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Proposition 1.8.1. Let \( x \) be a self-adjoint element of \( A \) . Then the following are equivalent:\n\n(i) \( x \geq 0 \) .\n\n(ii) There is an element \( y = {y}^{ \star } \) in \( A \) such that \( x = {y}^{2} \) .\n\n(iii) There is an element \( z \) in \( A \) such that \( x = {z}^{ \star }z \) . | Proof. That (ii) implies (iii) is a triviality, (iii) implies (i) is 1.7.1, and (i) implies (ii) is a simple exercise with the functional calculus which we leave for the reader. | No |
Proposition 2.1.3. Let \( \left\{ {{\mu }_{i} : i \in I}\right\} \) and \( \left\{ {{v}_{j} : j \in J}\right\} \) be two countable families of subrepresentations of \( \pi \), each satisfying the three conditions of 2.1.2, such that \( \pi = \sum {\mu }_{i} = \sum {v}_{j} \) . Then card \( I = \operatorname{card}J \) . | Proof. Let \( m = \operatorname{card}I, n = \operatorname{card}J \), and suppose, to the contrary, that \( 1 \leq m < n \leq {\aleph }_{0} \) . Choose \( {i}_{0} \in I \) and \( {j}_{0} \in J \), and let \( \mathcal{M} \) and \( \mathcal{N} \) be the respective commutants of \( m \cdot {\mu }_{{i}_{0}}\left( A\right) \... | Yes |
Proposition 2.1.4. Let \( {\pi }_{1} \) and \( {\pi }_{2} \) be two representations of \( A \) . Then the following are equivalent:\n\n(i) \( {\pi }_{1} \circ {\pi }_{2} \) .\n\n(ii) The only bounded operator \( T \) between the respective Hilbert spaces which satisfies \( T{\pi }_{1}\left( x\right) = {\pi }_{2}\left( ... | Proof. Let \( {\mathcal{H}}_{i} \) be the space on which \( {\pi }_{i} \) acts, \( i = 1,2 \) .\n\n(i) \( \Rightarrow \) (ii). Assume \( {\pi }_{1} \circ {\pi }_{2} \), and let \( T : {\mathcal{H}}_{1} \rightarrow {\mathcal{H}}_{2} \) satisfy the condition \( T{\pi }_{1}\left( x\right) = {\pi }_{2}\left( x\right) T, x ... | No |
Proposition 2.1.5. Let \( {\sigma }_{1} \) and \( {\sigma }_{2} \) be subrepresentations of a representation \( \pi \) .\n\n(i) If \( {\sigma }_{1} \) is equivalent to a subrepresentation of \( {\sigma }_{2} \), then \( {\bar{\sigma }}_{1} \leq {\bar{\sigma }}_{2} \) . | (i) Let \( {\mathfrak{M}}_{i} \) be the range space of \( {\sigma }_{i}, i = 1,2 \) . Then by hypothesis, there is a partial isometry \( U \in \pi {\left( A\right) }^{\prime } \) such that \( {U}^{ * }U \) is the projection on \( {\mathfrak{M}}_{1} \) and \( U{\mathfrak{M}}_{1} \subseteq {\mathfrak{M}}_{2} \) . Thus, \... | Yes |
Corollary 2.1.9. \( S \leftrightarrow {\pi }_{S} \) defines a bijective correspondence between equivalence classes of sequences \( S \) satisfying (i) and (ii) and the equivalence classes of type I representations of \( A \) on separable Hilbert spaces. | Thus the classification problem for type I separable representations is reduced to the problem of classifying multiplicity-free representations. We remark that with only minor variations the entire analysis of this section may be carried out for representations on Hilbert spaces which are inseparable. | No |
Theorem 2.2.1. \( \mathcal{L},{\mathcal{L}}^{\prime } \), and the strong closure of \( {\pi }_{\mu }\left( {C\left( X\right) }\right) \), are identical. | Proof. It is clear that \( {\pi }_{\mu }\left( {C\left( X\right) }\right) \subseteq \mathcal{Z} \subseteq {\mathcal{Z}}^{\prime } \) ; we will prove that \( {\mathcal{Z}}^{\prime } \subseteq \mathcal{Z} \) , and \( \mathcal{Z} \) is contained in the strong closure of \( {\pi }_{\mu }\left( {C\left( X\right) }\right) \)... | Yes |
Theorem 2.2.2. Let \( \mu \) and \( v \) be two finite Borel measures on \( X \) . Then \( {\pi }_{\mu } \) is equivalent to \( {\pi }_{v} \) iff \( \mu \) and \( v \) are equivalent measures. \( {\pi }_{\mu } \) is disjoint from \( {\pi }_{v} \) iff \( \mu \bot v \) . | Proof. Assume first that \( \mu \) and \( v \) are equivalent. We will find a unitary operator \( U : {L}^{2}\left( {X,\mu }\right) \rightarrow {L}^{2}\left( {X, v}\right) \) such that \( U{\pi }_{\mu } = {\pi }_{v}U \) . By the Radon-Nikodym theorem and the fact that \( \mu \ll v \), there is a nonnegative function \(... | Yes |
Lemma 2.2.3. Let \( A \) be a \( {C}^{ \star } \) -algebra and let \( \pi \) be a nondegenerate multiplicity-free representation of \( A \) on a separable Hilbert space \( \mathcal{H} \) . Then there is a vector \( \xi \in \mathcal{H} \) such that \( \left\lbrack {\pi \left( A\right) \xi }\right\rbrack = \mathcal{H} \)... | Note that every nonzero \( \pi \left( A\right) \) -invariant subspace \( \mathcal{M} \) contains a nonzero vector \( \xi \) such that \( \left\lbrack {\pi \left( A\right) \xi }\right\rbrack \subseteq \mathcal{M} \) (any \( \xi \neq 0 \) in \( \mathcal{M} \) will do). By Zorn’s lemma, then, we may find a set \( \left\{ ... | Yes |
Theorem 2.2.4. Let \( A \) be a commutative separable \( {C}^{ \star } \) -algebra and let \( \pi \) be a nondegenerate multiplicity-free representation of \( A \) on a separable Hilbert space. Then there is a finite Borel measure \( \mu \) on \( X \), the spectrum of \( A \), such that \( \pi \) is equivalent to \( {\... | Proof. We may identify \( A \) with \( C\left( X\right) \) via the Gelfand map. Let \( \mathcal{H} \) be the representation space of \( \pi \) . Then by 2.2.3 there is a nonzero vector \( \xi \in \mathcal{H} \) such that \( \mathcal{H} = \left\lbrack {\pi \left( A\right) \xi }\right\rbrack \) . Define a bounded linear ... | Yes |
Theorem 2.3.2. Let \( \mathcal{R} \) be a von Neumann algebra on a separable Hilbert space.\n\nThen the following are equivalent:\n\n(i) The identity representation of \( \mathcal{R} \) is type \( I \) .\n\n(ii) \( \mathcal{R} \) is a type I von Neumann algebra.\n\n(iii) The identity representation of \( {\mathcal{R}}^... | Proof. By symmetry and the double commutant theorem it suffices to prove (i) \( \Rightarrow \) (ii) \( \Rightarrow \) (iii). And since (ii) \( \Rightarrow \) (iii) follows from the preceding remarks, we are left with (i) \( \Rightarrow \) (ii).\n\nAssume (i), and let \( C \) be a nonzero central projection of \( \mathc... | Yes |
Lemma 3.1.1. An open subspace of a Polish space is Polish. | Proof. Let \( G \) be an open set in \( P \), and let \( d \) be a metric on \( P \) . Define a function \( f : G \rightarrow \mathbb{R} \) by \( f\left( x\right) = 1/d\left( {x, P \smallsetminus G}\right) \lbrack d\left( {x, S}\right) \) of course denotes the distance from \( x \) to the set \( S\rbrack \) . Since \( ... | Yes |
Theorem 3.1.2. Let \( P \) be a Polish space and let \( S \subseteq P \) be a \( {G}_{\delta } \). Then \( S \) is a Polish space in its relative topology. | SKETCH OF PROOF. Let \( {G}_{i} \subseteq P \) be a sequence of open sets such that \( S = \) \( \bigcap {G}_{i} \). By the lemma, we can find Polish spaces \( {P}_{i} \) and homeomorphisms \( {f}_{i} : {P}_{i} \rightarrow {G}_{i} \). Define \( Q \) to be the set of all sequences \( \left( {{p}_{1},{p}_{2},\ldots }\rig... | No |
Proposition 3.1.3. Let \( P \) be a Polish space and let \( d \) be a metric defining the topology of \( P \) . Then \( P \) admits an open sieve \( \left\{ {{A}_{{n}_{1}}\ldots {}_{{n}_{k}}}\right\} \) with the following properties:\n\n(iii) \( \operatorname{diam}\left( {A}_{{n}_{1}\cdots {n}_{k}}\right) \leq 1/k \), ... | Proof. Since \( P \) is separable we can find a sequence \( {A}_{1},{A}_{2},\ldots \) of nonempty open balls of diameter \( \leq 1 \) whose union is \( P \) (for example, let \( \left\{ {{x}_{1},{x}_{2},\ldots }\right\} \) be a countable dense set and let \( {A}_{i} \) be the open ball of radius \( \frac{1}{2} \) cente... | Yes |
Theorem 3.1.4. For every Polish space \( P \), there is a continuous open mapping of \( {N}^{\infty } \) onto \( P \) . | Proof. Let \( \left\{ {A}_{{n}_{1}\cdots {n}_{k}}\right\} \) be an open sieve for \( P \) having the properties of 3.1.3. We will define a function \( f : {N}^{\infty } \rightarrow P \) as follows. Choose \( v = \left( {{n}_{1},{n}_{2},\ldots }\right) \in {N}^{\infty } \) . By the remarks following 3.1.3, we may define... | Yes |
Theorem 3.2.1. Let \( P \) be a Polish space. Then for every Borel set \( E \subseteq P \) , there is a Polish space \( Q \) and a 1-1 continuous function \( f : Q \rightarrow P \) such that \( E = f\left( Q\right) \) . | Proof. Let \( \mathcal{F} \) be the class of all subsets of \( P \) having the stated property. We will show that family of all subsets \( E \), such that both \( E \) and \( P \smallsetminus E \) belong to \( \mathcal{F} \), contains the Borel sets. For that, according to the preceding lemma, it suffices to show that ... | Yes |
Corollary 2. Let \( P \) and \( Q \) be Polish spaces, and let \( f \) be a continuous \( 1 - 1 \) function of \( P \) onto \( Q \) . Then \( f \) maps closed sets to Borel sets (and therefore Borel sets to Borel sets). | Proof. Let \( F \subseteq P \) be closed. Because \( f \) is \( 1 - 1 \) and onto, \( f\left( F\right) \) and \( f\left( {P \smallsetminus F}\right) \) are complementary sets in \( Q \) . Since \( F \) and \( P \smallsetminus F \) are Polish subspaces (cf. 3.1.1) and since \( f \) is continuous, \( f\left( F\right) \) ... | Yes |
Theorem 3.2.3. Let \( P, Q \) be Polish spaces and let \( f : P \rightarrow Q \) be a 1-1 continuous function. Then \( f\left( P\right) \) is a Borel set in \( Q \) . | Proof. Choose a sequence \( {\mathcal{P}}_{1} \geq {\mathcal{P}}_{2} \geq \cdots \) of countable partitions of \( P \) as in the preceding remark, say \( {\mathcal{P}}_{n} = \left\{ {{A}_{n1},{A}_{n2},\ldots }\right\}, n \geq 1 \) . Now since each set \( {A}_{nk} \) is a \( {G}_{\delta } \) it is a Polish subspace of \... | Yes |
Proposition 3.3.1. Let \( X, Y \) be Borel spaces, with \( Y \) countably separated, and let \( f : X \rightarrow Y \) be a Borel function. Then the graph of \( f \) is a Borel subset of \( X \times Y \) . | Proof. Let \( {E}_{1},{E}_{2},\ldots \) be a separating family of Borel sets in \( Y \) . Let \( \Delta = \) \( \{ \left( {y, y}\right) : y \in Y\} \) be the diagonal in \( Y \times Y \) . Then \( \left( {Y \times Y}\right) \smallsetminus \Delta \) is the union of the two sets \( \mathop{\bigcup }\limits_{n}{E}_{n} \ti... | Yes |
Theorem 3.3.2. Let \( X \) be a standard Borel space, let \( Q \) be a Polish space, and let \( f \) be a 1-1 Borel map of \( X \) into \( Q \) . Then \( f\left( X\right) \) is a Borel set in \( Q \) and \( f \) is an isomorphism of \( X \) onto \( f\left( X\right) \) . | Proof. We will show that \( f\left( X\right) \) is a Borel set; the rest follows from this and the obvious fact that a Borel subset of the standard Borel space \( X \) defines a standard subspace.\n\nBy definition, we may assume that \( X \) is a Borel subset of a Polish space \( {P}_{0} \) ; and by 3.2.1. there is a 1... | Yes |
Theorem 3.3.4. Let \( X \) be an analytic Borel space, let \( Q \) be a Polish space, and let \( f \) be a Borel map of \( X \) into \( Q \) . Then \( f\left( X\right) \) is an analytic set in \( Q \) . | Proof. The argument is similar to that of 3.3.2. We can assume \( X \) is an analytic subspace of a Polish space \( {P}_{0} \) . Then by definition there is a continuous map of a Polish space \( P \) into \( {P}_{0} \) having range \( X \) . Thus we may assume (by considering the composition of the two functions) that ... | Yes |
Corollary 1. Let \( X \) be a subspace of a Polish space \( P \) . Then \( X \) is an analytic Borel space in its relative structure if, and only if, \( X \) is an analytic set in \( P \) . | Proof. The if part is obvious. Conversely, suppose there exists an analytic set \( Y \) in a Polish space \( Q \) and a Borel isomorphism \( f \) of \( X \) on \( Y \) . Now apply 3.3.4 to \( {f}^{-1} \) to conclude that \( X \) is an analytic set. | No |
Corollary 2. Let \( X \) and \( Y \) be analytic Borel spaces and let \( f \) be a \( 1 - 1 \) Borel map of \( X \) onto \( Y \) . Then \( f \) is a Borel isomorphism. | Proof. Again we may assume \( X \) and \( Y \) are analytic subspaces of Polish spaces \( P \) and \( Q \) . Let \( E \) be a Borel set in \( X \) ; we have to show that \( f\left( E\right) \) is a Borel set in \( Y.E \) has the form \( X \cap B \) for some Borel set \( B \subseteq P \) . Thus \( E \) is analytic (here... | Yes |
Theorem 3.3.5. Unique structure theorem. Let \( \left( {X,\mathcal{B}}\right) \) be an analytic Borel space and let \( {\mathcal{B}}_{0} \) be a countably generated sub- \( \sigma \) -field of \( \mathcal{B} \) which separates points in \( X \) . Then \( {\mathcal{B}}_{0} = \mathcal{B} \) . | Proof. By the lemma, there is a Borel isomorphism \( f \) of \( \left( {X,{\mathcal{B}}_{0}}\right) \) onto a subspace \( Y \) of a Polish space \( P \) . Note that the Borel structure on \( Y \) is \( f\left( {\mathcal{B}}_{0}\right) \) . Now \( f \) is also a 1-1 Borel map of \( \left( {X,\mathcal{B}}\right) \) into ... | Yes |
Corollary 1. Let \( X \) be an analytic Borel space, let \( Y \) be a countably separated Borel space, and let \( f \) be a Borel map of \( X \) onto \( Y \) . Then \( Y \) is an analytic Borel space. | Proof. Let \( \left\{ {{E}_{1},{E}_{2},\ldots }\right\} \) be a separating family of Borel sets in \( Y \) . We claim: \( \left\{ {{E}_{1},{E}_{2},\ldots }\right\} \) generates the Borel structure in \( Y \) . For this choose an arbitrary Borel set \( F \) in \( Y \), and let \( {\mathcal{B}}_{0} \) be the \( \sigma \)... | Yes |
Corollary 2. Let \( X \) be an analytic Borel space and let \( \sim \) be an equivalence relation in \( X \). Assume there is a sequence \( {f}_{1},{f}_{2},\ldots \) of real valued Borel functions on \( X \) such that for any pair of points \( x, y \) in \( X \) one has \( x \sim y \) iff \( {f}_{n}\left( x\right) = {f... | Proof. Let \( q : X \rightarrow X/ \sim \) be the natural projection. Define \( {g}_{n} : X/ \sim \rightarrow \mathbb{R} \) by \( {g}_{n}\left( {q\left( x\right) }\right) = {f}_{n}\left( x\right) \). Then each \( {g}_{n} \) is a Borel function and the \( g \)’s separate points in \( X/ \sim \). So if we let \( {r}_{1},... | Yes |
Theorem 3.4.1. Let \( P \) be a Polish space, let \( Y \) be a Borel space, and let \( f \) be a function from \( P \) onto \( Y \) satisfying\n\n(i) \( f \) maps open sets to Borel sets;\n\n(ii) the inverse image of each point of \( Y \) is a closed subset of \( P \) .\n\nThen \( f \) has a Borel cross section. | Proof. Assume first that the theorem has been proved for the special case \( P = {N}^{\infty } \) . Then for a general \( P \) we can find a continuous open map \( g \) of \( {N}^{\infty } \) onto \( P \) (3.1.4). Now the function \( f \circ g : {N}^{\infty } \rightarrow Y \) satisfies (i) because \( g \) is an open ma... | Yes |
Proposition 3.4.2. Let \( X \) and \( Y \) be Borel spaces, with \( X \) countably separated, and let \( f \) be a Borel map of \( X \) onto \( Y \) . Assume that \( f \) has a Borel cross section \( g : Y \rightarrow X \) . Then \( g\left( Y\right) \) is a Borel set in \( X \) and \( g \) is a Borel isomorphism of \( ... | Proof. Let \( F \subseteq X \) be the set of all fixed points of the Borel function \( g \circ f : X \rightarrow X \) . One sees easily that \( F = g\left( Y\right) \), and we claim that \( F \) is a Borel set in \( X \) . Indeed, because \( X \) is countably separated, the diagonal \( \Delta = \) \( \{ \left( {x, x}\r... | Yes |
Theorem 3.4.3. Let \( X \) be an analytic Borel space, let \( Y \) be a countably separated Borel space, and let \( f \) be a Borel map of \( X \) onto \( Y \) . Then \( f \) has an absolutely measurable cross section. | Proof. By Corollary 1 of 3.3.5 \( Y \) is an analytic Borel space, so there is no loss if we assume that \( X \) and \( Y \) are analytic subspaces of Polish spaces \( P \) and \( Q \) . The map \( x \mapsto \left( {x, f\left( x\right) }\right) \in P \times Q \) is a Borel map of \( X \) onto the graph of \( f \), so b... | Yes |
Theorem 4.1.1. Let \( A \) be a \( {C}^{ \star } \) -algebra, let \( \mathcal{H} \) be a separable Hilbert space, and let \( T \) be a bounded self-adjoint operator on \( \mathcal{H} \) . Then the set \( {\mathcal{S}}_{T} \) of all representations \( \pi \in \operatorname{rep}\left( {A,\mathcal{H}}\right) \) for which ... | Proof. Choose any real number \( r > \parallel T\parallel \) and choose \( \pi \in \operatorname{rep}\left( {A,\mathcal{H}}\right) \) . We claim: \( \pi \in {\mathcal{S}}_{T} \) iff \( T \) belongs to the weak closure of \( \pi \left( {A}_{r}\right) ,{A}_{r} \) denoting the ball of radius \( r \) in \( A \) . Indeed th... | No |
Corollary 1. The set of nondegenerate elements of \( \operatorname{rep}\left( {A,\mathcal{H}}\right) \) is a \( {G}_{\delta } \) . | Proof. This is \( {\mathcal{S}}_{I} \) where \( I \) is the identity operator on \( \mathcal{H} \) . | No |
Corollary 2. \( \operatorname{irr}\left( {A,\mathcal{H}}\right) \) is a \( {G}_{\delta } \) in \( \operatorname{rep}\left( {A,\mathcal{H}}\right) \) . | Proof. Let \( S \) and \( T \) be the real and imaginary parts of any particular irreducible operator on \( \mathcal{H} \) . Then a \( {C}^{ \star } \) -algebra on \( \mathcal{H} \) is irreducible iff its weak closure contains both \( S \) and \( T \) . Thus \( \operatorname{irr}\left( {A,\mathcal{H}}\right) = {\mathca... | Yes |
For every \( {\pi }_{0} \in \operatorname{rep}\left( {A,\mathcal{H}}\right) ,\left\{ {\sigma \in \operatorname{rep}\left( {A,\mathcal{H}}\right) : \sigma \propto {\pi }_{0}}\right\} \) is a \( {G}_{\delta } \) in \( \operatorname{rep}\left( {A,\mathcal{H}}\right) \) . | Proof. Let \( I \oplus 0 \) be the projection of \( \mathcal{H} \oplus \mathcal{H} \) onto its first coordinate space. It follows from 2.1.4 that \( \sigma \propto {\pi }_{0} \) iff \( I \oplus 0 \) belongs to the weak closure of \( \sigma \oplus {\pi }_{0}\left( A\right) \) (in \( \mathcal{L}\left( {\mathcal{H} \oplus... | Yes |
For every \( {\pi }_{0} \in \operatorname{irr}\left( {A,\mathcal{H}}\right) \), the equivalence class \( \{ \pi \in \operatorname{irr}\left( {A,\mathcal{H}}\right) \) : \( \left. {\pi \sim {\pi }_{0}}\right\} \) is an \( {F}_{\sigma } \) subset of the topological space \( \operatorname{irr}\left( {A,\mathcal{H}}\right)... | Proof. Let \( C \) be the complement of \( \left\{ {\pi : \pi \sim {\pi }_{0}}\right\} \) in \( \operatorname{irr}\left( {A,\mathcal{H}}\right) \) . It suffices to show that \( C \) is a \( {G}_{\delta } \) in \( \operatorname{irr}\left( {A,\mathcal{H}}\right) \) . Now two irreducible representations of \( A \) on \( \... | Yes |
Theorem 4.1.2. \( \widehat{A} \) is a quotient space of a standard Borel space, in which each singleton is a Borel set. | It is natural to ask at this point if the Borel space \( \widehat{A} \) is a good one or a pathological one. Since \( \widehat{A} \) is a quotient of a standard Borel space, the results of Chapter 3 show that the best one would hope for in general is that \( \widehat{A} \) be analytic, and Corollary 2 of 3.3.5 shows th... | No |
Theorem 4.1.3. Let \( A \) be a separable GCR algebra, fix \( n = \infty ,1,2,\ldots, a \) denumerable cardinal, and let \( q : \operatorname{irr}\left( {A,{\mathcal{H}}_{n}}\right) \rightarrow {\widehat{A}}_{n} \) be the canonical quotient map. Then \( q \) has a Borel cross section. | Proof. Since \( \operatorname{irr}\left( {A,{\mathcal{H}}_{n}}\right) \) is a Polish space and since the equivalence relation in \( \operatorname{irr}\left( {A,{\mathcal{H}}_{n}}\right) \) is defined by a group of homeomorphisms (the unitary group \( \mathcal{U} \) of \( {\mathcal{H}}_{n} \) acts on \( \operatorname{ir... | Yes |
Lemma 4.1.4. Let \( A \) be a separable \( {C}^{ * } \) -algebra, let \( \mathcal{H} \) be a separable Hilbert space, and let \( \Gamma \) be the graph of the unitary equivalence relation in \( \operatorname{irr}\left( {A,\mathcal{H}}\right) \), regarded as a subset of the Polish space \( \operatorname{irr}\left( {A,\m... | Proof. Define a map \( \phi \) of the Polish space \( \operatorname{irr}\left( {A,\mathcal{H}}\right) \times \mathcal{U} \) into the Polish \( \operatorname{space}\operatorname{irr}\left( {A,\mathcal{H}}\right) \times \operatorname{irr}\left( {A,\mathcal{H}}\right) \) by \( \phi \left( {\pi, V}\right) = \left( {\pi ,{V... | Yes |
Corollary 1. The map \( F \rightarrow {L}_{F} \) is a \( \star \) -homomorphism of \( \mathcal{B}\left( {X,\mathcal{K}}\right) \) onto \( {\mathcal{Z}}^{\prime } \) . | Note that \( {L}_{F} = 0 \) if and only if \( F\left( x\right) = 0 \) a.e. \( \left( \mu \right) \) ; thus these functions form a closed ideal \( \mathcal{N} \) in \( \mathcal{B}\left( {X,\mathcal{K}}\right) \) and the map \( F \rightarrow {L}_{F} \) induces an isometric *-isomorphism of \( \mathcal{B}\left( {X,\mathca... | Yes |
Corollary 2. Let \( \mathcal{A} \) be a separable \( {C}^{ \star } \) -subalgebra of \( {\mathcal{Z}}^{\prime } \) . Then there is an isometric \( \star \) -homomorphism \( \pi : \mathcal{A} \rightarrow \mathcal{B}\left( {X,\mathcal{K}}\right) \) such that \( {L}_{\pi \left( T\right) } = T \) for all \( T \in \mathcal{... | Proof. Let \( \mathcal{S} \) be a countable subset of \( \mathcal{A} \) which generates \( \mathcal{A} \) as a \( {C}^{ * } \) - algebra. We may assume \( {\mathcal{S}}^{ \star } = \mathcal{S} \), and that \( \mathcal{S} \) contains the identity if \( \mathcal{A} \) does. Let \( {\mathcal{A}}_{0} \) be the countable fa... | Yes |
Corollary 1. If \( F \in \mathcal{B}\left( {X,\mathcal{K}}\right) \) is such that \( {L}_{F} \in {\mathcal{A}}^{\prime \prime } \), then \( F\left( x\right) \in {\pi }_{x}{\left( \mathcal{A}\right) }^{\prime \prime } \) for almost every \( x \in X \) . | Proof. It suffices to prove the assertion separately for the real and imaginary parts of \( F \), so we can assume \( F \) (and therefore \( {L}_{F} \) ) is self-adjoint.\n\nBy the double commutant theorem and the corollary of Kaplansky's density theorem (1.2.2), we can find a sequence \( {T}_{n} = {T}_{n}^{ * } \in \m... | Yes |
Corollary 2. If \( \mathcal{A} \) is strongly dense in \( {\mathcal{Z}}^{\prime } \), then \( {\pi }_{x} \) is irreducible for almost every \( x \in X \) . | Proof. Choose any irreducible operator \( T \in \mathcal{L}\left( \mathcal{K}\right) \), and let \( F \) be the constant function \( F\left( x\right) = T, x \in X \) . Then \( {L}_{F} \in {\mathcal{L}}^{\prime } \), so by hypothesis \( {L}_{F} \in {\mathcal{A}}^{\prime \prime } \) . By Corollary 1 we conclude that \( T... | Yes |
Corollary 4. Assume \( \mathcal{Z} \subseteq {\mathcal{A}}^{\prime \prime } \subseteq {\mathcal{Z}}^{\prime } \). Then \( {\mathcal{A}}^{\prime \prime } \) coincides with the set of all multiplications \( {L}_{F}, F \in \mathcal{B}\left( {X,\mathcal{K}}\right) \), satisfying \( F\left( x\right) \in {\pi }_{x}{\left( \m... | Proof. The inclusion \( \subseteq \) is Corollary 1 above. For the other inclusion choose \( F \in \mathcal{B}\left( {X,\mathcal{K}}\right) \) satisfying \( F\left( x\right) \in {\pi }_{x}{\left( \mathcal{A}\right) }^{\prime \prime } \) a.e. We will show that \( {L}_{F} \) commutes with \( {\mathcal{A}}^{\prime } \). C... | Yes |
Corollary 1. \( {\pi }_{\mu }{\left( A\right) }^{\prime \prime } = {\mathcal{X}}^{\prime } \) . | Proof. It is obvious that \( {\pi }_{\mu }\left( A\right) \) commutes with \( \mathcal{Z} \), so by the above theorem we have \( \mathcal{Z} \subseteq {\pi }_{\mu }{\left( A\right) }^{\prime \prime } \subseteq {\mathcal{L}}^{\prime } \). Moreover, since the mapping which associates with each operator \( {\pi }_{\mu }\l... | Yes |
Corollary 2. \( {\pi }_{\mu } \) is multiplicity-free. | Proof. By Corollary \( 1,{\pi }_{\mu }{\left( A\right) }^{\prime } \) is the abelian von Neumann algebra \( {\mathcal{Z}}^{\prime \prime } \) . | Yes |
Theorem 4.3.2. Let \( \mu \) and \( v \) be two finite Borel measures concentrated on \( {\widehat{A}}_{n} \) . Then \( {\pi }_{\mu } \sim {\pi }_{v} \) iff \( \mu \) and \( v \) are equivalent measures, and \( {\pi }_{\mu } \circ {\pi }_{v} \) iff \( \mu \) and \( v \) are mutually singular. | Proof. We have arranged the proof of 2.2.2, so that it translates verbatim to a proof of this theorem, provided one uses 4.3.1. in place of 2.2.1. For example, to see that \( \mu \sim v \) implies \( {\pi }_{\mu } \sim {\pi }_{v} \), let \( h = {d\mu }/{dv} \) as in 2.2.2., and define \( U : {L}^{2}\left( {{\widehat{A}... | Yes |
Proposition 4.3.3. Let \( \mu \) and \( {\pi }_{\mu } = {\pi }_{{\mu }_{\infty }} \oplus {\pi }_{{\mu }_{1}} \oplus {\pi }_{{\mu }_{2}} \oplus \cdots \) be as above, and let \( {P}_{\infty },{P}_{1},{P}_{2},\ldots \) be the projections of the Hilbert space \( \mathcal{H} \) of \( {\pi }_{\mu } \) onto the respective co... | Proof. We claim first than \( {\pi }_{{\mu }_{m}} \) and \( {\pi }_{{\mu }_{n}} \) are disjoint if \( m \neq n \) . To prove this, it suffices to show that for every nonzero subrepresentation \( \sigma \) of \( {\pi }_{{\mu }_{n}},\sigma {\left( A\right) }^{\prime } \) is an abelian von Neumann algebra having multiplic... | Yes |
For every nondegenerate multiplicity-free representation \( \pi \) of \( A \), there is a finite Borel measure \( \mu \) on \( \widehat{A} \) such that \( \pi \) is equivalent to \( {\pi }_{\mu } \) . | Let \( \pi \) be a multiplicity-free representation of \( A \) on a Hilbert space \( \mathcal{H} \) . Note that since \( A \) is separable,2.2.3 implies that \( \mathcal{H} \) is a separable space. Let us first make a reduction. Decompose the center \( \mathcal{C} \) of \( \pi {\left( A\right) }^{\prime \prime } \) as ... | Yes |
Lemma 1.1. Let \( {A}^{\prime } \rightarrow A \rightarrow {A}^{\prime \prime } \) and \( {B}^{\prime } \rightarrow B \rightarrow {B}^{\prime \prime } \) be two short exact sequences. Suppose that in the commutative diagram\n\n are isomorphisms; we have to show that \( \alpha \) is an isomorphism.\n\nFirst we show that \( \ker \alpha = 0 \) . Let \( a \in \ker \alpha \), then \( 0 = {\varepsilo... | No |
Theorem 2.1. Let \( {B}^{\prime }\overset{\mu }{ \rightarrow }B\overset{\varepsilon }{ \rightarrow }{B}^{\prime \prime } \) be an exact sequence of \( \Lambda \) -modules. For every \( \Lambda \) -module \( A \) the induced sequence\n\n\[ 0 \rightarrow {\operatorname{Hom}}_{A}\left( {A,{B}^{\prime }}\right) \overset{{\... | Proof. First we show that \( {\mu }_{ * } \) is injective.\n\nAssume that \( {\mu \varphi } \) in the diagram\n\n\n\nis the zero map. Since \( \mu : {B}^{\prime } \rightarrowtail B \) is injective this implies that \( ... | Yes |
Proposition 3.1. Let \( M \) be a \( \Lambda \) -module, \( {\psi }_{A} : A \rightarrow M \) and \( {\psi }_{B} : B \rightarrow M \) A-module homomorphisms. Then there exists a unique map \[ \psi = \left\langle {{\psi }_{A},{\psi }_{B}}\right\rangle : A \oplus B \rightarrow M \] such that \( \psi {\iota }_{A} = {\psi }... | Proof. Define \( \psi \left( {a, b}\right) = {\psi }_{A}\left( a\right) + {\psi }_{B}\left( b\right) \) . This obviously is the only homomorphism \( \psi : A \oplus B \rightarrow M \) satisfying \( \psi {\iota }_{A} = {\psi }_{A} \) and \( \psi {\iota }_{B} = {\psi }_{B} \) . 1 | Yes |
Proposition 3.2. Let \( M \) be a \( \Lambda \) -module and let \( \left\{ {{\psi }_{j} : {A}_{j} \rightarrow M}\right\}, j \in J \) , be a family of \( \Lambda \) -module homomorphisms. Then there exists a unique homomorphism \( \psi = \left\langle {\psi }_{j}\right\rangle : {\bigoplus }_{j \in J}{A}_{j} \rightarrow M... | Proof. We define \( \psi \left( {\left( {a}_{j}\right) }_{j \in J}\right) = \mathop{\sum }\limits_{{j \in J}}{\psi }_{j}\left( {a}_{j}\right) \) . This is possible because \( {a}_{j} = 0 \) except for a finite number of indices. The map \( \psi \) so defined is obviously the only homomorphism \( \psi : {\bigoplus }_{j ... | Yes |
Proposition 3.3. Let \( M \) be a \( \Lambda \) -module and let \( \left\{ {{\varphi }_{j} : M \rightarrow {A}_{j}}\right\}, j \in J \) , be a family of \( \Lambda \) -module homomorphisms. Then there exists a unique homomorphism \( \varphi = \left\{ {\varphi }_{j}\right\} : M \rightarrow \mathop{\prod }\limits_{{j \in... | [] | Yes |
Proposition 3.4. Let \( B \) be a \( \Lambda \) -module and \( \left\{ {A}_{j}\right\}, j \in J \) be a family of \( \Lambda \) - modules. Then there is an isomorphism\n\n\[ \eta : {\operatorname{Hom}}_{A}\left( {{\bigoplus }_{j \in J}{A}_{j}, B}\right) \rightarrow \mathop{\prod }\limits_{{j \in J}}{\operatorname{Hom}}... | Proof. The proof reveals that this theorem is merely a restatement of the universal property of the direct sum. For \( \psi : {\bigoplus }_{j \in J}{A}_{j} \rightarrow B \), define \( \eta \left( \psi \right) = {\left( \psi {\iota }_{j} : {A}_{j} \rightarrow B\right) }_{j \in J} \) . Conversely a family \( \left\{ {{\p... | Yes |
Proposition 3.5. Let \( A \) be a \( \Lambda \) -module and \( \left\{ {B}_{j}\right\}, j \in J \) be a family of \( \Lambda \) -modules. Then there is an isomorphism\n\n\[ \zeta : {\operatorname{Hom}}_{\Lambda }\left( {A,\mathop{\prod }\limits_{{j \in J}}{B}_{j}}\right) \overset{ \sim }{ \rightarrow }\mathop{\prod }\l... | The proof is left to the reader. \( ▱ \) | No |
Proposition 4.1. Suppose the \( \Lambda \) -module \( P \) is free on the set \( S \) . Then \( P \cong {\bigoplus }_{s \in S}{\Lambda }_{s} \) where \( {\Lambda }_{s} = \Lambda \) as a left module for \( s \in S \) . Conversely, \( {\bigoplus }_{s \in S}{\Lambda }_{s} \) is free on the set \( \left\{ {{1}_{{A}_{s}}, s... | Proof. We define \( \varphi : P \rightarrow {\bigoplus }_{s \in S}{\Lambda }_{s} \) as follows: Every element \( a \in P \) is expressed uniquely in the form \( a = \mathop{\sum }\limits_{{s \in S}}{\lambda }_{s}s \) ; set \( \varphi \left( a\right) = {\left( {\lambda }_{s}\right) }_{s \in S} \) . Conversely, for \( s ... | Yes |
Proposition 4.2. Let \( P \) be free on the set \( S \) . To every \( \Lambda \) -module \( M \) and to every function \( f \) from \( S \) into the set underlying \( M \), there is a unique A-module homomorphism \( \varphi : P \rightarrow M \) extending \( f \) . | Proof. Let \( f\left( s\right) = {m}_{s} \) . Set \( \varphi \left( a\right) = \varphi \left( {\mathop{\sum }\limits_{{s \in S}}{\lambda }_{s}s}\right) = \mathop{\sum }\limits_{{s \in S}}{\lambda }_{s}{m}_{s} \) . This obviously is the only homomorphism having the required property. | Yes |
Proposition 4.3. Every A-module \( A \) is a quotient of a free module \( P \) . | Proof. Let \( S \) be a set of generators of \( A \) . Let \( P = {\bigoplus }_{s \in S}{\Lambda }_{s} \) with \( {\Lambda }_{s} = \Lambda \) and define \( \varphi : P \rightarrow A \) to be the extension of the function \( f \) given by \( f\left( {1}_{{\Lambda }_{s}}\right) = s \) . Trivially \( \varphi \) is surject... | Yes |
Proposition 4.4. Let \( P \) be a free \( \Lambda \) -module. To every surjective homomorphism \( \varepsilon : B \rightarrow C \) of \( \Lambda \) -modules and to every homomorphism \( \gamma : P \rightarrow C \) there exists a homomorphism \( \beta : P \rightarrow B \) such that \( {\varepsilon \beta } = \gamma \) . | Proof. Let \( P \) be free on \( S \) . Since \( \varepsilon \) is surjective we can find elements \( {b}_{s} \in B, s \in S \) with \( \varepsilon \left( {b}_{s}\right) = \gamma \left( s\right), s \in S \) . Define \( \beta \) as the extension of the function \( f : S \rightarrow B \) given by \( f\left( s\right) = {b... | Yes |
Proposition 4.5. A direct sum \( {\bigoplus }_{i \in I}{P}_{i} \) is projective if and only if each \( {P}_{i} \) is. | Proof. We prove the proposition only for \( A = P \oplus Q \) . The proof in the general case is analogous. First assume \( P \) and \( Q \) projective. Let \( \varepsilon : B \rightarrow C \) be surjective and \( \gamma : P \oplus Q \rightarrow C \) a homomorphism. Define \( {\gamma }_{P} = \gamma {\iota }_{P} : P \ri... | Yes |
Lemma 4.6. Suppose that \( \sigma : C \rightarrow B \) is a splitting for the short exact sequence \( A\overset{\mu }{ \rightarrow }B\overset{\varepsilon }{ \rightarrow }C \) . Then \( B \) is isomorphic to the direct sum \( A \oplus C \) . Under this isomorphism, \( \mu \) corresponds to \( {\iota }_{A} \) and \( \sig... | Proof. By the universal property of the direct sum we define a map \( \psi \) as follows\n\n\n\n\n\nThen the diagram\n\n the following statements are equivalent:\n\n(1) \( P \) is projective;\n\n(2) for every short exact sequence \( A\overset{\mu }{ \rightarrow }B\overset{\varepsilon }{ \rightarrow }C \) of \( \Lambda \) -modules the induced sequence\n\n\[ 0 \rightarrow {\operatorname{Hom}}_{A}\left( {... | Proof. (1) \( \Rightarrow \) (2). By Theorem 2.1 we only have to show exactness at \( {\operatorname{Hom}}_{A}\left( {P, C}\right) \), i.e. that \( {\varepsilon }_{ * } \) is surjective. But since \( \varepsilon : B \rightarrow C \) is surjective this is asserted by the fact that \( P \) is projective.\n\n\( \left( 2\r... | Yes |
Proposition 6.1. \( \varepsilon : B \rightarrow C \) is epimorphic if and only if it is surjective. | Proof. Let \( B\overset{\varepsilon }{ \rightarrow }C\xrightarrow[{\alpha }_{2}]{{\alpha }_{1}}M \) . If \( \varepsilon \) is surjective then clearly \( {\alpha }_{1}{\varepsilon b} = {\alpha }_{2}{\varepsilon b} \) for all \( b \in B \), implies \( {\alpha }_{1}c = {\alpha }_{2}c \) for all \( c \in C \) . Conversely,... | Yes |
Proposition 6.2. \( \mu : A \rightarrow B \) is monomorphic if and only if it is injective. | Proof. If \( \mu \) is injective, then \( \mu {\alpha }_{1}x = \mu {\alpha }_{2}x \) for all \( x \in M \) implies \( {\alpha }_{1}x = {\alpha }_{2}x \) for all \( x \in M \) . Conversely, suppose \( \mu \) monomorphic and \( {a}_{1},{a}_{2} \in A \) such that \( \mu {a}_{1} = \mu {a}_{2} \) . Choose \( M = \Lambda \) ... | Yes |
Proposition 7.2. Every quotient of a divisible module is divisible. | Proof. Let \( \varepsilon : D \rightarrow E \) be an epimorphism and let \( D \) be divisible. For \( e \in E \) and \( 0 \neq \lambda \in \Lambda \) there exists \( d \in D \) with \( \varepsilon \left( d\right) = e \) and \( {d}^{\prime } \in D \) with \( \lambda {d}^{\prime } = d \) . Setting \( {e}^{\prime } = \var... | Yes |
Proposition 7.4. Every abelian group may be embedded in a divisible (hence injective) abelian group. | Proof. We shall define a monomorphism of the abelian group \( A \) into a direct product of copies of \( \mathbb{Q}/\mathbb{Z} \) . By Proposition 6.3 this will suffice. Let \( 0 \neq a \in A \) and let \( \left( a\right) \) denote the subgroup of \( A \) generated by \( a \) . Define \( \alpha : \left( a\right) \right... | Yes |
Proposition 8.1. Let \( A \) be a left \( A \) -module and let \( G \) be an abelian group. Regard \( {\operatorname{Hom}}_{\mathbf{Z}}\left( {\Lambda, G}\right) \) as a left \( \Lambda \) -module via the right \( \Lambda \) -module structure of \( \Lambda \) . Then there is an isomorphism of abelian groups\n\n\[ \eta ... | Proof. Let \( \varphi : A \rightarrow {\operatorname{Hom}}_{\mathbf{z}}\left( {\Lambda, G}\right) \) be a \( \Lambda \) -module homomorphism. We define a homomorphism of abelian groups \( {\varphi }^{\prime } : A \rightarrow G \) by\n\n\[ {\varphi }^{\prime }\left( a\right) = \left( {\varphi \left( a\right) }\right) \l... | Yes |
Theorem 8.2. Let \( G \) be a divisible abelian group. Then \( \bar{\Lambda } = {\operatorname{Hom}}_{\mathbb{Z}}\left( {\Lambda, G}\right) \) is an injective \( \Lambda \) -module. | Proof. Let \( \mu : A \rightarrow B \) be a monomorphism of \( \Lambda \) -modules, and let \( \alpha : A \rightarrow \bar{A} \) a homomorphism of \( \Lambda \) -modules. We have to show that there exists \( \beta : B \rightarrow \bar{\Lambda } \) such that \( {\beta \mu } = \alpha \) . To prove this, we remark that \(... | Yes |
Proposition 8.3. Every A-module \( A \) is a submodule of a cofree, hence injective, \( \Lambda \) -module. | Proof. Let \( 0 \neq a \in A \) and let \( \left( a\right) \) denote the submodule of \( A \) generated by \( a \) . By the remarks preceeding Theorem 8.2 there exists a nonzero \( \Lambda \) -homomorphism \( \alpha : \left( a\right) \rightarrow {\Lambda }^{ * } \) . Since \( {\Lambda }^{ * } \) is injective there exis... | Yes |
Theorem 8.4. For a Λ-module I the following statements are equivalent :\n\n(1) \( I \) is injective;\n\n(2) for every exact sequence \( A\overset{\mu }{ \rightarrow }B\overset{\varepsilon }{ \rightarrow }C \) of \( \Lambda \) -modules the induced sequence \( 0 \rightarrow {\operatorname{Hom}}_{A}\left( {C, I}\right) \o... | The proof is dual to the proof of Theorem 4.7. For the step (3) \( \Rightarrow \) (4) one needs the dual of Lemma 4.6. The details are left to the reader. | No |
Proposition 9.1. B is an essential extension of \( A \) if and only if, for every \( 0 \neq b \in B \), there exists \( \lambda \in \Lambda \) such that \( {\lambda b} \in A \) and \( {\lambda b} \neq 0 \) . | Proof. Let \( B \) be an essential extension of \( A \), and let \( H \) be the submodule generated by \( b \in B \) . Since \( H \neq 0 \) it follows that \( H \cap A \neq 0 \), i.e. there exists \( \lambda \in \Lambda \) such that \( 0 \neq {\lambda b} \in A \) . Conversely, let \( H \) be a nontrivial submodule of \... | Yes |
Theorem 9.2. Let \( A \) be a submodule of the injective module I. Let \( E \) be a maximal essential extension of \( A \) contained in \( I \) . Then \( E \) is injective. | Proof. First we show that \( E \) does not admit any nontrivial essential monomorphism.\n\nLet \( \mu : E \rightarrow X \) be an essential monomorphism. Since \( I \) is injective, there exists a homomorphism \( \xi : X \rightarrow I \) completing the diagram\n\n be two maximal essential extensions of \( A \) contained in injective modules \( {I}_{1},{I}_{2} \) . Then \( {E}_{1} \cong {E}_{2} \) and every injective module \( I \) containing \( A \) also contains a submodule isomorphic to \( {E}_{1} \) . | Proof. Consider the diagram\n\n\n\nSince \( {E}_{2} \) is injective there exists \( \xi : {E}_{1} \rightarrow {E}_{2} \) completing the diagram. As in the proof of Theorem 9.2 one shows that \( \xi \) is monomorphic. B... | No |
Proposition 4.1. Let \( \tau \) be a natural transformation from the functor \( \mathfrak{C}\left( {A, - }\right) \) to the functor \( F \) from \( \mathfrak{C} \) to \( \mathfrak{S} \) . Then \( \tau \mapsto {\tau }_{A}\left( {1}_{A}\right) \) sets up a one-one correspondence between the set \( \left\lbrack {\mathfrak... | Proof. We show first that \( \tau \) is entirely determined by the element \( {\tau }_{A}\left( {1}_{A}\right) \in F\left( A\right) \) . Let \( \varphi : A \rightarrow B \) and consider the commutative diagram\n\n\n\nT... | Yes |
Corollary 4.2. The set of morphisms \( \mathfrak{C}\left( {{A}^{\prime }, A}\right) \) and the set of natural transformations \( \left\lbrack {\mathfrak{C}\left( {A, - }\right) ,\mathfrak{C}\left( {{A}^{\prime }, - }\right) }\right\rbrack \) are in one-to-one correspondence, the correspondence being given by \( \psi \m... | Proof. If \( \tau \) is such a natural transformation, let \( \psi = {\tau }_{A}\left( {1}_{A}\right) \), so that \( \psi : {A}^{\prime } \rightarrow A \) . Then, by (4.1) \( \tau \) is given by\n\n\[{\tau }_{B}\left( \varphi \right) = {\varphi }_{ * }\left( \psi \right) = {\varphi \psi } = {\psi }^{ * }\left( \varphi ... | Yes |
Proposition 4.3. Let \( \tau : \mathfrak{C}\left( {A, - }\right) \rightarrow \mathfrak{C}\left( {{A}^{\prime }, - }\right) ,{\tau }^{\prime } : \mathfrak{C}\left( {{A}^{\prime }, - }\right) \rightarrow \mathfrak{C}\left( {{A}^{\prime \prime }, - }\right) \) . Then if \( \tau \mapsto \psi ,{\tau }^{\prime } \mapsto {\ps... | Proof. \( {\left( {\tau }^{\prime }\tau \right) }_{B} = \left( {\tau }_{B}^{\prime }\right) \left( {\tau }_{B}\right) = {\psi }^{\prime * }{\psi }^{ * } = {\left( \psi {\psi }^{\prime }\right) }^{ * } \). | Yes |
Theorem 5.1. Let \( \left( {X;{p}_{i}}\right) ,\left( {{X}^{\prime };{p}_{i}^{\prime }}\right) \) both be products of the family \( \left\{ {X}_{i}\right\}, i \in I \) , in \( \mathfrak{C} \) . Then there exists a unique isomorphism \( \xi : X \rightarrow {X}^{\prime } \) such that \( {p}_{i}^{\prime }\xi = {p}_{i} \) ... | Proof. By the universal property for \( X \) there exists a (unique) morphism \( \eta : {X}^{\prime } \rightarrow X \) such that \( {p}_{i}\eta = {p}_{i}^{\prime } \) . Similarly there exists a (unique) morphism \( \xi : X \rightarrow {X}^{\prime } \) such that \( {p}_{i}^{\prime }\xi = {p}_{i} \) . Then\n\n\[ \n{p}_{i... | Yes |
Proposition 5.2. Let \( \mathfrak{C} \) be a category in which \( \mathfrak{C}\left( {X, Y}\right) \) is non-empty for all \( X, Y \) (e.g., a category with zero object). Then if \( \mathop{\prod }\limits_{i}{X}_{i} \) exists it admits each \( {X}_{i} \) as a retract. Thus, in particular, each \( {p}_{i} \) is an epimo... | Proof. In the definition of \( \mathop{\prod }\limits_{i}{X}_{i} \), take \( Y = {X}_{j} \), for a fixed \( j \in I \), and \( {f}_{j} = 1 : {X}_{j} \rightarrow {X}_{j} \) . For \( i \neq j \) let \( {f}_{i} \) be arbitrary. Then \( {p}_{j}f = 1 : {X}_{j} \rightarrow {X}_{j} \) so that \( \mathop{\prod }\limits_{i}{X}_... | Yes |
Proposition 5.3. Given two families \( \left\{ {X}_{i}\right\} ,\left\{ {Y}_{i}\right\} \) of objects of \( \mathfrak{C} \), indexed by the same indexing set \( I \), then if the products \( \mathop{\prod }\limits_{i}{X}_{i},\mathop{\prod }\limits_{i}{Y}_{i} \) exist, and if \( {f}_{i} : {X}_{i} \rightarrow {Y}_{i}, i ... | Proof. The first assertion is merely an application of the universal property of \( \mathop{\prod }\limits_{i}{Y}_{i} \) . The proof of the second is left to the reader. (It should\n\nbe clear what we understand by the category \( {\mathfrak{C}}^{I} \) ; see Exercise 4.7.) \( ▱ \) | No |
Proposition 5.4. Let \( {f}_{i} : Z \rightarrow {X}_{i}, g : W \rightarrow Z,{g}_{i} : {X}_{i} \rightarrow {Y}_{i}, i \in I \) . Then, if the products in question exist,\n\n\[ \text{(i)}\left\{ {f}_{i}\right\} g = \left\{ {{f}_{i}g}\right\} \text{,(ii)}\left( {\mathop{\prod }\limits_{i}{g}_{i}}\right) \left\{ {f}_{i}\r... | Proof. We leave the proof to the reader, with the hint that it is sufficient to prove that each side projects properly onto the \( i \) -component under the projection \( {p}_{i} \) . | No |
Proposition 5.5. Let \( \mathfrak{C} \) be a category in which any two objects admit a product. Thus given objects \( X, Y, Z \) in \( \mathfrak{C} \) we have projections\n\n\[ \n{p}_{1} : X \times Y \rightarrow X,\;{q}_{1} : \left( {X \times Y}\right) \times Z \rightarrow X \times Y, \]\n\n\[ \n{p}_{2} : X \times Y \r... | Proof. Given \( {f}_{1} : W \rightarrow X,{f}_{2} : W \rightarrow Y,{f}_{3} : W \rightarrow Z \), we form \( g : W \rightarrow X \times Y \) such that \( {p}_{1}g = {f}_{1},{p}_{2}g = {f}_{2} \) . We then form \( h : W \rightarrow \left( {X \times Y}\right) \times Z \) such that \( {q}_{1}h = g,{q}_{2}h = {f}_{3} \) . ... | Yes |
Proposition 5.6. If any two objects in \( \\mathfrak{C} \) admit a product, so does any finite collection of objects. | Proof. We argue by induction, using an obvious generalization of the proof of Proposition 5.5. | No |
Proposition 6.1. \( \left( {\Delta ;\alpha ,\beta }\right) \) is the product of \( \varphi \) and \( \psi \) in \( \mathfrak{C}/X \) . | This means that \( \alpha ,\beta \) play the roles of \( {p}_{1},{p}_{2} \) in the definition of a product, when interpreted as morphisms \( \alpha : \Delta \rightarrow \varphi ,\beta : \Delta \rightarrow \psi \) in \( \mathfrak{C}/X \) . | No |
Theorem 6.2. Let (6.1) be a pull-back diagram in a category \( \mathfrak{C} \) with zero object. Then\n\n(i) if \( \left( {J,\mu }\right) \) is the kernel of \( \beta ,\left( {J,{\alpha \mu }}\right) \) is the kernel of \( \varphi \) ;\n\n(ii) if \( \left( {J, v}\right) \) is the kernel of \( \varphi, v \) may be facto... | Proof. (i)\n\n\n\nSet \( v = {\alpha \mu } \) . We first show that \( v \) is monomorphic; for \( \mu \) and \( \{ \alpha ,\beta \} \) are monomorphic, so \( \{ \alpha ,\beta \} \mu = \{ v,0\} : J \rightarrow A \times ... | Yes |
Proposition 7.1. Let \( F : \mathfrak{C} \rightarrow \mathfrak{D},{F}^{\prime } : \mathfrak{D} \rightarrow \mathfrak{E}, G : \mathfrak{D} \rightarrow \mathfrak{C},{G}^{\prime } : \mathfrak{E} \rightarrow \mathfrak{D} \) be functors and let \( \eta : F \dashv G,{\eta }^{\prime } : {F}^{\prime } \dashv {G}^{\prime } \) b... | We leave the proof as an exercise. [] | No |
Proposition 7.2. Let \( F : \mathfrak{C} \rightarrow \mathfrak{D}, G : \mathfrak{D} \rightarrow \mathfrak{C} \) be functors and let \( \varepsilon : 1 \rightarrow {GF} \) , \( \delta : {FG} \rightarrow 1 \) be natural transformations such that \( {\delta F} \circ {F\varepsilon } = 1,{G\delta } \circ {\varepsilon G} = 1... | Proof. First, \( \eta \) is natural. For\n\n\[ \eta \left( {\beta \circ \varphi \circ {F\alpha }}\right) = G\left( {\beta \circ \varphi \circ {F\alpha }}\right) \circ {\varepsilon }_{{X}^{\prime }} \]\n\n\[ = {G\beta } \circ {G\varphi } \circ {GF\alpha } \circ {\varepsilon }_{{X}^{\prime }} \]\n\n\[ = {G\beta } \circ {... | Yes |
Proposition 7.3. Suppose \( F \dashv G \) . Then \( F \) determines \( G \) up to natural equivalence and \( G \) determines \( F \) up to natural equivalence. | Proof. It is plainly sufficient to establish the first assertion. Suppose then that \( \eta : F \dashv G,{\eta }^{\prime } : F \dashv {G}^{\prime } \) . Consider the natural equivalence of functors\n\n\[ \mathfrak{C}\left( {-,{GY}}\right) \xrightarrow[]{{\eta }_{ \sim }^{-1}}\mathfrak{D}\left( {F-, Y}\right) \overset{{... | Yes |
Proposition 7.4. Under the same hypotheses as in Proposition 7.3, with \( \theta ,\bar{\theta } \) defined as in (7.4),(7.5), we have\n\n(i) \( {\theta F} \circ \varepsilon = {\varepsilon }^{\prime };\delta \circ F\bar{\theta } = {\delta }^{\prime } \);\n\n(ii) \( {\theta }_{Y} \circ \eta \left( \varphi \right) = {\eta... | \[ \text{Proof. (i)}\;{\theta F} \circ \varepsilon = {G}^{\prime }{\delta F} \circ {\varepsilon }^{\prime }{GF} \circ \varepsilon \]\n\[ = {G}^{\prime }{\delta F} \circ {G}^{\prime }{F\varepsilon } \circ {\varepsilon }^{\prime },\;\text{ by the naturality of }{\varepsilon }^{\prime }, \]\n\[ = {\varepsilon }^{\prime }.... | No |
Proposition 7.5. If \( F : \mathfrak{C} \rightarrow \mathfrak{D} \) is full and faithful and if \( F \dashv G \), then the unit \( \varepsilon : 1 \rightarrow {GF} \) is a natural equivalence. | Proof. Let \( \delta : {FG} \rightarrow 1 \) be the counit. Then \( {\delta F} : {FGF} \rightarrow F \) . Since \( F \) is full and faithful we may define a transformation \( \varrho : {GF} \rightarrow 1 \) by\n\n\[ F{\varrho }_{X} = {\delta }_{FX} \]\n\nand it is plain that \( \varrho \) is natural. We show that \( \v... | Yes |
Proposition 7.6. If \( F : \mathfrak{C} \rightarrow \mathfrak{D} \) is a full embedding and if \( F \dashv G \), then there exists \( {G}^{\prime } \) with \( F \dashv {G}^{\prime } \) where the unit \( {\varepsilon }^{\prime } : 1 \rightarrow {G}^{\prime }F \) is the identity. | Proof. We construct the functor \( {G}^{\prime } \) as follows\n\n\[ \n{G}^{\prime }\left( Y\right) = G\left( Y\right) \;\text{ if }\;Y \notin \operatorname{Im}F, \]\n\n\[ \n{G}^{\prime }F\left( X\right) = X\text{.} \]\nFor \( \beta : {Y}_{1} \rightarrow {Y}_{2} \)\n\n\[ \n{G}^{\prime }\left( \beta \right) = G\left( \b... | Yes |
Theorem 7.7. If \( G : \mathfrak{D} \rightarrow \mathfrak{C} \) has a left adjoint then \( G \) preserves products, pull-backs and kernels. | Proof. We must show that if \( \left\{ {Y;{p}_{i}}\right\} \) is the product of objects \( {Y}_{i} \) in \( \mathfrak{D} \), then \( \left\{ {{GY};G\left( {p}_{i}\right) }\right\} \) is the product of the objects \( G\left( {Y}_{i}\right) \) in \( \mathfrak{C} \) . Suppose given \( {f}_{i} : X \rightarrow G{Y}_{i} \) .... | No |
Proposition 8.1. The product of the objects \( {X}_{i} \) is \( \left( {X;{p}_{i}}\right) \) where \( X = G\left( \left\{ {X}_{i}\right\} \right) \) | Proof. Given \( {f}_{i} : Y \rightarrow {X}_{i} \), we have a morphism \( f = \left\{ {f}_{i}\right\} : P\left( Y\right) \rightarrow \left\{ {X}_{i}\right\} \) . Then \( \eta \left( f\right) \) is a morphism \( Y \neq X \) such that, by (7.3), \n\n\[ \n\delta \circ P\left( {\eta \left( f\right) }\right) = f. \n\] \n\nB... | No |
Proposition 8.2. \( \pi : {FG}\left( {\varphi ,\psi }\right) \rightarrow \left( {\varphi ,\psi }\right) \) is the pull-back of \( \left( {\varphi ,\psi }\right) \) . | Proof. Let \( G\left( {\varphi ,\psi }\right) = Y \) . Then \( \pi : F\left( Y\right) \rightarrow \left( {\varphi ,\psi }\right) \) is a pair of morphisms \( \alpha : Y \rightarrow A,\beta : Y \rightarrow B \) such that \( {\varphi \alpha } = {\psi \beta } \) . Moreover, if \( \left( {\gamma ,\delta }\right) : F\left( ... | Yes |
Theorem 8.3. Let \( \mathfrak{P} \) be a small connected category and let \( F : \mathfrak{C} \rightarrow {\mathfrak{C}}^{\mathfrak{P}} \) be the constant functor. Then \( F \) is a full embedding. | Proof. The point at issue is that \( F \) is full. Let \( f : X \rightarrow Y \) in \( \mathfrak{C} \) and let \( P, Q \) be objects of \( \mathfrak{P} \) . We have a chain in \( \mathfrak{P} \)\n\n\[ P \rightarrow \cdot \leftarrow \cdot \rightarrow \cdots \leftarrow \cdot \rightarrow Q \]\n\nand hence must show that, ... | No |
Lemma 8.5. If \( P : \mathfrak{C} \rightarrow {\mathfrak{C}}^{\mathfrak{P}} \) is the constant functor (for any \( \mathfrak{C} \) ), then the diagram\n\n\n\ncommutes. | [] | No |
Lemma 9.2. \( {i}_{1}{p}_{1} + {i}_{2}{p}_{2} = 1 : {A}_{1} \oplus {A}_{2} \rightarrow {A}_{1} \oplus {A}_{2} \) . | Proof. Now \( {p}_{1}\left( {{i}_{1}{p}_{1} + {i}_{2}{p}_{2}}\right) = {p}_{1}{i}_{1}{p}_{1} + {p}_{1}{i}_{2}{p}_{2} = {p}_{1} \), since \( {p}_{1}{i}_{1} = 1 \) , \( {p}_{1}{i}_{2} = 0 \) . Similarly \( {p}_{2}\left( {{i}_{1}{p}_{1} + {i}_{2}{p}_{2}}\right) = {p}_{2} \) . Thus, by the uniqueness property of the produc... | Yes |
Proposition 9.3. Given\n\n\[ \nA\xrightarrow[]{\{ \varphi ,\psi \} }B \oplus C\xrightarrow[]{\langle \gamma ,\delta \rangle }D, \n\]\n\nwe have\n\n\[ \n\langle \gamma ,\delta \rangle \{ \varphi ,\psi \} = {\gamma \varphi } + {\delta \psi } \n\] | Proof.\n\n\[ \n\langle \gamma ,\delta \rangle \{ \varphi ,\psi \} = \left( {\gamma {p}_{1} + \delta {p}_{2}}\right) \{ \varphi ,\psi \} = \gamma {p}_{1}\{ \varphi ,\psi \} + \delta {p}_{2}\{ \varphi ,\psi \} \n\]\n\n\[ \n= {\gamma \varphi } + {\delta \psi } \n\] | Yes |
Corollary 9.4. The addition in the set \( \mathfrak{A}\left( {A, B}\right) \) is determined by the category \( \mathfrak{A} \) . | Proof. If \( {\varphi }_{1},{\varphi }_{2} : A \rightarrow B \) then \( {\varphi }_{1} + {\varphi }_{2} = \left\langle {{\varphi }_{1},{\varphi }_{2}}\right\rangle \{ 1,1\} \) . | Yes |
Proposition 9.5. Let \( F : \mathfrak{A} \rightarrow \mathfrak{B} \) be a functor from the additive category \( \mathfrak{A} \) to the additive category \( \mathfrak{B} \) . Then the following conditions are equivalent:\n\n(i) \( F \) preserves sums (of two objects);\n\n(ii) \( F \) preserves products (of two objects);... | Proof. (i) \( \Rightarrow \) (ii). This is not quite trivial since we are required to show that \( F\langle 1,0\rangle = \langle 1,0\rangle \) and \( F\langle 0,1\rangle = \langle 0,1\rangle \) . Thus we must show that \( F\left( 0\right) = 0 \) and for this it is plainly sufficient to show that \( F \) maps zero objec... | Yes |
Lemma 9.7. Suppose \( {v\eta } \) and \( \eta \) have the same kernel and \( \eta \) is an epimorphism. Then \( v \) is a monomorphism. | Proof. Use property (iii) of an abelian category to write \( v = {o\sigma } \) , with \( \sigma \) epimorphic, \( \varrho \) monomorphic. Then \( {v\eta } = \varrho {\sigma \eta } \) and if \( \mu \) is the kernel of \( {\sigma \eta } \), then \( \mu \) is the kernel of \( \varrho {\sigma \eta } = {v\eta } \) and hence... | Yes |
Proposition 10.2. Let \( F : \mathfrak{C} \rightarrow \mathfrak{D} \) and \( F \dashv G \) . If \( G \) maps epimorphisms to epimorphisms, then \( F \) maps projectives to projectives. | Proof. Let \( P \) be a projective object of \( \mathfrak{C} \) and consider the diagram, in \( \mathfrak{D} \) ,  Applying the adjugant, this gives rise to a diagram , where \( U : \mathfrak{C} \rightarrow \mathfrak{S} \) is the underlying functor. Then the counit \( \delta : \operatorname{Fr}U\left( A\right) \rightarrow A \) is an epimorphism. | Proof. Suppose \( \alpha ,{\alpha }^{\prime } : A \rightarrow B \) and \( \alpha \circ \delta = {\alpha }^{\prime } \circ \delta \) . Applying the adjugant we find \( U\left( \alpha \right) = U\left( {\alpha }^{\prime }\right) \) . But \( U \) is injective on morphisms so \( \alpha = {\alpha }^{\prime } \) . | Yes |
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