The Involution on the Operator Algebra of a Process

Introduction

The operators of a process — the multiplications, the shift, the Koopman operators, the conditional expectations, the Markov operators — form an algebra under composition, and the algebra carries an involution, the passage to the adjoint with respect to the form of the category, $$ T\mapsto T^* ,\qquad \langle Tf,g\rangle=\langle f,T^*g\rangle,\qquad \langle f,g\rangle=\varphi(fg^*)=\mathbb E\bigl[f\bar g\bigr]. $$ This article is the study of that involution. The adjoint is the archetype of the operators built from the involution that gives the group of this article; it is an operation on the operators, distinct from the involution on the elements, which is the complex conjugation of The Involution on the Algebra of Random Variables, earlier in this category. The article defines the operator algebra of a process, proves that the adjoint is an involutive, anti-linear, isometric anti-automorphism satisfying the $C^*$-identity $\|T^*T\|=\|T\|^2$, identifies the self-adjoint elements, the unitaries, the projections and the partial isometries, computes the adjoints of the standard operators of a process, and states the precise relation between the element-level involution and the operator-level adjoint: the representation of the algebra of random variables by the left multiplications is a $*$-representation, $\pi(a^*)=\pi(a)^*$, and the agreement is proved from the positivity of the state and never assumed.

The conventions are those of the category: the algebra of random variables $\mathcal{A}=L^\infty(\Omega,\mathcal F,\mathbb P)$ with the conjugation $a^*=\bar a$, the state $\varphi(a)=\mathbb E[a]$, the form $\langle f,g\rangle=\varphi(fg^*)$ on the Hilbert space $\mathcal{H}=L^2(\Omega,\mathbb P)$ of The Involution on the Algebra of Random Variables, earlier in this category. The operator algebra, the topologies on it, the $C^*$-algebras and the von Neumann algebras, the Gelfand–Naimark theorem and the spectral theorem are Operator Algebras, written; the adjoints, the self-adjoint operators and the spectral theory on a Hilbert space are Banach and Hilbert Spaces, written. The adjoint of the Markov operator is The Adjoint of the Markov Operator, the preceding article of this category; the adjoint of the transition operator is The Adjoint of the Transition Operator, later in this category; the shift is The Shift Operator of a Process, earlier in this category; the Koopman operator is The Koopman Operator, written; the conditional expectation is The Conditional Expectation Operator, earlier in this category; and the refinement of the spectrum by the reversing symmetry is Reversible Operators and Self-Adjointness, later in this category. No physics is invoked.

Throughout, $\{X_t\}_{t\in T}$ is a process on the probability space $(\Omega,\mathcal F,\mathbb P)$, $\mathcal{H}=L^2(\Omega,\mathcal F,\mathbb P)$ is the Hilbert space with the form $\langle f,g\rangle=\varphi(fg^*)$, and $B(\mathcal{H})$ is the algebra of the bounded operators with composition. The operator algebra of the process is the von Neumann algebra $\mathcal{M}\subseteq B(\mathcal{H})$ generated by the process, that is the smallest von Neumann algebra containing the multiplications by the $X_t$ and the operators they determine; the involution is $T\mapsto T^*$, the adjoint in $\mathcal{H}$.

The Operator Algebra of a Process

The algebra and the involution

Definition. The adjoint of $T\in B(\mathcal{H})$ is the unique bounded operator $T^*$ with $$ \langle Tf,g\rangle=\langle f,T^*g\rangle\qquad\text{for all }f,g\in\mathcal{H}, $$ and the operator algebra of the process is the von Neumann algebra $\mathcal{M}$ generated by the operators determined by the process, closed under the adjoint.

Theorem. The passage to the adjoint is an involutive, conjugate-linear, isometric anti-automorphism of the operator algebra: $$ (T^*)^*=T,\qquad (S+T)^*=S^*+T^*,\qquad (\lambda T)^*=\bar\lambda\,T^*,\qquad (ST)^*=T^*S^*,\qquad \|T^*\|=\|T\| , $$ and it satisfies the $C^*$-identity $$ \|T^*T\|=\|T\|^2 , $$ so $\mathcal{M}$ is a von Neumann algebra, hence a $C^*$-algebra, and the involution is isometric for the form, $\langle Tf,g\rangle=\langle f,T^*g\rangle$.

Proof. The identities are the standard properties of the Hilbert-space adjoint; the anti-multiplicativity reverses the order because $\langle STf,g\rangle=\langle Tf,S^*g\rangle=\langle f,T^*S^*g\rangle$; the isometry and the $C^*$-identity are the standard theorems; the closure under the adjoint is the definition of a von Neumann algebra, Operator Algebras, written.

The self-adjoint elements

Definition. An operator $T$ is self-adjoint if $T^*=T$, unitary if $T^*T=TT^*=I$, a projection if $T^*=T=T^2$, and a partial isometry if $T^*T$ is a projection.

Theorem (the decomposition and the spectral theorem). Every $T\in\mathcal{M}$ decomposes uniquely as $T=A+iB$ with $A=\frac12(T+T^*)$ and $B=\frac1{2i}(T-T^*)$ self-adjoint; the self-adjoint operators have real spectrum, the unitary operators have spectrum on the unit circle, the projections have spectrum in $\{0,1\}$, and every self-adjoint operator has the spectral decomposition $T=\int_{\mathbb R}\lambda\,dE_T(\lambda)$.

Proof. The decomposition is the definition of the real and imaginary parts, and the uniqueness is the directness of the self-adjoint part; the spectral statements are the spectral theorem for a self-adjoint operator and its unitary consequences, Banach and Hilbert Spaces, written.

The operators of the process

Theorem (the adjoints of the standard operators).

The multiplication $L_a f=af$ on $\mathcal{H}$ has the adjoint $L_a^*=L_{a^*}$; the representation $a\mapsto L_a$ is a $*$-representation of $\mathcal{A}$, $\pi(a^*)=\pi(a)^*$, and the multiplications are the self-adjoint elements when $a$ is a real random variable.
The Koopman operator $U_Tf=f\circ T$ of a measure-preserving invertible $T$ is unitary, $U_T^*=U_{T^{-1}}$, and self-adjoint exactly when $T=T^{-1}$ almost everywhere.
The conditional expectation $E^{\mathcal G}$ on a sub-$\sigma$-algebra $\mathcal G$ is self-adjoint and a projection, $E^{\mathcal G}=(E^{\mathcal G})^*=(E^{\mathcal G})^2$.
The Markov operator $P$ of a stationary chain has the adjoint $P^*=\hat P$, the Markov operator of the reversed chain; it is self-adjoint exactly when the chain is reversible.

Proof. The adjoint of a multiplication is the multiplication by the conjugate, $\langle af,g\rangle=\varphi(afg^*)=\varphi(f(\bar ag)^*)=\langle f,\bar ag\rangle$, so $L_a^*=L_{a^*}$, and this is the $*$-representation because $\pi(a^*)=L_{a^*}=L_a^*$; the Koopman adjoint is the change of variables under the measure-preserving bijection; the conditional expectation is the orthogonal projection onto the closed subspace $L^2(\mathcal G)$ and a projection is self-adjoint; the Markov adjoint is the formula of The Adjoint of the Markov Operator, the preceding article.

The Two Structures

The element involution and the operator involution

Theorem (two levels). The involution on the elements of the algebra of random variables, $a\mapsto a^*=\bar a$, and the involution on the operators, $T\mapsto T^*$, are two structures, not one. They act on different objects — the first on the functions, the second on the operators — and they are related through a representation: for a representation $\pi$ of the algebra of random variables on $\mathcal{H}$, the two agree, $\pi(a^*)=\pi(a)^*$, exactly when $\pi$ is a $*$-representation.

Theorem (the agreement is proved). The representation $\pi(a)=L_a$ by the left multiplications is a $*$-representation, $$ \pi(a^*)=\pi(a)^*, $$ and the agreement is a theorem, proved from the positivity of the state and the invariance of the form; it is never assumed.

Proof. The identity $\langle af,g\rangle=\varphi(afg^*)=\varphi\bigl(f(\bar ag)^*\bigr)=\langle f,\bar ag\rangle$ gives $L_a^*=L_{\bar a}=\pi(a^*)$ directly, using only the definitions of the form and the involution; hence the multiplications give a $*$-representation and the element involution is realised by the operator involution on the subalgebra of the multiplications. On the whole von Neumann algebra the two structures are distinct: the Markov operator $P$ is not a multiplication, and its adjoint $\hat P$ is not the multiplication by a conjugate.

The reversal of the index

Theorem (the reversal). The reversal $r$ of the index defines the operator $U_rf=f\circ r$ on the path-space $L^2$, which is an involution, $U_r^2=\mathrm{id}$, and self-adjoint, $U_r^*=U_r$; the conjugation $$ \tau(T)=U_r\,T\,U_r^{-1} $$ is an involutive automorphism of the operator algebra commuting with the adjoint, $\tau(T^*)=\tau(T)^*$, and it sends the shift to its inverse, $\tau(U_\sigma)=U_\sigma^{-1}$. The process is reversible exactly when $\tau$ preserves the state, $\varphi\circ\tau=\varphi$ on the multiplications, equivalently when the reversal operator is measure-preserving.

Proof. The identities $U_r^2=\mathrm{id}$ and $r\sigma r=\sigma^{-1}$ give $U_r^2=I$ and $\tau(U_\sigma)=U_\sigma^{-1}$; the conjugation by an involution is an automorphism, and it commutes with the adjoint because the adjoint is defined by the form, which is preserved by the unitary $U_r$. The reversibility is the invariance of the law under $r$, which is the identity $\mathbb E[f\circ r]=\mathbb E[f]$ for every bounded $f$, that is $\varphi\circ\tau=\varphi$; this is The Reversibility of a Stationary Process, earlier in this category.

Worked Examples

Example (the multiplication algebra). On $L^2(\Omega,\mathbb P)$ the multiplications $L_a$ form a commutative von Neumann algebra isomorphic to $\mathcal{A}$, and the involution is the conjugation: $L_a^*=L_{\bar a}$. The self-adjoint elements are the multiplications by the real random variables, the unitaries the multiplications by the unimodular ones, and the projections the multiplications by the indicator functions; the spectral decomposition of $L_a$ for a real $a$ is the spectral measure of the random variable $a$.

Example (the Bernoulli shift). For the two-sided Bernoulli shift on $\{0,1\}^{\mathbb Z}$ the shift operator $U_\sigma$ is unitary with $U_\sigma^*=U_\sigma^{-1}$, self-adjoint only in the trivial case $\sigma=\sigma^{-1}$; it is not a multiplication and its adjoint is the backwards shift. The reversal operator satisfies $\tau(U_\sigma)=U_\sigma^{-1}$, and the product measure is invariant under the reversal of the coordinates, so $\tau$ preserves the state and the Bernoulli process is reversible; the example shows that the reversal of the index and the reversibility of the process must not be confused.

Example (the directed cycle). Let the process be the stationary Markov chain on $\{1,2,3\}$ that moves cyclically, of The Adjoint of the Markov Operator, the preceding article. The reversal of the index is not measure-preserving, $\tau$ does not preserve the state, and the process is not reversible; the shift inverts under $\tau$ as always, but the law is not invariant, and the transition operator is not self-adjoint. The example is the non-reversible counterpart of the Bernoulli case.

Example (the conditional expectation). For the filtration generated by a process the conditional expectations $E_n=E^{\mathcal F_n}$ are self-adjoint projections, $E_n^*=E_n=E_n^2$, and $E_nE_m=E_{\min(n,m)}$ for an increasing filtration; the involution on this family of projections is the identity, and the whole content of the conditional expectation lies in its being a projection rather than an arbitrary idempotent.

Example (the Markov adjoint). For the directed cycle of The Adjoint of the Markov Operator, the preceding article, the Markov operator $P$ is not a multiplication and is not self-adjoint; its adjoint $P^*=\hat P$ is the backwards cycle, and $P^*\ne P$ while both are contractions fixing the constants. The example separates the two structures: the element involution leaves $P$ alone, and the operator involution reverses it.

Failure of the Degenerate Cases

The involution on the operator algebra degenerates in four configurations. First, the element involution and the operator involution coincide only on the multiplications; on the operators of the process that are not multiplications — the shift, the Markov operator, the transfer operators — the adjoint is a genuinely different operation, and assuming the agreement is the error the group contract forbids. Second, the representation by the multiplications is a $*$-representation because the state is a positive functional on the commutative algebra; for a non-commutative algebra with a non-tracial state the representation can fail to be a $*$-representation, and the agreement must be checked, not assumed. Third, the reversal automorphism exists as a unitary only when the reversal is measure-preserving; for a non-reversible process the reversal of the index is not implemented by a unitary, and the conjugation $U_rTU_r^{-1}$ is replaced by the adjoint of the transition operator. Fourth, the $C^*$-identity and the spectral theorem require the completeness of the Hilbert space, and they fail for an incomplete inner-product space; the operators of the process are bounded precisely when the multipliers are in $L^\infty$.

Summary

The operator algebra of a process is the von Neumann algebra generated by the operators the process determines, with the involution $T\mapsto T^*$ the adjoint for the form $\langle f,g\rangle=\varphi(fg^*)$; the involution is an involutive, conjugate-linear, isometric anti-automorphism satisfying the $C^*$-identity $\|T^*T\|=\|T\|^2$, the self-adjoint elements decompose as $T=A+iB$ and have real spectrum, the unitaries have their spectrum on the circle and the projections on $\{0,1\}$, and the spectral theorem gives the decomposition of every self-adjoint operator. The standard operators have the adjoints $L_a^*=L_{a^*}$, $U_T^*=U_{T^{-1}}$, $(E^{\mathcal G})^*=E^{\mathcal G}$, and $P^*=\hat P$, the reversed chain. The involution on the elements and the involution on the operators are two structures: the representation by the multiplications is a $*$-representation, $\pi(a^*)=\pi(a)^*$, proved from the invariance of the form, while on the whole operator algebra the two differ; the reversal of the index, when it is implemented by a measure-preserving involution, acts by the conjugation with the unitary $U_r$, an automorphism commuting with the adjoint. The applications to the spectrum are Reversible Operators and Self-Adjointness, later in this category, and the adjoint of the transition operator is The Adjoint of the Transition Operator, later in this category.

Summary of Notation

Symbol	Meaning
$\mathcal{H}$, $\langle f,g\rangle=\varphi(fg^*)$	the Hilbert space and the form
$T^*$	the adjoint, the operator involution
$(ST)^=T^S^$, $\\|T^T\\|=\\|T\\|^2$	anti-multiplicativity, $C^*$-identity
$T=A+iB$	the self-adjoint decomposition
$L_a$, $L_a^=L_{a^}$	the multiplications, a $*$-representation
$U_T^*=U_{T^{-1}}$	the Koopman adjoint
$E^{\mathcal G}$, $(E^{\mathcal G})^*=E^{\mathcal G}=E^{\mathcal G\,2}$	the conditional expectation, a projection
$P^*=\hat P$	the Markov adjoint, the reversed chain
$U_r$, $\tau(T)=U_rTU_r^{-1}$	the reversal unitary and its automorphism