( \mathcal{K}\right) \) . Then \( \alpha \) is an irreducible representation of \( \mathcal{C}\left( \mathcal{H}\right) \), and the conclusion follows from Corollary 2.\n\nNow suppose \( \alpha \) is a \( \star \... | Yes |
Proposition 1.5.2. Let \( A \) be a CCR algebra and let \( \pi \) and \( \sigma \) be irreducible representations of \( A \) such that \( \ker \pi \subseteq \ker \sigma \) . Then \( \pi \) and \( \sigma \) are equivalent. | Proof. Suppose \( \pi \) and \( \sigma \) act on Hilbert spaces \( \mathcal{H} \) and \( \mathcal{K} \) . By the preceding remarks \( \pi \left( A\right) = \mathcal{C}\left( \mathcal{H}\right) \), and by the hypothesis ker \( \pi \subseteq \ker \sigma \), the mapping \( \lambda : \pi \left( x\right) \rightarrow \sigma ... | Yes |
Proposition 1.5.4. Let \( A \) be a GCR algebra. Then for every irreducible representation \( \pi \) of \( A \) on \( \mathcal{H},\pi \left( A\right) \) contains \( \mathcal{C}\left( \mathcal{H}\right) \) . Two irreducible representations of \( A \) which have the same kernel are equivalent. | Proof. Now \( \pi \left( A\right) \) is \( \star \) -isomorphic with \( A/\ker \pi \) and therefore it contains a nonzero CCR ideal. Since the identity representation of the ideal is irreducible (1.3.4) this implies in particular that \( \pi \left( A\right) \) contains a nonzero compact operator. Therefore the first co... | Yes |
Theorem 1.5.5. Every GCR algebra \( A \) has exactly one composition series \( \left\{ {{J}_{\alpha } : 0 \leq \alpha \leq {\alpha }_{0}}\right\} \) with the property that \( {J}_{\alpha + 1}/{J}_{\alpha } \) is the largest CCR ideal in \( A/{J}_{\alpha } \) for every \( \alpha ,0 \leq \alpha < {\alpha }_{0} \) . Conve... | Proof. Let \( A \) be a GCR algebra. We define the series \( \left\{ {J}_{\alpha }\right\} \) by transfinite induction. Put \( {J}_{0} = 0 \) . Note that \( A = A/{J}_{0} \) contains a nonzero CCR ideal. Inductively, let \( \beta \) be an ordinal such that \( {J}_{\alpha } \) has been defined for all \( \alpha < \beta ... | Yes |
Lemma 1. If \( x \) and \( y \) are self-adjoint elements of \( A \) with \( \operatorname{sp}\left( x\right) \geq 0 \) and \( \operatorname{sp}\left( y\right) \geq 0 \), then \( \operatorname{sp}\left( {x + y}\right) \geq 0 \) . | Proof. By multiplying the sum \( x + y \) by a sufficiently small positive scalar, if necessary, we may assume \( \parallel x\parallel \leq 1 \) and \( \parallel y\parallel \leq 1 \) . Now since the spectral radius does not exceed the norm, the spectrum of \( x \) must be contained in the real interval \( \left\lbrack ... | Yes |
Lemma 2. Let \( z \in A \) be such that \( \operatorname{sp}\left( {{z}^{ * }z}\right) \leq 0 \) . Then \( z = 0 \) . | Proof. Assume that \( \operatorname{sp}\left( {{z}^{ * }z}\right) \leq 0 \) . Then by the preceding remark we also have \( \operatorname{sp}\left( {z{z}^{ * }}\right) \leq 0 \) . Lemma 1, applied to the elements \( - {z}^{ * }z \) and \( - z{z}^{ * } \), implies that \( \operatorname{sp}\left( {{z}^{ \star }z + z{z}^{ ... | Yes |
Theorem 1.7.1. For every \( z \) in \( A \), one has \( \operatorname{sp}\left( {{z}^{ \star }z}\right) \geq 0 \) . | Proof. We claim first that there exist self-adjoint elements \( u, v \) in \( A \) for which \( {uv} = {vu} = 0 \) and \( {z}^{ \star }z = {u}^{2} - {v}^{2} \) . Indeed, if we consider the real-valued continuous functions \( f, g \) defined on \( \mathbb{R} \) by\n\n\[ f\left( t\right) = \left\{ \begin{array}{ll} \sqrt... | Yes |
Theorem 1.7.2. Let \( x \) be a self-adjoint element of \( A \) . Then there is a pure state \( f \) such that \( \left| {f\left( x\right) }\right| = \parallel x\parallel \) . | Proof. We claim first that there is a state \( f \) with the property \( \left| {f\left( x\right) }\right| = \parallel x\parallel \) . Consider the \( {C}^{ \star } \) -subalgebra \( B \) generated by \( x \) and \( e \) . Since \( B \) is commutative, it follows from 1.1.1 that there is a complex homomorphism \( \omeg... | Yes |
Theorem 1.7.3. Gelfand-Naimark theorem. Every abstract \( {C}^{ \star } \) -algebra with identity is isometrically \( \star \) -isomorphic to a \( {C}^{ \star } \) -algebra of operators. | Proof. We have to exhibit an isometric representation \( \pi \) of \( A \) . By the preceding corollary we may choose, for each \( x \neq 0 \) in \( A \), a representation \( {\pi }_{x} \) such that \( \begin{Vmatrix}{{\pi }_{x}\left( x\right) }\end{Vmatrix} = \parallel x\parallel \) . Now simply let \( \pi \) be the d... | Yes |
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