Process dimension of classical and non-commutative processes (Q2884799)

Recall the observable operator model (OOM), i.e., a finite set \(\Delta\), a \(\mathbb{R}\)-vector space \(V\), \(v \in V\), \(T _{d } : V \rightarrow V\) linear, \(d \in \Delta\), \(r : V \rightarrow \mathbb{R}\) linear, \(r ( v ) = 1\), \(r ( \sum _{d \in \Delta } T _{d } ) = r\), \(( r \circ T _{d _{n }} \circ \cdots \circ T _{d _{1 }} ) ( v ) \geq 0\). It determines a probability \(\operatorname{P}\) on \(\Delta ^{\mathbb{N}}\), with marginals \(\operatorname{P} _{d _{1 } , \dotsc , d _{n }} = ( r \circ T _{d _{n }} \circ \dotsb \circ T _{d _{1 }} ) ( v )\) on the \( \{ 1 , \dotsc , n \} \)-cylinder \([ d _{1 } , \dotsc , d _{n }]\) with basis \(\{ ( d _{1 }, \dotsc , d _{n } ) \}\). Conversely, if the probability \(\operatorname{P}\) on \(\Delta ^{\mathbb{N}}\) is given, then, defining, on the space of signed measures on \(\Delta ^{\mathbb{N}}\), \(T _{d } ( \mu ) =\mu ( [ d ] \cap \sigma ^{- 1 } ( \cdot ))\), whereby \(\sigma ( d _{1 } , d _{2 }, \dots ) = ( d _{2 }, \dots )\), \(V _{\operatorname{P}}\) as the smallest linear subspace containing \(P\) and invariant to all \(T _{d }\), \(v = P\) and \(r ( \mu ) = \mu ( \Delta ^{\mathbb{N}} )\), we obtain an OOM which determines \(\operatorname{P}\). The dimension of \(\operatorname{P}\) is defined as the dimension of \(V _{\operatorname{P}}\). If \(\operatorname{P}\) is a shift invariant probability on \(\Delta^{\mathbb{Z}}\), then its conditional distribution of its projection on \(\Delta ^{\mathbb{N}}\) with respect to \(\Delta ^{ \{ 0 \} \cup ( - \mathbb{N} ) }\), is a function from \(\Delta ^{\mathbb{Z}}\) to the set \({\mathcal P} ( \Delta ^{\mathbb{N}} )\) of probabilities on \(\Delta ^{\mathbb{N}}\). The image of \(\operatorname{P}\) under this map is denoted by \(\mu _{{\mathcal E}} ^{\operatorname{P} }\) and its support by \({\mathcal E}_{\operatorname{P} } \subset {\mathcal P} ( \Delta ^{\mathbb{N}})\). The authors prove that \(\dim \operatorname{P}\) is the dimension of the linear hull of \({\mathcal E}_{\operatorname{P}}\) in \({\mathcal P} ( \Delta ^{\mathbb{N}} )\) by proving the equality of their weak closures (they show by examples that no corresponding inclusion holds when \(\dim \operatorname{P} = \infty\)). Also, (Proposition 1) they prove that \(\dim \operatorname{P}\) is representable as the sum of the dimensions of the probabilities in the support of the probability realizing the ergodic decomposition of \(\operatorname{P}\). In the noncommutative case, \(\Delta\) is replaced by a finite dimensional \(C^{ \ast }\)-algebra \(A\), \(T _{d }\) is supposed to be linear in \(d\) also and the construction leads to a state of \(A^{ \otimes ^{\mathbb{N}}}\), the converse also works similarly and leads to an analogous definition of a NC-OOM and of \(\dim\). The cases when \(A\) is commutative are in bijection with the OOMs. One main result is the weak lower-semicontinuity of the dimension, as a function of a state with values in \(\mathbb{N}\cup \{ \infty \} \). Another is the noncommutative generalization of Proposition 1.