Packets and transfer graphs

Proof of Lemma 7.3

Pairwise annihilation of the \(P_\alpha\) gives

Summing over \(j\) recovers \(\sum_\alpha P_\alpha=I\). The range statement follows from the directness forced by pairwise annihilation.

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Proof of Theorem 7.5

For \(x\in X\),

If \(x_\alpha\in X_\alpha\), then \(P_\beta x_\alpha=\delta_{\alpha\beta}x_\alpha\), so

Thus analysis and synthesis are inverse linear maps.

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Proof of Corollary 7.6

The forward implication is immediate. For the reverse implication, sum the packet equalities and use \(\sum_\alpha P_\alpha=I\).

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Proof of Corollary 7.7

For \(x=\sum_\alpha P_\alpha x\), any extension of the \(f_\alpha\) must satisfy

which proves both existence and uniqueness of the first formula. For the second, the sum \(g=\sum_\alpha g_\alpha\) satisfies

because \(g_\alpha(Y)\subseteq X_\alpha\). Uniqueness follows from \(\sum_\alpha P_\alpha=I\).

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Proof of Theorem 7.8

For every pair of packet indices,

If \((\alpha,\gamma)\neq(\beta,\delta)\), then either \(P_\alpha P_\beta=0\) or \(Q_\gamma Q_\delta=0\), so the corresponding tensor idempotents annihilate each other. Finally,

The range is generated by pure tensors \(P_\alpha x\otimes Q_\gamma y\), hence equals \(X_\alpha\otimes Y_\gamma\). Exact reconstruction follows from Theorem 7.5.

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Proof of Proposition 7.9

Because dualization reverses products,

Also \(\sum_\alpha P_\alpha^*=I_{X^*}\). Thus the dual maps form a packet system. Restriction gives a map from \(P_\alpha^*X^*\) to \(X_\alpha^*\). It is injective because a covector in the range of \(P_\alpha^*\) vanishes on every other packet, and it is surjective because any covector on \(X_\alpha\) extends to \(X\) through the direct sum decomposition.

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Proof of Theorem 7.10

The range of a bounded idempotent is its space of fixed points,

which is closed. Because \(I\) is finite, analysis and synthesis are bounded for the maximum and sum norms, respectively, and therefore for every equivalent finite product norm. Their inverse identities are algebraic and were proved in Theorem 7.5.

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Proof of Proposition 7.12

Since \(I=\mathcal S_{{\mathcal P}}\mathcal A_{{\mathcal P}}\),

The first displayed estimate follows from submultiplicativity. The inverse identity

gives the second estimate.

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Proof of Corollary 7.13

For \(x\in X\), choose \(m_n\in M\) with \(m_n\to x\). Continuity gives \(P_\alpha m_n\to P_\alpha x\), so \(P_\alpha X\subseteq{\overline{P_\alpha M}}\). The reverse inclusion follows because \(P_\alpha X\) is closed and contains \(P_\alpha M\). The bounded extensions remain pairwise annihilating idempotents whose finite sum is the identity.

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Proof of Lemma 7.15

Let \(x^{(k)}=(x_n^{(k)})\) be Cauchy in \(\|\cdot\|_u\). For every fixed \(n\),

so \(x_n^{(k)}\) converges to some \(x_n\in X_n\) because \(X_n\) is closed. Given \(\varepsilon>0\), choose \(k_0\) such that

For finite \(F\) and \(|a_n|\leq1\), pass \(\ell\to\infty\) in the finite sum to obtain

Hence \(\|x^{(k)}-x\|_u\leq\varepsilon\) once \(x\) is known to be unconditionally summable.

Choose such a \(k\geq k_0\). Since \(x^{(k)}\) is unconditionally summable, there is a finite \(F_0\) such that every finite \(F\subseteq\mathbb N\setminus F_0\) and every \(|a_n|\leq1\) satisfy

The preceding uniform estimate then gives

Thus the tails of \(\sum_nx_n\) are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion. Therefore \(x\in\bigoplus^uX_n\), and the earlier estimate proves \(x^{(k)}\to x\) in \(\|\cdot\|_u\).

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Proof of Theorem 7.16

Each \(P_nX={\operatorname{Ker}}(I-P_n)\) is closed. For \(x\in X\),

For \(y=(y_n)\in\bigoplus^uP_nX\), define

Since the finite partial sums with every coefficient equal to one are among those used in \(\|y\|_u\),

The assumed expansion gives \(\mathcal S_{{\mathcal P}}\mathcal A_{{\mathcal P}}x=x\). Conversely, boundedness of \(P_m\) permits termwise application to the unconditionally convergent series:

Thus \(\mathcal A_{{\mathcal P}}\mathcal S_{{\mathcal P}}=I\). Completeness of the coordinate space is Lemma 7.15.

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Proof of Theorem 8.2

Insert the finite packet identity on both sides:

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Proof of Theorem 8.3

Insert \(I=\sum_\beta P_\beta\) between \(S\) and \(T\):

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Proof of Corollary 8.4

Induct on \(m\) using Theorem 8.3. For a product of different operators, use the same induction with the specified operator at each step.

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Proof of Corollary 8.5

Because \(P_\alpha^*=P_\alpha\),

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Proof of Corollary 8.6

By definition,

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Proof of Theorem 8.7

Apply \(Q_\gamma\) on the left and \(P_\alpha\) on the right. Inserting \(I_X=\sum_\beta P_\beta\) gives

Inserting \(I_Y=\sum_\delta Q_\delta\) gives

Thus \(UT=SU\) implies every block equation. Conversely, if every block equation holds, then all blocks of \(UT-SU\) vanish. Exact operator reconstruction yields \(UT-SU=0\).

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Proof of Theorem 8.8

Injectivity follows from exact reconstruction. Given arbitrary linear maps \(B_{\beta\alpha}:X_\alpha\to X_\beta\), define

The packet decomposition is unique and the index set is finite, so this defines a linear operator whose blocks are the prescribed maps. Thus \(\Theta_{{\mathcal P}}\) is surjective. Compatibility with multiplication is Theorem 8.3.

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Proof of Corollary 8.9

For every target packet \(\beta\),

If every block in this sum is zero, then the coordinate with index \(\beta\) vanishes. This proves the inclusion. Equality is exactly the assertion that the displayed active sum is nonzero for every target in the set on the right. The final sufficient condition leaves at most one nonzero summand at each target, and kernel avoidance makes that summand nonzero.

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Proof of Corollary 8.10

Take dimensions in the block algebra isomorphism and use \(\dim{\operatorname{Hom}}(X_\alpha,X_\beta)=d_\alpha d_\beta\).

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Proof of Proposition 8.11

Pairwise annihilation gives \(\Delta_{{\mathcal P}}^2=\Delta_{{\mathcal P}}\) and identifies its range with diagonal block families. If \(T\) commutes with all \(P_\alpha\), then for \(\beta\neq\alpha\),

Thus (i) implies (ii). Exact reconstruction shows (ii) implies (iii). If \(T=\sum_\alpha P_\alpha TP_\alpha\), then

so (iii) implies (i).

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Proof of Theorem 8.13

The identity has diagonal blocks \(I_{X_\alpha}\) and no blocks outside the diagonal, so it belongs to \(\mathfrak M_R\) exactly when \((\alpha,\alpha)\in R\) for every \(\alpha\).

Assume \(R\) is transitive. If a summand

in the \((\gamma,\alpha)\) block of \(ST\) is nonzero, then \((\gamma,\beta)\) and \((\beta,\alpha)\) lie in \(R\). Transitivity gives \((\gamma,\alpha)\in R\), so every forbidden block of \(ST\) vanishes.

Conversely, suppose transitivity fails. Choose \(\alpha,\beta,\gamma\) with

Choose nonzero vectors \(u\in X_\alpha\), \(v\in X_\beta\), and \(w\in X_\gamma\). Choose linear functionals \(f\in X_\alpha^*\) and \(g\in X_\beta^*\) with \(f(u)=g(v)=1\). Let \(T\) have the sole block \(x\mapsto f(x)v\) from \(X_\alpha\) to \(X_\beta\), and let \(S\) have the sole block \(y\mapsto g(y)w\) from \(X_\beta\) to \(X_\gamma\). Then \(S,T\in\mathfrak M_R\), but \(ST\) has a nonzero forbidden \((\gamma,\alpha)\) block. This proves necessity. Combining reflexivity and transitivity gives (iii).

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Proof of Corollary 8.14

Adjoint reversal sends an allowed \((\beta,\alpha)\) block to an \((\alpha,\beta)\) block. Symmetry is therefore sufficient. If symmetry fails, choose a nonzero map of rank one in an allowed nonsymmetric block. Its adjoint occupies the forbidden reverse block, proving necessity. A reflexive, transitive, symmetric relation is an equivalence relation.

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Proof of Theorem 8.15

A nonzero block in \(\mathfrak N_R\) moves from one equivalence class to a strictly larger class in the quotient partial order. The block multiplication law shows that products concatenate such moves. Therefore left or right multiplication by an element of \(\mathfrak M_R\) preserves the property of having no blocks within a single class, so \(\mathfrak N_R\) is an ideal on both sides. A product of \(h+1\) elements of \(\mathfrak N_R\) would require a strict chain of \(h+1\) steps in \(\overline I\), which does not exist. This proves nilpotence.

Let \(P_C=\sum_{\alpha\in C}P_\alpha\). Define

If an intermediate class \(D\) contributes to \(P_CSTP_C\), then the preorder contains transfers \(C\to D\) and \(D\to C\), forcing \(C=D\) in the quotient poset. Hence

so \(\rho\) is an algebra homomorphism. Every collection of endomorphisms of the \(X_C\) defines an allowed operator that is block diagonal with respect to the classes, so \(\rho\) is surjective. Its kernel is exactly \(\mathfrak N_R\). The first isomorphism theorem gives the quotient formula.

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Proof of Theorem 9.2

It is enough to treat a word in the generators and packet projectors. Insert

between consecutive nonprojector generators. Every resulting \((\beta,\alpha)\) block is a finite sum of products

A nonzero product requires every consecutive block to be nonzero, hence gives a directed path

Packet projectors only insert stationary steps. Thus a word has no block from \(\alpha\) to an unreachable \(\beta\). Linear combinations preserve the same zero pattern.

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Proof of Proposition 9.3

For \(s\in S\), write \(s=\sum_{\alpha\in I_0}P_\alpha s\). If \(\beta\) is unreachable from \(I_0\), then for every \(A\in{\mathcal A}\),

by Theorem 9.2. The same holds for every finite linear combination of orbit vectors.

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Proof of Proposition 9.5

Every packet pure generating family is an unrestricted generating family, which gives the first inequality. For each \(\alpha\), choose a minimal generating family \(S_\alpha\subseteq X_\alpha\) for the corner action \(P_\alpha A P_\alpha\). The union \(S=\bigcup_\alpha S_\alpha\) is packet pure. Since every \(x\in X\) has the unique decomposition \(x=\sum_\alpha x_\alpha\) and every \(x_\alpha\) lies in \((P_\alpha A P_\alpha)S_\alpha\subseteq AS\), the union generates \(X\). Its cardinality is at most the displayed sum.

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Proof of Theorem 9.7

Let \(S\) be a packet pure generating family for \(X\). Discard zero seeds, which do not change the generated hull, and fix a source component \(C\). Seeds outside \(C\) cannot reach \(C\) by Proposition 9.3. Moreover, for \(s\in X_C\) and \(A\in{\mathcal A}\),

Thus the seeds of \(S\) lying in \(X_C\) must generate \(X_C\) under \({\mathcal A}_C\), and there are at least \(r_C\) of them. Summing over disjoint source components gives

For the reverse inequality, choose for each source component \(C\) a packet pure family \(S_C\) of size \(r_C\) generating \(X_C\) under \({\mathcal A}_C\), and put \(S=\bigcup_CS_C\). Then every source component lies in \({\mathcal A} S\). Traverse the finite condensation graph in a topological order. Suppose all components preceding a component \(D\) that is not a source have already been shown to lie in \({\mathcal A} S\). Choose an incoming condensation edge from such a component to \(D\), represented by a graph edge \(\alpha\to\beta\) with \(\alpha\) already generated and \(\beta\in D\). Surjectivity along edges gives a pure block

Because \(X_\alpha\subseteq{\mathcal A} S\) and \({\mathcal A} S\) is invariant under \({\mathcal A}\), surjectivity implies \(X_\beta\subseteq{\mathcal A} S\). For any other vertex \(\gamma\in D\), choose a directed path inside the strongly connected component from \(\beta\) to \(\gamma\). Successive surjective edge blocks propagate the full packet space along the path, so \(X_\gamma\subseteq{\mathcal A} S\). Hence \(X_D\subseteq{\mathcal A} S\). Induction over the condensation graph gives \(X={\mathcal A} S\), proving the upper bound.

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Proof of Corollary 9.9

Fix a source component \(C\) and a vertex \(\alpha\in C\). Choose a cyclic vector \(s\in X_\alpha\) for \(P_\alpha{\mathcal A} P_\alpha\). It generates all of \(X_\alpha\). Strong connectivity and surjectivity along edges then propagate the whole packet space to every vertex of \(C\). Thus \(r_C=1\) for every source component. Apply Theorem 9.7.

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