Proof of Monotonicity of Odd Powers under a Tracial State

Each result cited is applied with the data of its own statement. Throughout, $\lambda$, $a$, $b$ and $k$ are as in the statement, $n=2k-1$ and $D=a-b$. Products of three or more polynomials are written without brackets, which is legitimate by the associativity $(pq)r=p(qr)$ of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra. Since $\lambda$ is linear, it commutes with finite sums in the complex vector space $\mathcal{P}_{m}$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §vector-space), by induction along the recursion of Finite Sum Notation in a Vector Space; the resulting finite sums in $\mathbb{C}$, regarded as a vector space over itself, are those of Finite Sum Notation in a Field, the two definitions prescribing the same recursion for the same addition of $\mathbb{C}$ and hence the same map by the uniqueness in Existence and Uniqueness of Iterates of a Binary Operation; and a finite sum in $\mathbb{C}$ of real numbers is real and equals their finite sum in $\mathbb{R}$, by induction along that recursion, since sums of real numbers formed in $\mathbb{C}$ and in $\mathbb{R}$ coincide by condition 1 of The Complex Numbers. Likewise the distributive laws of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra extend to finite sums. The results of Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples are applied with $d=m$ and the tracial state $\lambda$.

Exponents and summation indices are integers $\ge0$ in the sense of The Integers as a Subset of the Real Numbers: by claim 1 of Arithmetic, Order, Discreteness and Intervals of the Integers each is $0$ or $\iota(q)$ for some $q\in\mathbb{N}$, unique by claim 7 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, where $\iota$ is the canonical map of $\mathbb{R}$, and $\iota(q)$ is identified with $q$ (sums corresponding by claim 4 there); sums, differences and products of integers are integers by claim 2 of Arithmetic, Order, Discreteness and Intervals of the Integers; every subtraction written below ($r-1-s$ for $s\le r-1$, $2k-2-s$ for $s\le2k-2$, $k-1-u$ for $u\le k-1$, $k-2-u$ for $u\le k-2$) has a result $\ge0$, by claim 3 of Elementary Arithmetic in an Ordered Field; $c^{0}=1$ (Step 1); and for integers $0\le\alpha\le\beta$, $\sum_{s=\alpha}^{\beta}t_{s}$ denotes the finite sum $\sum_{j=1}^{\beta-\alpha+1}t_{\alpha+j-1}$ of Finite Sum Notation in a Vector Space, respectively Finite Sum Notation in a Field, that is, the sum of the terms with $s=\alpha,\alpha+1,\dots,\beta$ in increasing order after the shift $s=\alpha+j-1$ onto $[\beta-\alpha+1]$, these being exactly the integers $s$ with $\alpha\le s\le\beta$ by claim 3 of Arithmetic, Order, Discreteness and Intervals of the Integers.

Step 1 (Powers, including exponent $0$). For an integer $s\ge0$ let $w_{s}\in W_{1}$ be the empty word $\varnothing$ if $s=0$, and the word of length $s$ all of whose letters are $1$ if $s\ge1$. For $c\in\mathcal{P}_{m}$ put $c^{s}=c_{w_{s}}$, the product along $w_{s}$ for the $1$-tuple $(c)$ as in Substitution of Noncommutative Polynomials into the Variables §word-products; for $s\ge1$ this is the power of the statement, and $c^{0}=c_{\varnothing}=1$ by the same definition. Let $\sigma:\mathcal{P}_{1}\to\mathcal{P}_{m}$ be the substitution of $(c)$. Three facts are used.

(P1) $c^{s}c^{t}=c^{s+t}$ for all integers $s,t\ge0$. Indeed $w_{s}w_{t}=w_{s+t}$ by Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §concatenation (by the clauses for the empty word if $s=0$ or $t=0$; otherwise $w_{s}w_{t}$ is the word of length $s+t$ all of whose letters are $1$), and $c_{w_{s}w_{t}}=c_{w_{s}}c_{w_{t}}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values.

(P2) $c^{1}=c$. Indeed $w_{1}$ is the letter $(1)$, whose monomial is the variable $x_{1}$ of $\mathcal{P}_{1}$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials, and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values gives $c_{w_{1}}=\sigma(x_{w_{1}})=\sigma(x_{1})=c$.

(P3) If $c\in\mathcal{P}_{m,\mathrm{sa}}$, then $(c^{s})^{*}=c^{s}$. Indeed all letters of $w_{s}$ equal $1$, so $w_{s}^{\mathrm{rev}}=w_{s}$ by Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §reversal, hence $(x_{w_{s}})^{*}=x_{w_{s}}$ in $\mathcal{P}_{1}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint; as $c$ is self-adjoint, Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values give $(c^{s})^{*}=\sigma(x_{w_{s}})^{*}=\sigma\bigl((x_{w_{s}})^{*}\bigr)=\sigma(x_{w_{s}})=c^{s}$.

Moreover $D=a+(-1)b\in\mathcal{P}_{m,\mathrm{sa}}$, since $\mathcal{P}_{m,\mathrm{sa}}$ is closed under sums and real multiples by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint; so (P3) applies to $a$, $b$ and $D$. Finally $(pq)^{*}=q^{*}p^{*}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, and applying it twice, $(pqr)^{*}=r^{*}q^{*}p^{*}$ for all $p,q,r\in\mathcal{P}_{m}$.

Step 2 (Telescoping). For every $r\in\mathbb{N}$,

$$a^{r}-b^{r}=\sum_{s=0}^{r-1}a^{s}Db^{r-1-s},$$

the sum in $\mathcal{P}_{m}$ of the terms indexed by $s=0,\dots,r-1$ in increasing order (the finite sum of Finite Sum Notation in a Vector Space after the index shift $s\mapsto s+1$). We argue by induction on $r$. For $r=1$ the right side is $a^{0}Db^{0}=1D1=D=a-b=a^{1}-b^{1}$, by Step 1 and the unit law $1p=p1=p$ of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials. Assume the identity for $r$. By (P1) and (P2), $a^{r+1}=a^{r}a$ and $b^{r+1}=b^{r}b$, so by the distributive laws and the scalar clause $(cp)q=c(pq)=p(cq)$ of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra (with $c=-1$, differences $p-q$ being $p+(-1)q$)

$$a^{r+1}-b^{r+1}=a^{r}a-a^{r}b+a^{r}b-b^{r}b=a^{r}D+(a^{r}-b^{r})b .$$

By the induction hypothesis, distributivity over finite sums (with the scalar clause of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra for the coefficient $-1$), and (P2) and (P1) in the form $b^{r-1-s}b=b^{r-1-s}b^{1}=b^{r-s}$, the second summand equals $\sum_{s=0}^{r-1}a^{s}Db^{r-s}$, and $a^{r}D=a^{r}Db^{0}$ by the unit law. By the recursion of the finite sum, adding the term with index $s=r$ gives $a^{r+1}-b^{r+1}=\sum_{s=0}^{r}a^{s}Db^{r-s}$, the identity for $r+1$.

Step 3 (Expansion of the trace). Take $r=n=2k-1$ in Step 2 and multiply on the right by $D$. Distributivity and the linearity of $\lambda$ give

$$S:=\lambda\bigl((a^{n}-b^{n})D\bigr)=\sum_{s=0}^{2k-2}\tau_{s},\qquad\tau_{s}=\lambda\bigl(a^{s}Db^{2k-2-s}D\bigr).$$

For integers $0\le u\le k-1$ put $e_{u}=\tau_{2u}=\lambda\bigl(a^{2u}Db^{2(k-1-u)}D\bigr)$, and, when $k\ge2$, for integers $0\le u\le k-2$ put $o_{u}=\tau_{2u+1}=\lambda\bigl(a^{2u+1}Db^{2(k-2-u)+1}D\bigr)$, using $2k-2-(2u+1)=2(k-2-u)+1$. Thus $S$ is the sum of Sum over a Finite Index Set over the index set $I=\{0,\dots,2k-2\}$, computed with the bijection $[2k-1]\to I$, $j\mapsto j-1$ (the shift of the convention above). Let $I_{\mathrm{e}}\subseteq I$ be the set of even indices $2u$ ($0\le u\le k-1$) and, when $k\ge2$, $I_{\mathrm{o}}\subseteq I$ the set of odd indices $2u+1$ ($0\le u\le k-2$). The maps $[k]\to I_{\mathrm{e}}$, $u+1\mapsto2u$, and, when $k\ge2$, $[k-1]\to I_{\mathrm{o}}$, $u+1\mapsto2u+1$, are bijections, so by Sum over a Finite Index Set, computed with them, $\sum_{s\in I_{\mathrm{e}}}\tau_{s}=\sum_{u=0}^{k-1}e_{u}$ and $\sum_{s\in I_{\mathrm{o}}}\tau_{s}=\sum_{u=0}^{k-2}o_{u}$. If $k=1$, then $I=\{0\}=I_{\mathrm{e}}$; if $k\ge2$, then $I$ is the disjoint union of $I_{\mathrm{e}}$ and $I_{\mathrm{o}}$, each integer of $I$ being even or odd and none both. As a sum over a finite index set does not depend on the bijection used to enumerate it (Sum over a Finite Index Set), we obtain (for $k\ge2$ by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §disjoint-union)

$$S=\sum_{u=0}^{k-1}e_{u}\quad\text{if }k=1,\qquad S=\sum_{u=0}^{k-1}e_{u}+\sum_{u=0}^{k-2}o_{u}\quad\text{if }k\ge2.$$

Step 4 (Even terms). Let $0\le u\le k-1$ and $v=k-1-u$, and put $g=b^{v}Da^{u}$. By Step 1, $g^{*}=(a^{u})^{*}D^{*}(b^{v})^{*}=a^{u}Db^{v}$, and by (P1) $g^{*}g=a^{u}Db^{2v}Da^{u}$. By condition (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state with $p=a^{u}$ and $q=Db^{2v}Da^{u}$, then (P1), then condition (c) with $p=Db^{2v}D$ and $q=a^{2u}$,

$$\lambda(g^{*}g)=\lambda(Db^{2v}Da^{u}a^{u})=\lambda(Db^{2v}Da^{2u})=\lambda(a^{2u}Db^{2v}D)=e_{u}.$$

By condition (b) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, $e_{u}$ is a real number and $e_{u}\ge0$.

Step 5 (Odd terms). Let $k\ge2$, let $0\le u\le k-2$ and $v=k-2-u\ge0$, and put $A=a^{u}Db^{v+1}$ and $B=a^{u+1}Db^{v}$. By Step 1, $A^{*}=b^{v+1}Da^{u}$ and $B^{*}=b^{v}Da^{u+1}$. Each of the following chains uses (P1) to merge adjacent powers, then condition (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state to move the leading power of $b$ to the end, then (P1) again, then condition (c) to move the leading factor $D$ to the end:

$$\lambda(B^{*}A)=\lambda(b^{v}Da^{2u+1}Db^{v+1})=\lambda(Da^{2u+1}Db^{2v+1})=\lambda(a^{2u+1}Db^{2v+1}D)=o_{u},$$ $$\lambda(A^{*}B)=\lambda(b^{v+1}Da^{2u+1}Db^{v})=\lambda(Da^{2u+1}Db^{2v+1})=\lambda(a^{2u+1}Db^{2v+1}D)=o_{u},$$ $$\lambda(A^{*}A)=\lambda(b^{v+1}Da^{2u}Db^{v+1})=\lambda(Da^{2u}Db^{2(v+1)})=\lambda(a^{2u}Db^{2(v+1)}D)=e_{u},$$ $$\lambda(B^{*}B)=\lambda(b^{v}Da^{2(u+1)}Db^{v})=\lambda(Da^{2(u+1)}Db^{2v})=\lambda(a^{2(u+1)}Db^{2v}D)=e_{u+1},$$

where the identifications with $o_{u}$, $e_{u}$ and $e_{u+1}$ use $v=k-2-u$, $v+1=k-1-u$ and $v=k-1-(u+1)$. By Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §adjoint with $p=A$ and $q=B$, $\lambda(B^{*}A)=\overline{\lambda(A^{*}B)}$, that is, $o_{u}=\overline{o_{u}}$; so $o_{u}$ is real by claim 1 of Properties of Complex Conjugation and Modulus. By Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §cauchy-schwarz with $p=A$ and $q=B$,

$$|o_{u}|^{2}=|\lambda(B^{*}A)|^{2}\le\lambda(A^{*}A)\,\lambda(B^{*}B)=e_{u}e_{u+1}.$$

Step 6 (Arithmetic-geometric bound). Let $k\ge2$ and $0\le u\le k-2$. Put $x=|o_{u}|$ and $y_{u}=\tfrac12(e_{u}+e_{u+1})$, where $\tfrac12$ is the inverse of $1+1$. Since $o_{u}$ is real, $x$ is its absolute value by claim 8 of Properties of the Absolute Value in an Ordered Field, so $0\le x$ by claim 1 there. Next $0\le e_{u}+e_{u+1}$ by Step 4 and claim 2 of Elementary Arithmetic in an Ordered Field, and $0\le\tfrac12$ by claim 4 there, as $0\le1+1$ by claims 1 and 2 there and $1+1\neq0$, because $1+1=\iota(1)+1=\iota(2)$ by claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field and $0<\iota(2)$ by claim 3 there; so $0\le y_{u}$ by claim 5 there. Expanding in the field $\mathbb{R}$,

$$y_{u}^{2}-e_{u}e_{u+1}=\tfrac14\bigl((e_{u}+e_{u+1})^{2}-4e_{u}e_{u+1}\bigr)=\tfrac14(e_{u}-e_{u+1})^{2}.$$

For a real $z$ one has $z^{2}=|z|^{2}$, since $|z|$ equals $z$ or $-z$, and $0\le|z|^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field applied to $0\le|z|$ (claim 1 of Properties of the Absolute Value in an Ordered Field); with $z=e_{u}-e_{u+1}$ and claim 5 of Elementary Arithmetic in an Ordered Field (multiplying by $\tfrac14=\tfrac12\cdot\tfrac12\ge0$) the right side is $\ge0$. Thus $x^{2}\le e_{u}e_{u+1}\le y_{u}^{2}$ by Step 5 and claim 3 of Elementary Arithmetic in an Ordered Field, and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field (with $\alpha=x$, $\beta=y_{u}$) gives $|o_{u}|\le y_{u}$.

Step 7 (Conclusion). If $k=1$, then $S=e_{0}$, which is real and nonnegative by Step 4. Let $k\ge2$. Then $S$ is real, being a sum of the real numbers $e_{u}$ (Step 4) and $o_{u}$ (Step 5). By Step 6, $|o_{u}|\le y_{u}$, so $0\le y_{u}-|o_{u}|=(-|o_{u}|)-(-y_{u})$ and hence $-y_{u}\le-|o_{u}|$, by claim 3 of Elementary Arithmetic in an Ordered Field applied twice; with claim 3 of Properties of the Absolute Value in an Ordered Field, $-y_{u}\le-|o_{u}|\le o_{u}$, so $0\le o_{u}+y_{u}$ for every $0\le u\le k-2$ (claim 3 of Elementary Arithmetic in an Ordered Field). By claims 5 and 2 of Properties of Finite Sums, $0\le\sum_{u=0}^{k-2}o_{u}+\sum_{u=0}^{k-2}y_{u}$. Put $E=\sum_{i=0}^{k-1}e_{i}$. By claims 2 and 3 of Properties of Finite Sums and the reindexing $i=u+1$,

$$\sum_{u=0}^{k-2}y_{u}=\tfrac12\Bigl(\sum_{i=0}^{k-2}e_{i}+\sum_{i=1}^{k-1}e_{i}\Bigr).$$

Now $\{0,\dots,k-1\}$ is the disjoint union of $\{0,\dots,k-2\}$ and $\{k-1\}$, and also of $\{0\}$ and $\{1,\dots,k-1\}$; the sums over $\{0,\dots,k-1\}$, $\{0,\dots,k-2\}$ and $\{1,\dots,k-1\}$ are the ones written above, by Sum over a Finite Index Set computed with the bijections $[k]\to\{0,\dots,k-1\}$ and $[k-1]\to\{0,\dots,k-2\}$, $j\mapsto j-1$, and $[k-1]\to\{1,\dots,k-1\}$, $j\mapsto j$; and the sum over a singleton $\{s\}$, computed with $[1]\to\{s\}$, $1\mapsto s$, equals its term $e_{s}$ by claim 1 of Properties of Finite Sums. So Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §disjoint-union gives $E=\sum_{i=0}^{k-2}e_{i}+e_{k-1}=e_{0}+\sum_{i=1}^{k-1}e_{i}$, whence $\sum_{i=0}^{k-2}e_{i}=E-e_{k-1}\le E$ and $\sum_{i=1}^{k-1}e_{i}=E-e_{0}\le E$, because $e_{k-1},e_{0}\ge0$ (Step 4). Hence $\sum_{u=0}^{k-2}y_{u}\le\tfrac12(E+E)=E$ by claims 2, 3 and 5 of Elementary Arithmetic in an Ordered Field. Combining with Step 3,

$$S=E+\sum_{u=0}^{k-2}o_{u}\ge E-\sum_{u=0}^{k-2}y_{u}\ge E-E=0 .$$

So $\lambda\bigl((a^{n}-b^{n})(a-b)\bigr)=S$ is real and nonnegative.