C Peripheral Channel Structure and Transfer-Operator Gaps: Supporting Results
This appendix supports Chapter 7. It proves the algebraic mixed-transfer identities, the Frobenius-norm apparatus behind the eigenvalue bounds, the peripheral intertwiners behind the gauge-rigidity theorems, the Banach-algebra convergence facts behind the overlap-decay theorems, the rank-one Perron projection behind the primitive overlap limit, and the cyclic-decomposition machinery that removes peripheral periodicity.
C.1 Mixed-transfer powers and overlap traces
This section proves the self-transfer identity, the word expansion of the mixed transfer operator, and the trace identities used in Theorem 7.3.1 and Theorem 7.3.2. Recall the mixed transfer operator \(F_{AB}\) of Definition 7.2.1 and its rectangular form \(F_{AB}^{\mathrm{rect}}\) of Definition 7.2.2.
Setting \(B=A\) recovers the ordinary transfer map: \(F_{AA}=\mathcal{E}_A\).
Substituting \(B=A\) into (??) gives \(F_{AA}(X)=\sum _iA^iX(A^i)^\dagger =\mathcal{E}_A(X)\) by (??).
For every \(X\in M_{D}(\mathbb {C})\) and \(N\geq 0\),
where \(A^\sigma =A^{\sigma _1}\cdots A^{\sigma _N}\) denotes the word evaluation of Definition 2.1.2.
Induct on \(N\). The base case is \(F_{AB}^0(X)=X\). For the inductive step, apply \(F_{AB}\) to each summand in (??) and reindex using \(\{ 0,\ldots ,d{-}1\} ^{N+1}\cong \{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} ^N\).
For any linear endomorphism \(T\) on \(M_{D_1\times D_2}(\mathbb {C})\), the operator trace expands as
where \(E_{pq}\in M_{D_1\times D_2}(\mathbb {C})\) has entry \(1\) in position \((p,q)\) and zero elsewhere.
The operator trace is \(\operatorname{Tr}(T)=\sum _{p,q}\operatorname{tr}(T(E_{pq})E_{qp})\), and \(\operatorname{tr}(ME_{qp})=M_{pq}\).
Theorem 7.3.1 and Theorem 7.3.2 combine Lemma C.1.2 and Lemma C.1.3 the same way, over the square and rectangular matrix-unit bases respectively: expanding \(\operatorname{Tr}(T)\) by (??), applying (??) to each \(T(E_{pq})\) for \(T=F_{AB}^N\) or \(T=(F_{AB}^{\mathrm{rect}})^N\), and reassembling the sums over matrix-unit indices gives
C.2 Frobenius estimates and eigenvalue bounds
This section proves the trace positivity fact (Lemma C.2.1) and the detailed Frobenius-norm estimate behind the rectangular eigenvalue bound (Theorem 7.4.2), of which Theorem 7.4.1 is the equal-bond-dimension case \(D_1=D_2=D\). The estimate uses the Frobenius (Hilbert–Schmidt) norm \(\| {\cdot }\| _F\) on rectangular matrices and an isometric embedding into Euclidean space.
If \(A,B\geq 0\), then \(\operatorname{tr}(AB)\geq 0\).
Write \(B=U\Lambda U^\dagger \) with \(U\) unitary and \(\Lambda \) diagonal. Then
because \(U^\dagger AU\geq 0\) and \(\Lambda _{ii}\geq 0\) for every \(i\).
For a possibly rectangular matrix \(X\in M_{m\times n}(\mathbb {C})\), its Frobenius norm squared is
One has \(\| X\| _F^2=\operatorname{Re}\operatorname{tr}(X^\dagger X)\).
Expand the matrix product and trace entry by entry.
Flattening matrix entries defines an isometric embedding \(\operatorname{vec}:M_{m\times n}(\mathbb {C})\to \mathbb {C}^{mn}\) with respect to the Frobenius norm: \(\| \operatorname{vec}(X)\| ^2=\| X\| _F^2\).
One has \(\| \operatorname{vec}(X)\| ^2=\| X\| _F^2\).
Both sides equal \(\sum _{i,j}\lVert X_{ij}\rVert ^2\): the left side by the Euclidean entry-norm formula and the right side by (??).
The following internal estimate is folded into a private supporting lemma in the formalization and is recorded here, untagged, purely as an exposition step toward Theorem 7.4.2.
Let \(A\) and \(B\) be normalized MPS tensors of bond dimensions \(D_1\geq 1\) and \(D_2\geq 1\). For every \(X\in M_{D_1\times D_2}(\mathbb {C})\) and \(n\geq 0\),
The word expansion (??) writes \((F_{AB}^{\mathrm{rect}})^n(X)=\sum _\sigma A^\sigma (X(B^\sigma )^\dagger )\), a sum over words \(\sigma \) of length \(n\). Embedding into Euclidean space by \(\operatorname{vec}\) (Definition C.2.4), the triangle inequality and the Frobenius submultiplicativity \(\| MN\| _F\leq \| M\| _F\| N\| _F\) give
Cauchy–Schwarz over the word index bounds the right side by \(\left(\sum _\sigma \| A^\sigma \| _F^2\right)^{1/2} \left(\sum _\sigma \| X(B^\sigma )^\dagger \| _F^2\right)^{1/2}\). The trace-preserving normalization of \(A\) gives \(\sum _\sigma (A^\sigma )^\dagger A^\sigma =\mathbb {1}_{D_1}\), so \(\sum _\sigma \| A^\sigma \| _F^2=D_1\); the same normalization for \(B\) gives \(\sum _\sigma (B^\sigma )^\dagger B^\sigma =\mathbb {1}_{D_2}\), which forces the exact identity \(\sum _\sigma \| X(B^\sigma )^\dagger \| _F^2=\| X\| _F^2\) (both sides equal \(\operatorname{Re}\operatorname{tr}(X^\dagger X)\) after expanding by the same normalization). Hence \(\| (F_{AB}^{\mathrm{rect}})^n(X)\| _F\leq \sqrt{D_1} \| X\| _F\leq D_1\, \| X\| _F\), giving (??).
Theorem 7.4.2 follows directly from Lemma C.2.6: if \(F_{AB}^{\mathrm{rect}}(X)=\mu X\) with \(X\neq 0\), iterating gives \((F_{AB}^{\mathrm{rect}})^n(X)=\mu ^nX\), and Lemma C.2.6 gives \(|\mu |^{2n}\| X\| _F^2\leq D_1^2\| X\| _F^2\) for every \(n\); since \(\| X\| _F{\gt}0\), if \(|\mu |{\gt}1\) then \(|\mu |^{2n}\to \infty \), contradicting the uniform bound, so \(|\mu |\leq 1\).
For normalized tensors of bond dimensions \(D_1\) and \(D_2\), \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}})\leq 1\).
On a finite-dimensional space every spectral value is an eigenvalue, and each satisfies \(|\mu |\leq 1\) by Theorem 7.4.2.
C.3 Peripheral intertwiners and gauge rigidity
This section proves the intertwining relations behind Theorem 7.7.1 and Theorem 7.7.4: transporting a modulus-one mixed-transfer eigenvector through separate unital gauges produces Kraus-level intertwiners, and these intertwiners force equal bond dimension once they are invertible.
Let \(A\) be a normalized tensor of bond dimension \(D_1\) and \(B\) a normalized tensor of bond dimension \(D_2\). Suppose \(\rho _A,\rho _B\) are fixed points of the respective transfer maps and \(S_A,S_B\) are invertible matrices with \(S_AS_A^\dagger =\rho _A\) and \(S_BS_B^\dagger =\rho _B\) (for instance the positive definite Perron–Frobenius fixed points of Theorem 6.4.2 and their square roots, as in the right-canonical gauge of Theorem 6.5.1). Gauge to unital families \(A'^i=S_A^{-1}A^iS_A\) and \(B'^i=S_B^{-1}B^iS_B\). If \(X\in M_{D_1\times D_2}(\mathbb {C})\setminus \{ 0\} \) satisfies \(F_{AB}^{\mathrm{rect}}(X)=\mu X\) with \(|\mu |=1\), then \(X'=S_A^{-1}X(S_B^\dagger )^{-1}\) is nonzero, the families \(A'\) and \(B'\) are unital, and, for every \(i\),
Embed the gauged families and transported eigenvector into the block matrices
on \(M_{D_1+D_2}(\mathbb {C})\); the matrices \(\{ C^i\} \) form a unital Kraus family, and \(E_C(Y)=\mu Y\). The positive definite fixed points of the adjoint maps give a faithful weighted trace for the block map. Applying the Kadison–Schwarz inequality to \(Y\) and pairing its positive semidefinite gap with this fixed point shows that the gap vanishes. The Kraus relation of Theorem 5.2.2 then gives \(Y(C^i)^\dagger =\mu (C^i)^\dagger Y\); applying the same argument to the complementary product gives \(C^iY=\mu YC^i\). Expanding these block relations gives (??).
Let \(A\) and \(B\) be tensors of the same bond dimension. Suppose invertible gauges take them to \(A'\) and \(B'\), and suppose there are an invertible matrix \(X'\) and a scalar \(\mu \) with \(|\mu |=1\) such that \(A'^iX'=\mu X'B'^i\) for every physical index \(i\). Then \(A\) and \(B\) are gauge-phase equivalent.
Compose the two gauge matrices with the intertwiner to obtain one invertible matrix conjugating \(A\) to a scalar multiple of \(B\).
Combining these two lemmas proves Theorem 7.7.1: Lemma C.3.1 supplies the intertwining relations (??) for the separately gauged, unital families. The Kadison–Schwarz equality used inside its own proof makes \(X'X'^\dagger \) and \(X'^\dagger X'\) nonzero positive semidefinite fixed points of the gauged transfer maps; under irreducibility, Theorem 6.2.4 upgrades these to positive definite matrices, so the transported intertwiner \(X'\) is invertible. Lemma C.3.2 then converts the invertible intertwiner, together with the two gauge matrices, into a single invertible matrix realizing the gauge-phase relation \(B^i=\overline\mu XA^iX^{-1}\) via the Skolem–Noether theorem (Theorem 3.2.4).
Lemma 7.5.1 gives the companion kernel-invariance argument, independent of irreducibility: if a modulus-one rectangular intertwiner (??) exists between injective tensors of bond dimensions \(D_1\) and \(D_2\), then the first relation shows that \(\ker (X)\) is invariant under every \((B^i)^\dagger \); since the \(\{ B^i\} \) span the full matrix algebra, so do the \(\{ (B^i)^\dagger \} \), so \(\ker (X)\) is trivial and \(D_2\leq D_1\). Taking adjoints of the second relation and applying the same argument to \(X^\dagger \) shows that \(\ker (X^\dagger )\) is invariant under every \((A^i)^\dagger \) and hence trivial, giving \(D_1\leq D_2\). Combined with Lemma C.3.1, this proves \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}}){\lt}1\) whenever \(A\) and \(B\) are injective tensors of unequal bond dimension, which is exactly Theorem 7.5.2.
C.4 Spectral-radius decay and overlap limits
This section collects the Banach-algebra convergence facts underlying Theorem 7.6.2, Theorem 7.6.3, and the block-separation consequences of the transfer-operator gap.
Let \(a\) be an element of a complex Banach algebra with \(\rho _{\operatorname{spec}}(a){\lt}1\). Then \(a^n\to 0\) as \(n\to \infty \).
Choose \(r\) with \(\rho _{\operatorname{spec}}(a){\lt}r{\lt}1\). The Gelfand formula \(\| a^n\| ^{1/n}\to \rho _{\operatorname{spec}}(a)\) gives \(\| a^n\| \leq r^n\) for all sufficiently large \(n\), and \(r^n\to 0\).
Theorem 7.6.2 combines the gap \(\rho _{\operatorname{spec}}(F_{AB}){\lt}1\) of Theorem 7.4.6 with Lemma C.4.1 to give \(\| F_{AB}^n\| _{\mathrm{op}}\to 0\), hence \(F_{AB}^n(X)\to 0\) for every \(X\). Since \(\operatorname{Tr}\) is continuous on the finite-dimensional matrix-endomorphism algebra, this gives \(\operatorname{Tr}(F_{AB}^N)\to 0\) term by term over the finitely many matrix-unit entries of (??); combined with the overlap–trace identity (??), this proves Theorem 7.6.3.
If \(A\) and \(B\) are injective normalized tensors that are not gauge-phase equivalent, then \(\operatorname{tr}(F_{AB}^N(X))\to 0\) as \(N\to \infty \) for every \(X\in M_{D}(\mathbb {C})\).
Theorem 7.6.2 gives \(F_{AB}^N(X)\to 0\) in operator norm, and continuity of the trace gives the conclusion.
If \(\rho \) is a fixed point of \(\mathcal{E}_A\), then \(\operatorname{tr}(\mathcal{E}_A^N(\rho ))=\operatorname{tr}(\rho )\) for every \(N\geq 0\). In particular, this applies to the distinguished Perron–Frobenius fixed point from Theorem 6.4.2.
Since \(\mathcal{E}_A(\rho )=\rho \), one has \(\mathcal{E}_A^N(\rho )=\rho \) for every \(N\). Taking the trace proves the claim.
C.5 Rank-one Perron projection
This section proves the rank-one Perron limit (Theorem 7.11.1) underlying the primitive overlap convergence (Theorem 7.11.2). Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be trace-preserving with a fixed point \(\rho \neq 0\), \(\operatorname{tr}(\rho )\neq 0\). Write \(P:=P_\rho \) for the fixed-point projection of Definition 4.9.1 (Chapter 4),
and \(N:=E-P\) for the complementary part. The power decomposition (Theorem 4.9.2, Chapter 4) gives \(E^n=P+N^n\) for \(n\geq 1\).
The fixed-point projection \(P\) has operator trace \(\operatorname{Tr}(P)=1\).
\(P\) has the rank-one form \(P(X)=f(X)\rho \) for the linear functional \(f(X)=\operatorname{tr}(X)/\operatorname{tr}(\rho )\). The operator trace of a rank-one endomorphism \(X\mapsto f(X)\rho \) equals \(f(\rho )\), and \(f(\rho )=\operatorname{tr}(\rho )/\operatorname{tr}(\rho )=1\).
Under the hypothesis \(\rho _{\operatorname{spec}}(N){\lt}1\) of Theorem 7.11.1, Lemma C.4.1 gives \(N^n\to 0\), so \(\operatorname{Tr}(E^n)=\operatorname{Tr}(P)+\operatorname{Tr}(N^n)\to \operatorname{Tr}(P)=1\) by Lemma C.5.1. This is exactly the mechanism recorded in Theorem 7.11.1.
When \(E=\mathcal{E}_A\) is the transfer map of a normalized MPS tensor and \(\rho \) is its Perron–Frobenius fixed point (Theorem 6.4.2, Chapter 6), the overlap–trace identity (??) with \(A=B\) gives \(O_{AA}(N)=\operatorname{Tr}(\mathcal{E}_A^N)\), so Theorem 7.11.1 yields the primitive self-overlap limit of Theorem 7.11.2. The hypothesis \(\rho _{\operatorname{spec}}(N){\lt}1\) is exactly the complementary transfer-map gap that the complementary spectral-radius theorem for primitive maps (Theorem 4.11.3, Chapter 4) supplies: for a primitive \(E\) all eigenvalues bounded by \(1\) in modulus and no trace-zero fixed point, every eigenvalue of \(N=E-P\) has modulus strictly less than \(1\), and since \(M_{D}(\mathbb {C})\) is finite-dimensional this bounds the spectral radius \(\rho _{\operatorname{spec}}(N)\) itself.
Let \(A\) be an irreducible normalized MPS tensor whose transfer map \(\mathcal{E}_A\) is primitive. Then there is a nonzero positive semidefinite fixed point \(\rho \) of \(\mathcal{E}_A\) with \(\operatorname{tr}(\rho )\neq 0\) such that, for \(P:=P_\rho \) and \(N:=\mathcal{E}_A-P\), \(\rho _{\operatorname{spec}}(N){\lt}1\).
The channel fixed-point existence theorem (Theorem 6.4.1) supplies a nonzero positive semidefinite fixed point \(\rho \). To see that every trace-zero fixed point of \(\mathcal{E}_A\) vanishes, let \(X\) be a (not necessarily Hermitian) fixed point with \(\operatorname{tr}(X)=0\). Since \(\mathcal{E}_A(X)^\dagger =\mathcal{E}_A(X^\dagger )\), the matrices \(H_1:=X+X^\dagger \) and \(H_2:=\mathrm{i}(X-X^\dagger )\) are Hermitian fixed points of \(\mathcal{E}_A\) with \(\operatorname{tr}(H_1)=\operatorname{tr}(X)+\overline{\operatorname{tr}(X)}=0\) and \(\operatorname{tr}(H_2)=\mathrm{i}(\operatorname{tr}(X)-\overline{\operatorname{tr}(X)})=0\). A nonzero trace-zero Hermitian fixed point would decompose, by the positive-semidefinite splitting of Hermitian fixed points ( [ Wol12 , Proposition 6.8 ] ), into two nonzero positive semidefinite fixed points; irreducibility of \(\mathcal{E}_A\) forces both to be proportional to \(\rho \) (Theorem 6.3.1), and their difference having zero trace forces the proportionality constants to agree, so the Hermitian fixed point itself vanishes. Applying this to \(H_1\) and \(H_2\) gives \(H_1=H_2=0\), and \(X=\tfrac 12(H_1-\mathrm{i}H_2)=0\). The complementary spectral-radius theorem for primitive maps (Theorem 4.11.3) then applies with this trace-zero-fixed-point triviality and the eigenvalue bound \(|\mu |\le 1\) from the trace-preserving normalization, giving \(\rho _{\operatorname{spec}}(N){\lt}1\).
This concretely instantiates the hypothesis of Theorem 7.11.1 and, via Theorem 7.11.2, gives the primitive self-overlap limit for every irreducible peripherally primitive normalized MPS tensor.
C.6 Periodicity removal
This section proves the cyclic-decomposition machinery of Theorem 7.9.1.1 and the divisibility statements of Theorem 7.9.1 and Theorem 7.9.2: normalizing a peripheral eigenvector to a unitary, forming its discrete Fourier (cyclic) spectral projections, and tracking how the channel shifts the resulting corners.
Let \(E\) be an irreducible unital Kraus map on \(M_{D}(\mathbb {C})\) with a positive-definite adjoint fixed point. If \(X \ne 0\) satisfies \(E(X) = \mu X\) with \(|\mu | = 1\), then \(X\) is invertible.
By the Kadison–Schwarz inequality (??), the gap \(G=E(X^\dagger X)-E(X)^\dagger E(X)\) is positive semidefinite. Pairing \(G\) with the positive-definite adjoint fixed point and using \(E(X)=\mu X\) and \(|\mu |=1\) gives weighted trace zero, hence \(G=0\). Therefore \(E(X^\dagger X)=E(X)^\dagger E(X)=X^\dagger X\), so \(X^\dagger X\) is a nonzero positive semidefinite fixed point. Irreducibility upgrades it to a positive-definite, hence invertible, matrix, so \(\det (X)\ne 0\).
Let \(E\) be an irreducible unital Kraus map on \(M_{D}(\mathbb {C})\) with a positive-definite adjoint fixed point. If \(\mu , \nu \) are peripheral eigenvalues of \(E\), then so is \(\mu \nu \).
Take eigenvectors \(X\) and \(Y\) for \(\mu \) and \(\nu \), respectively. The Kadison–Schwarz equality at \(X\) places \(X\) in the multiplicative domain. Thus the right multiplicative identity (??) gives \(E(YX)=E(Y)E(X)=(\nu \mu )YX\). Since \(X\) and \(Y\) are invertible by Lemma C.6.1, \(YX\ne 0\), and \(|\mu \nu |=1\).
If \(E\) is irreducible and unital, then every fixed point of \(E\) is a scalar multiple of \(\mathbb {1}\).
If \(E(X)=X\), then the Hermitian matrices \(X+X^\dagger \) and \(\mathrm{i}(X-X^\dagger )\) are also fixed by \(E\). The positive-semidefinite fixed-point uniqueness theorem gives \(X+X^\dagger =a\mathbb {1}\) and \(\mathrm{i}(X-X^\dagger )=b\mathbb {1}\), hence \(X=\frac{1}{2}(a-\mathrm{i}b)\mathbb {1}\).
For an irreducible unital Schwarz map with faithful adjoint fixed point, each peripheral eigenvalue admits a unitary eigenvector.
For an eigenvector \(X\) at a peripheral eigenvalue \(\gamma \), the Kadison–Schwarz equality gives \(E(X^\dagger X)=E(X)^\dagger E(X)=X^\dagger X\), so \(X^\dagger X\) is a nonzero positive semidefinite fixed point. Irreducible uniqueness of the positive semidefinite fixed point makes it a positive scalar \(c\mathbb {1}\); rescaling \(X\) by \(c^{-1/2}\) produces a unitary eigenvector.
If \(E(U)=\mu U\) with \(|\mu |=1\) and \(U\) unitary, then \(E(U^n)=\mu ^nU^n\) for every \(n\).
The Kadison–Schwarz equality at the unitary eigenvector places \(U\) in the multiplicative domain, so the right multiplicative identity (??) gives \(E(U^n)=E(U)^n=\mu ^nU^n\) by induction on \(n\).
Peripheral unitary eigenvectors can be normalized compatibly with the faithful fixed-point state.
Given a unitary eigenvector \(U\) for a primitive \(m\)-th root \(\gamma \), Lemma C.6.5 gives \(E(U^m)=U^m\), a fixed point that is scalar by Lemma C.6.3: \(U^m=\alpha \mathbb {1}\). Unitarity of \(U^m\) forces \(|\alpha |=1\); multiplying \(U\) by an \(m\)-th root \(\beta \) of \(\alpha ^{-1}\) produces \((\beta U)^m=1\) while preserving the eigenvector equation.
A primitive \(m\)-th peripheral root yields \(m\) orthogonal projections summing to \(\mathbb {1}\) and cyclically permuted by the channel.
Given a unitary \(U\) of order \(m\) with \(E(U)=\gamma U\) for a primitive \(m\)-th root \(\gamma \), the Fourier projections \(P_k:=m^{-1}\sum _{j=0}^{m-1}\overline\gamma ^{kj}U^j\) are mutually orthogonal and sum to \(\mathbb {1}\) by discrete Fourier orthogonality of the characters \(j\mapsto \gamma ^{kj}\); since \(E(U^j)=\gamma ^jU^j\) (Lemma C.6.5), each Fourier mode is shifted by one step, giving \(E(P_{k+1})=P_k\).
Combining these lemmas proves Theorem 7.9.1.1: Lemma C.6.6 supplies a unitary generator of exact order \(m\) for a primitive \(m\)-th peripheral root, and Lemma C.6.7 builds the \(m\) Fourier projections out of its powers, using Lemma C.6.5 to identify how the channel shifts them.
C.6.1 Cyclic corners and restricted primitivity
For an orthogonal projection \(P\) and linear map \(E\), the corner \(PM_{D}(\mathbb {C})P\) is preserved if \(E(PXP)=PE(PXP)P\) for every \(X\).
The corner subspace associated with \(P\) is \(\{ PXP:X\in M_{D}(\mathbb {C})\} \).
The restriction of \(E\) to an invariant corner \(PM_{D}(\mathbb {C})P\).
Irreducibility of the corner-restricted map.
The rank of an orthogonal projection \(P\in M_{D}(\mathbb {C})\), equivalently the dimension of its range.
Let \(P\in M_{D}(\mathbb {C})\) be an orthogonal projection of rank \(n\). The corner algebra \(PM_{D}(\mathbb {C})P\) is linearly isomorphic to the full matrix algebra \(M_{n}(\mathbb {C})\). The isomorphism is built from the spectral diagonalisation of \(P\) by conjugating the top-left \(n\times n\) block by the unitary diagonalising \(P\).
The rank of an orthogonal projection \(P\) equals its trace: \(n=\operatorname{tr}(P)\).
The eigenvalues of an orthogonal (Hermitian idempotent) projection are \(0\) or \(1\): the eigenvalues \(\lambda \) satisfy \(\lambda ^2=\lambda \) from \(P^2=P\). The trace is the sum of the eigenvalues, which counts the eigenvalue-\(1\) multiplicities, i.e. the rank \(n\).
Appropriate powers of the channel preserve each sector corner.
Induction on the exponent, using the left- and right-multiplicative domain identities on each sector projection \(P_k\), shows \(T^n(P_{k+n\bmod m}XP_{k+n\bmod m})=P_k\, T^n(X)\, P_k\); specializing to \(n=m\), where \(P_{k+m}=P_k\), gives that \(T^m\) preserves the corner \(P_kM_{D}(\mathbb {C})P_k\).
Let \(P\in M_{D}(\mathbb {C})\) be a nonzero idempotent and let \(E\) be a linear map on \(M_{D}(\mathbb {C})\) that preserves the corner \(PM_{D}(\mathbb {C})P\), is primitive, and fixes the corner projection, \(E(P)=P\). Then the restriction of \(E\) to the corner \(PM_{D}(\mathbb {C})P\) is again primitive.
The corner projection \(P\) lies in its own corner and is fixed by the restriction, so \(1\) is a peripheral eigenvalue of the restriction. Conversely, every eigenvalue of the restriction of unit modulus arises from a nonzero corner element \(X\) with \(E(X)=\mu X\); viewing \(X\) as an element of \(M_{D}(\mathbb {C})\) exhibits \(\mu \) as a peripheral eigenvalue of \(E\). Primitivity of \(E\) forces \(\mu =1\), so the peripheral spectrum of the restriction is exactly \(\{ 1\} \) and the restriction is primitive.
Let \(T\) be irreducible, let \(P_0,\ldots ,P_{m-1}\) be orthogonal projections summing to \(\mathbb {1}\) and cyclically permuted by \(T\), and suppose that every orthogonal projection \(Q\) with \(QP_k=P_kQ=Q\) that is invariant under the corner restriction of \(T^m\) to \(P_kM_{D}(\mathbb {C})P_k\) admits an orthogonal projection \(R\) invariant under \(T\) on the full algebra with \(Q=0\iff R=0\) and \(Q=P_k\iff R=\mathbb {1}\). Then, for every \(k\), the restriction of \(T^m\) to the corner \(P_kM_{D}(\mathbb {C})P_k\) is irreducible.
Let \(Q\) be an invariant projection for the corner restriction of \(T^m\) on \(P_kM_{D}(\mathbb {C})P_k\). The orbit-sum hypothesis supplies an ambient invariant projection \(R\) for \(T\) with \(Q=0\iff R=0\) and \(Q=P_k\iff R=\mathbb {1}\). Irreducibility of \(T\) forces \(R=0\) or \(R=\mathbb {1}\), hence \(Q=0\) or \(Q=P_k\) by the two biconditionals.
Each sector restriction is primitive.
Every peripheral eigenvalue of \(T\) is an \(m\)-th root of unity, so the peripheral spectrum of \(T^m\) is exactly \(\{ 1\} \); combined with \(T^m(P_k)=P_k\), Lemma C.6.1.9 makes the corner restriction of \(T^m\) to \(P_kM_{D}(\mathbb {C})P_k\) primitive.
The channel raised to the period preserves every cyclic corner.
The same induction as in Lemma C.6.1.8, with the cyclic shift \(k\mapsto k+1\) replaced by a permutation \(\sigma \) of the sector index set, shows that \(T^{\mathrm{ord}(\sigma )}\) preserves each corner \(P_kM_{D}(\mathbb {C})P_k\).
C.6.2 Group structure and divisibility
The product of two peripheral eigenvalues is peripheral.
Peripheral eigenvalues admit unitary eigenvectors (Lemma C.6.4). If \(U_\alpha ,U_\beta \) are unitary eigenvectors for \(\alpha ,\beta \), the multiplicative-domain identity gives \(E(U_\alpha U_\beta )=E(U_\alpha )E(U_\beta )=\alpha \beta \, U_\alpha U_\beta \), and \(U_\alpha U_\beta \neq 0\) since both factors are unitary.
The inverse of a peripheral eigenvalue is peripheral.
For a unitary eigenvector \(U_\alpha \) with \(|\alpha |=1\), conjugating the eigenvector equation gives \(E(U_\alpha ^\dagger )=\overline\alpha \, U_\alpha ^\dagger =\alpha ^{-1}U_\alpha ^\dagger \), and \(U_\alpha ^\dagger \neq 0\).
Theorem 7.9.1.3 follows the same pattern: given a peripheral unitary eigenvector \(U\) for \(\gamma \) (Lemma C.6.4), the multiplicative-domain identity gives \(E(XU^\dagger )=XU^\dagger \) for any \(\gamma \)-eigenvector \(X\), so the scalar fixed-point lemma for irreducible unital maps (Lemma C.6.3) gives \(XU^\dagger =c\mathbb {1}\), hence \(X=cU\) and the eigenspace is one-dimensional.
The divisibility statements of Theorem 7.9.1 and Theorem 7.9.2 follow from Lemma C.6.7 and Theorem C.6.1.7: the \(m\) cyclic Fourier projections \(P_0,\ldots ,P_{m-1}\) sum to \(\mathbb {1}\) and are cyclically permuted by \(E\), so trace preservation gives \(\operatorname{tr}(P_{k+1})=\operatorname{tr}(P_k)\) for every \(k\), hence \(D=\operatorname{tr}(\mathbb {1})=\sum _{k=0}^{m-1}\operatorname{tr}(P_k)=m\, \operatorname{tr}(P_0)\). Since the trace of an orthogonal projection is a natural number (Theorem C.6.1.7), \(m\mid D\).
This periodicity-removal chain — unitary normalization, Fourier projections, cyclic Kraus shift (Lemma 7.9.1.2), the blocked cyclic corners of this section, and restricted primitivity — is exactly the machinery behind the period-removing blocking theorems of Chapter 9: [ Wol12 , Theorem 6.6 ] supplies the cyclic peripheral spectrum of the adjoint transfer map, and Theorem 9.3.1.1 (Section 9.3.1) and Theorem 9.11.1 apply it to realize the cyclic-sector decomposition of a blocked irreducible tensor with rectangular support isometries.