Rolf Gohm - Academia.edu (original) (raw)

Papers by Rolf Gohm

Rocky Mountain Journal of Mathematics

To a semi-cosimplicial object (SCO) in a category, we associate a system of partial shifts on the... more To a semi-cosimplicial object (SCO) in a category, we associate a system of partial shifts on the inductive limit. We show how to produce an SCO from an action of the infinite braid monoid B + ∞ and provide examples. In categories of (noncommutative) probability spaces, SCOs correspond to spreadable sequences of random variables; hence, SCOs can be considered as the algebraic structure underlying spreadability. 2010 AMS Mathematics subject classification. Primary 18G30, 20F36, 46L53.

arXiv (Cornell University), May 31, 2010

We provide an operator algebraic proof of a classical theorem of Thoma which characterizes the ex... more We provide an operator algebraic proof of a classical theorem of Thoma which characterizes the extremal characters of the infinite symmetric group S∞. Our methods are based on noncommutative conditional independence emerging from exchangeability [GK09, Kös10] and we reinterpret Thoma's theorem as a noncommutative de Finetti type result. Our approach is, in parts, inspired by Jones' subfactor theory and by Okounkov's spectral proof of Thoma's theorem [Oko99], and we link them by inferring spectral properties from certain commuting squares.

Operators and Matrices, 2018

We prove that the symbol of the characteristic function of a minimal contractive lifting is an in... more We prove that the symbol of the characteristic function of a minimal contractive lifting is an injective map and that the converse also holds, using explicit computation and functional models. We discuss when the characteristic function of a lifting is a polynomial and present a series representation for the characteristic functions of liftings.

Communications in Mathematical Physics, 2017

We introduce a notion of universal preparability for a state of a system, more precisely: for a n... more We introduce a notion of universal preparability for a state of a system, more precisely: for a normal state on a von Neumann algebra. It describes a situation where from an arbitrary initial state it is possible to prepare a target state with arbitrary precision by a repeated interaction with a sequence of copies of another system. For B(H) we give criteria sufficient to ensure that all normal states are universally preparable which can be verified for a class of non-commutative birth and death processes realized, in particular, by the interaction of a micromaser with a stream of atoms. As a tool, the theory of tight sequences of states and of stationary states is further developed and we show that in the presence of stationary faithful normal states universal preparability of all normal states is equivalent to asymptotic completeness, a notion studied earlier in connection with the scattering theory of non-commutative Markov processes.

Key Words: Science, IMAPS, UG Year 3, Lecture Based Learning, Problem-Based Learning, Reflective ... more Key Words: Science, IMAPS, UG Year 3, Lecture Based Learning, Problem-Based Learning, Reflective Learning, Feedback

Lecture Notes in Mathematics, 2005

Key Words: Science, IMAPS, UG Year 2, Student Engagement, Continous Assessment, Formative Feedbac... more Key Words: Science, IMAPS, UG Year 2, Student Engagement, Continous Assessment, Formative Feedback, Problem-Based Learning

Key Words: Science, IMAPS, UG Year 3, Problem-Based Learning, Small Group, Feedback, Student-Lect... more Key Words: Science, IMAPS, UG Year 3, Problem-Based Learning, Small Group, Feedback, Student-Lecturer Interaction

JOURNAL OF OPERATOR THEORY

We discuss a class of endomorphisms of the hyperfinite II1-factor which are adapted in a certain ... more We discuss a class of endomorphisms of the hyperfinite II1-factor which are adapted in a certain way to a tower C1 ⊂ C p ⊂ Mp ⊂ Mp⊗C p ⊂ • • • so that for p = 2 we get Bogoljubov transformations of a Clifford algebra. Results are given about surjectivity, innerness, Jones index and the shift property.

JOURNAL OF OPERATOR THEORY

A local version of the concept of polynomial boundedness for operators on Banach spaces is define... more A local version of the concept of polynomial boundedness for operators on Banach spaces is defined and its relations to functional calculi are examined. For certain positive operators on L ∞-spaces, especially for endomorphisms, lack of local polynomial boundedness corresponds to mixing properties. In particular, we give a new characterization of the weak mixing property. Some results extend to more general C *-algebras. This is done by constructing certain topological embeddings of the unit vector base of l 1 (IN0) into the orbits of an operator. To analyze the underlying structure we introduce the concept of a transition set. We compute transition sets for the shift operator on l 1 (Z) and show how to define a corresponding similarity invariant. Contents: 1 Introduction 2 Local Functional Calculi 3 Relations to Ergodic Theory 4 The Transition Set.

The probabilistic index of a completely positive map is defined in analogy with a formula of M.Pi... more The probabilistic index of a completely positive map is defined in analogy with a formula of M.Pimsner and S.Popa for conditional expectations. As an application, we describe a new strategy for com- puting the Jones index of the range of certain endomorphisms. We use extended transition operators to associate to an endomorphism a unital completely positive map acting on a

Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, 2012

To a pair of subspaces wandering with respect to a row isometry we associate a transfer function ... more To a pair of subspaces wandering with respect to a row isometry we associate a transfer function which in general is multi-Toeplitz and in interesting special cases is multi-analytic. Then we describe in an expository way how characteristic functions from operator theory as well as transfer functions from noncommutative Markov chains fit into this scheme.

Banach Center Publications, 2011

We define tensor product decompositions of E 0-semigroups with a structure analogous to a classic... more We define tensor product decompositions of E 0-semigroups with a structure analogous to a classical theorem of Beurling. Such decompositions can be characterized by adaptedness and exactness of unitary cocycles. For CCR-flows we show that such cocycles are convergent. Introduction. A well-known theorem of Beurling characterizes invariant subspaces of the right shift on ℓ 2 (N) by inner functions in the unit disc. In this case the restriction of the right shift to a nontrivial invariant subspace is automatically conjugate (unitarily equivalent) to the original shift. This interesting self-similar structure is fundamental in many respects. It is the prototype of a very fruitful interaction between operator theory and function theory, see for example [Ni, FF]. In this paper we want to study a somewhat analogous self-similar structure for operators on a different level. While the original setting concerns isometries and decompositions of the Hilbert space into direct sums, we want to study E 0-semigroups, i.e., pointwise weak *-continuous semigroups of unital *-endomorphisms of B(H) for some complex separable Hilbert space H (cf. [Ar]), and decompositions of the Hilbert space into tensor products. To make the analogy visible, we present in Section 1 the Nagy-Foiaş functional model for *-stable contractions and their characteristic functions in a suitable way and in particular we emphasize a limit formula for the characteristic function which is not made explicit in the standard presentations. This analogy motivates the definition of decompositions of Beurling type for E 0-semigroups in Section 2. It is then shown that there is a reformulation in terms of unitary cocycles for amplifications of the E 0-semigroup. Relevant properties of the cocycles are adaptedness and exactness.

Mathematics of Control, Signals, and Systems, 2015

A noncommutative Fornasini-Marchesini system (a multi-variable version of a linear system) can be... more A noncommutative Fornasini-Marchesini system (a multi-variable version of a linear system) can be realized within a weak Markov process (a model for quantum evolution). For a discrete time parameter the resulting structure is worked out systematically and some quantum mechanical interpretations are given. We introduce subprocesses and quotient processes and then the notion of a γ-extension for processes which leads to a complete classification of all the ways in which processes can be built from subprocesses and quotient processes. We show that within a γ-extension we have a cascade of noncommutative Fornasini-Marchesini systems. We study observability in this setting and as an application we gain new insights into stationary Markov chains where observability for the system is closely related to asymptotic completeness in a scattering theory for the chain.

Contemporary Mathematics, 2003

For normal unital completely positive maps on von Neumann algebras respecting distinguished state... more For normal unital completely positive maps on von Neumann algebras respecting distinguished states, we consider the problem to find normal unital completely positive extensions acting on all bounded operators on the GNS-spaces and respecting the corresponding cyclic vectors. We show that there exists a duality relating this problem to a certain dilation problem on the commutants. Some explicit computations for low dimensions are presented.

Quantum Probability, 2006

We define tensor product decompositions of E 0-semigroups with a structure analogous to a classic... more We define tensor product decompositions of E 0-semigroups with a structure analogous to a classical theorem of Beurling. Such decompositions can be characterized by adaptedness and exactness of unitary cocycles. For CCR-flows we show that such cocycles are convergent. Introduction. A well-known theorem of Beurling characterizes invariant subspaces of the right shift on ℓ 2 (N) by inner functions in the unit disc. In this case the restriction of the right shift to a nontrivial invariant subspace is automatically conjugate (unitarily equivalent) to the original shift. This interesting self-similar structure is fundamental in many respects. It is the prototype of a very fruitful interaction between operator theory and function theory, see for example [Ni, FF]. In this paper we want to study a somewhat analogous self-similar structure for operators on a different level. While the original setting concerns isometries and decompositions of the Hilbert space into direct sums, we want to study E 0-semigroups, i.e., pointwise weak *-continuous semigroups of unital *-endomorphisms of B(H) for some complex separable Hilbert space H (cf. [Ar]), and decompositions of the Hilbert space into tensor products. To make the analogy visible, we present in Section 1 the Nagy-Foiaş functional model for *-stable contractions and their characteristic functions in a suitable way and in particular we emphasize a limit formula for the characteristic function which is not made explicit in the standard presentations. This analogy motivates the definition of decompositions of Beurling type for E 0-semigroups in Section 2. It is then shown that there is a reformulation in terms of unitary cocycles for amplifications of the E 0-semigroup. Relevant properties of the cocycles are adaptedness and exactness.

Lecture Notes in Mathematics, 2004

Contents. 1.1 An Example with 2 ×22 \times 2 - Matrices 1.1.1 A Stochastic Map 1.1.2 Direct Appro... more Contents. 1.1 An Example with 2 ×22 \times 2 - Matrices 1.1.1 A Stochastic Map 1.1.2 Direct Approach 1.1.3 Computations 1.1.4 Parametrization of the Set of Extensions 1.2 An Extension Problem 1.2.1 The Set Z(S,f</font >B){\cal Z}(S,{\phi_{{\cal B}}}) of Extensions 1.2.2 Z{\cal Z} as a Convex Set 1.2.3 Discussion 1.3 Weak Tensor Dilations 1.3.1 The Definition 1.3.2 Representations 1.3.3 Construction of Examples 1.3.4 The Associated Isometry 1.3.5 The Minimal Version of an Associated Isometry 1.3.6 Minimal Part of the Stinespring Representation 1.4 Equivalence of Weak Tensor Dilations 1.4.1 An Equivalence Relation 1.4.2 Equivalence and Unitary Equivalence 1.5 Duality 1.5.1 Dual Stochastic Maps 1.5.2 From Dilation to Extension 1.5.3 From Extension to Dilation 1.5.4 One-to-One Correspondence 1.5.5 Discussion 1.6 The Automorphic Case 1.6.1 Conditional Expectations 1.6.2 Adjoints 1.6.3 Automorphic Tensor Dilations 1.6.4 Duality for Automorphic Tensor Dilations 1.7 Examples 1.7.1 Example 1.1 Revisited 1.7.2 Further Discussion 1.7.3 A Class of Maps on M 2 1.7.4 Maps on M n

Lecture Notes in Mathematics, 2004

ABSTRACT

Lecture Notes in Mathematics, 2004

4.1 Commutative Stationarity 4.1.1 Stationarity in Classical Probability 4.1.2 Backward and Forwa... more 4.1 Commutative Stationarity 4.1.1 Stationarity in Classical Probability 4.1.2 Backward and Forward Transition 4.1.3 Classification of Stationary Extensions 4.1.4 Transition Operators 4.1.5 Adapted Endomorphisms 4.1.6 Counterexample 4.1.7 Stationary Extensions and Extending Factors 4.1.8 Elementary Tensor Representations 4.1.9 Construction of Extending Factors by Partitions 4.1.10 Discussion 4.1.11 Two-Valued Processes 4.1.12 Markov Processes 4.1.13 Markovian Extensions and Entropy 4.2 Prediction Errors for Commutative Processes 4.2.1 The Problem 4.2.2 Prediction for Finite-Valued Processes 4.2.3 A Guessing Game 4.2.4 A Combinatorial Formula for Prediction Errors 4.2.5 Asymptotics 4.2.6 Example 4.3 Low-Dimensional Examples 4.3.1 Qubits 4.3.2 Associated Stochastic Maps 4.3.3 Determinism and Unitarity 4.3.4 Complete Invariants 4.3.5 Probabilistic Interpretations 4.4 Clifford Algebras and Generalizations 4.4.1 Clifford Algebras 4.4.2 The Clifford Functor 4.4.3 Generalized Clifford Algebras 4.5 Tensor Products of Matrices 4.5.1 Matrix Filtrations 4.5.2 LPR Is Automatic 4.5.3 Associated Stochastic Maps 4.5.4 Non-surjectivity 4.5.5 Adapted Endomorphisms in the Literature 4.6 Noncommutative Extension of Adaptedness 4.6.1 Extending Adaptedness 4.6.2 Three Points of Time 4.6.3 Criteria for the Extendability of Adaptedness 4.6.4 An Example: The Fredkin Gate 4.6.5 Discussion

Rocky Mountain Journal of Mathematics

To a semi-cosimplicial object (SCO) in a category, we associate a system of partial shifts on the... more To a semi-cosimplicial object (SCO) in a category, we associate a system of partial shifts on the inductive limit. We show how to produce an SCO from an action of the infinite braid monoid B + ∞ and provide examples. In categories of (noncommutative) probability spaces, SCOs correspond to spreadable sequences of random variables; hence, SCOs can be considered as the algebraic structure underlying spreadability. 2010 AMS Mathematics subject classification. Primary 18G30, 20F36, 46L53.

arXiv (Cornell University), May 31, 2010

We provide an operator algebraic proof of a classical theorem of Thoma which characterizes the ex... more We provide an operator algebraic proof of a classical theorem of Thoma which characterizes the extremal characters of the infinite symmetric group S∞. Our methods are based on noncommutative conditional independence emerging from exchangeability [GK09, Kös10] and we reinterpret Thoma's theorem as a noncommutative de Finetti type result. Our approach is, in parts, inspired by Jones' subfactor theory and by Okounkov's spectral proof of Thoma's theorem [Oko99], and we link them by inferring spectral properties from certain commuting squares.

Operators and Matrices, 2018

We prove that the symbol of the characteristic function of a minimal contractive lifting is an in... more We prove that the symbol of the characteristic function of a minimal contractive lifting is an injective map and that the converse also holds, using explicit computation and functional models. We discuss when the characteristic function of a lifting is a polynomial and present a series representation for the characteristic functions of liftings.

Communications in Mathematical Physics, 2017

We introduce a notion of universal preparability for a state of a system, more precisely: for a n... more We introduce a notion of universal preparability for a state of a system, more precisely: for a normal state on a von Neumann algebra. It describes a situation where from an arbitrary initial state it is possible to prepare a target state with arbitrary precision by a repeated interaction with a sequence of copies of another system. For B(H) we give criteria sufficient to ensure that all normal states are universally preparable which can be verified for a class of non-commutative birth and death processes realized, in particular, by the interaction of a micromaser with a stream of atoms. As a tool, the theory of tight sequences of states and of stationary states is further developed and we show that in the presence of stationary faithful normal states universal preparability of all normal states is equivalent to asymptotic completeness, a notion studied earlier in connection with the scattering theory of non-commutative Markov processes.

Key Words: Science, IMAPS, UG Year 3, Lecture Based Learning, Problem-Based Learning, Reflective ... more Key Words: Science, IMAPS, UG Year 3, Lecture Based Learning, Problem-Based Learning, Reflective Learning, Feedback

Lecture Notes in Mathematics, 2005

Key Words: Science, IMAPS, UG Year 2, Student Engagement, Continous Assessment, Formative Feedbac... more Key Words: Science, IMAPS, UG Year 2, Student Engagement, Continous Assessment, Formative Feedback, Problem-Based Learning

Key Words: Science, IMAPS, UG Year 3, Problem-Based Learning, Small Group, Feedback, Student-Lect... more Key Words: Science, IMAPS, UG Year 3, Problem-Based Learning, Small Group, Feedback, Student-Lecturer Interaction

JOURNAL OF OPERATOR THEORY

We discuss a class of endomorphisms of the hyperfinite II1-factor which are adapted in a certain ... more We discuss a class of endomorphisms of the hyperfinite II1-factor which are adapted in a certain way to a tower C1 ⊂ C p ⊂ Mp ⊂ Mp⊗C p ⊂ • • • so that for p = 2 we get Bogoljubov transformations of a Clifford algebra. Results are given about surjectivity, innerness, Jones index and the shift property.

JOURNAL OF OPERATOR THEORY

A local version of the concept of polynomial boundedness for operators on Banach spaces is define... more A local version of the concept of polynomial boundedness for operators on Banach spaces is defined and its relations to functional calculi are examined. For certain positive operators on L ∞-spaces, especially for endomorphisms, lack of local polynomial boundedness corresponds to mixing properties. In particular, we give a new characterization of the weak mixing property. Some results extend to more general C *-algebras. This is done by constructing certain topological embeddings of the unit vector base of l 1 (IN0) into the orbits of an operator. To analyze the underlying structure we introduce the concept of a transition set. We compute transition sets for the shift operator on l 1 (Z) and show how to define a corresponding similarity invariant. Contents: 1 Introduction 2 Local Functional Calculi 3 Relations to Ergodic Theory 4 The Transition Set.

The probabilistic index of a completely positive map is defined in analogy with a formula of M.Pi... more The probabilistic index of a completely positive map is defined in analogy with a formula of M.Pimsner and S.Popa for conditional expectations. As an application, we describe a new strategy for com- puting the Jones index of the range of certain endomorphisms. We use extended transition operators to associate to an endomorphism a unital completely positive map acting on a

Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, 2012

To a pair of subspaces wandering with respect to a row isometry we associate a transfer function ... more To a pair of subspaces wandering with respect to a row isometry we associate a transfer function which in general is multi-Toeplitz and in interesting special cases is multi-analytic. Then we describe in an expository way how characteristic functions from operator theory as well as transfer functions from noncommutative Markov chains fit into this scheme.

Banach Center Publications, 2011

We define tensor product decompositions of E 0-semigroups with a structure analogous to a classic... more We define tensor product decompositions of E 0-semigroups with a structure analogous to a classical theorem of Beurling. Such decompositions can be characterized by adaptedness and exactness of unitary cocycles. For CCR-flows we show that such cocycles are convergent. Introduction. A well-known theorem of Beurling characterizes invariant subspaces of the right shift on ℓ 2 (N) by inner functions in the unit disc. In this case the restriction of the right shift to a nontrivial invariant subspace is automatically conjugate (unitarily equivalent) to the original shift. This interesting self-similar structure is fundamental in many respects. It is the prototype of a very fruitful interaction between operator theory and function theory, see for example [Ni, FF]. In this paper we want to study a somewhat analogous self-similar structure for operators on a different level. While the original setting concerns isometries and decompositions of the Hilbert space into direct sums, we want to study E 0-semigroups, i.e., pointwise weak *-continuous semigroups of unital *-endomorphisms of B(H) for some complex separable Hilbert space H (cf. [Ar]), and decompositions of the Hilbert space into tensor products. To make the analogy visible, we present in Section 1 the Nagy-Foiaş functional model for *-stable contractions and their characteristic functions in a suitable way and in particular we emphasize a limit formula for the characteristic function which is not made explicit in the standard presentations. This analogy motivates the definition of decompositions of Beurling type for E 0-semigroups in Section 2. It is then shown that there is a reformulation in terms of unitary cocycles for amplifications of the E 0-semigroup. Relevant properties of the cocycles are adaptedness and exactness.

Mathematics of Control, Signals, and Systems, 2015

A noncommutative Fornasini-Marchesini system (a multi-variable version of a linear system) can be... more A noncommutative Fornasini-Marchesini system (a multi-variable version of a linear system) can be realized within a weak Markov process (a model for quantum evolution). For a discrete time parameter the resulting structure is worked out systematically and some quantum mechanical interpretations are given. We introduce subprocesses and quotient processes and then the notion of a γ-extension for processes which leads to a complete classification of all the ways in which processes can be built from subprocesses and quotient processes. We show that within a γ-extension we have a cascade of noncommutative Fornasini-Marchesini systems. We study observability in this setting and as an application we gain new insights into stationary Markov chains where observability for the system is closely related to asymptotic completeness in a scattering theory for the chain.

Contemporary Mathematics, 2003

For normal unital completely positive maps on von Neumann algebras respecting distinguished state... more For normal unital completely positive maps on von Neumann algebras respecting distinguished states, we consider the problem to find normal unital completely positive extensions acting on all bounded operators on the GNS-spaces and respecting the corresponding cyclic vectors. We show that there exists a duality relating this problem to a certain dilation problem on the commutants. Some explicit computations for low dimensions are presented.

Quantum Probability, 2006

We define tensor product decompositions of E 0-semigroups with a structure analogous to a classic... more We define tensor product decompositions of E 0-semigroups with a structure analogous to a classical theorem of Beurling. Such decompositions can be characterized by adaptedness and exactness of unitary cocycles. For CCR-flows we show that such cocycles are convergent. Introduction. A well-known theorem of Beurling characterizes invariant subspaces of the right shift on ℓ 2 (N) by inner functions in the unit disc. In this case the restriction of the right shift to a nontrivial invariant subspace is automatically conjugate (unitarily equivalent) to the original shift. This interesting self-similar structure is fundamental in many respects. It is the prototype of a very fruitful interaction between operator theory and function theory, see for example [Ni, FF]. In this paper we want to study a somewhat analogous self-similar structure for operators on a different level. While the original setting concerns isometries and decompositions of the Hilbert space into direct sums, we want to study E 0-semigroups, i.e., pointwise weak *-continuous semigroups of unital *-endomorphisms of B(H) for some complex separable Hilbert space H (cf. [Ar]), and decompositions of the Hilbert space into tensor products. To make the analogy visible, we present in Section 1 the Nagy-Foiaş functional model for *-stable contractions and their characteristic functions in a suitable way and in particular we emphasize a limit formula for the characteristic function which is not made explicit in the standard presentations. This analogy motivates the definition of decompositions of Beurling type for E 0-semigroups in Section 2. It is then shown that there is a reformulation in terms of unitary cocycles for amplifications of the E 0-semigroup. Relevant properties of the cocycles are adaptedness and exactness.

Lecture Notes in Mathematics, 2004

Contents. 1.1 An Example with 2 ×22 \times 2 - Matrices 1.1.1 A Stochastic Map 1.1.2 Direct Appro... more Contents. 1.1 An Example with 2 ×22 \times 2 - Matrices 1.1.1 A Stochastic Map 1.1.2 Direct Approach 1.1.3 Computations 1.1.4 Parametrization of the Set of Extensions 1.2 An Extension Problem 1.2.1 The Set Z(S,f</font >B){\cal Z}(S,{\phi_{{\cal B}}}) of Extensions 1.2.2 Z{\cal Z} as a Convex Set 1.2.3 Discussion 1.3 Weak Tensor Dilations 1.3.1 The Definition 1.3.2 Representations 1.3.3 Construction of Examples 1.3.4 The Associated Isometry 1.3.5 The Minimal Version of an Associated Isometry 1.3.6 Minimal Part of the Stinespring Representation 1.4 Equivalence of Weak Tensor Dilations 1.4.1 An Equivalence Relation 1.4.2 Equivalence and Unitary Equivalence 1.5 Duality 1.5.1 Dual Stochastic Maps 1.5.2 From Dilation to Extension 1.5.3 From Extension to Dilation 1.5.4 One-to-One Correspondence 1.5.5 Discussion 1.6 The Automorphic Case 1.6.1 Conditional Expectations 1.6.2 Adjoints 1.6.3 Automorphic Tensor Dilations 1.6.4 Duality for Automorphic Tensor Dilations 1.7 Examples 1.7.1 Example 1.1 Revisited 1.7.2 Further Discussion 1.7.3 A Class of Maps on M 2 1.7.4 Maps on M n

Lecture Notes in Mathematics, 2004

ABSTRACT

Lecture Notes in Mathematics, 2004

4.1 Commutative Stationarity 4.1.1 Stationarity in Classical Probability 4.1.2 Backward and Forwa... more 4.1 Commutative Stationarity 4.1.1 Stationarity in Classical Probability 4.1.2 Backward and Forward Transition 4.1.3 Classification of Stationary Extensions 4.1.4 Transition Operators 4.1.5 Adapted Endomorphisms 4.1.6 Counterexample 4.1.7 Stationary Extensions and Extending Factors 4.1.8 Elementary Tensor Representations 4.1.9 Construction of Extending Factors by Partitions 4.1.10 Discussion 4.1.11 Two-Valued Processes 4.1.12 Markov Processes 4.1.13 Markovian Extensions and Entropy 4.2 Prediction Errors for Commutative Processes 4.2.1 The Problem 4.2.2 Prediction for Finite-Valued Processes 4.2.3 A Guessing Game 4.2.4 A Combinatorial Formula for Prediction Errors 4.2.5 Asymptotics 4.2.6 Example 4.3 Low-Dimensional Examples 4.3.1 Qubits 4.3.2 Associated Stochastic Maps 4.3.3 Determinism and Unitarity 4.3.4 Complete Invariants 4.3.5 Probabilistic Interpretations 4.4 Clifford Algebras and Generalizations 4.4.1 Clifford Algebras 4.4.2 The Clifford Functor 4.4.3 Generalized Clifford Algebras 4.5 Tensor Products of Matrices 4.5.1 Matrix Filtrations 4.5.2 LPR Is Automatic 4.5.3 Associated Stochastic Maps 4.5.4 Non-surjectivity 4.5.5 Adapted Endomorphisms in the Literature 4.6 Noncommutative Extension of Adaptedness 4.6.1 Extending Adaptedness 4.6.2 Three Points of Time 4.6.3 Criteria for the Extendability of Adaptedness 4.6.4 An Example: The Fredkin Gate 4.6.5 Discussion