## Correspondences vs Correspondences

Without doubt, who has read the title will already have guessed that we are speaking of two different types of correspondences we are going to confront. Classically, a correspondence from a set $B$ to a set $A$ is simply a subset $K$ of $B\times A$. These include relations on a set $A$ (as correspondences from $A$ to $A$, or, meaning the same, on $A$), and they include functions from $B$ to $A$ (represented by their graphs, clearly subsets of $B\times A$). In graph theory, the directed graphs with vertex set $V$ are in one-to-one correspondence with the correspondences on $V$ (each indicating the set of edges of the directed graph). Directed graphs occur in the theory of Markov chains on the (discrete) state space $S$, when we ask which points in $S$ are connected with a nonzero transition probability. A $C^*$-correspondence from a $C^*$-algebra $\cal A$ to a $C^*$-algebra $\cal B$ is an $\cal A$-$\cal B$-bimodule $E$ with a $\cal B$-valued inner product turning $E$ into a Hilbert $\cal B$-module and such that the left action of $\cal A$ defines a nondegenerate $*$-homomorphism into the adjointable operators on $E$. (It is worth noting how asymmetric this is in the roles played by $\cal B$ and by $\cal A$ in this definition.) $C^*$-Correspondences arise in purely operator-algebraic contexts such as Kasparov's KK-theory. Particularly rich sources of $C^*$-correspondences arise from reversible and irreversible dynamics both classical and quantum, where the latter includes the former as the special where the $C^*$-algebras standing for a classical system are commutative algebras of functions. For instance, reversible dynamics give rise to so-called product systems of $C^*$-correspondences; they can be classified by their product systems (in a specific sense), and they can be constructed from product systems. Irreversible dynamics such as a (quantum or classical) Markov semigroup give rise, first, to so-called subproduct systems which, then, are transformed by an inductive limits construction into a product system that, by a further inductive limit construction, gives rise to a reversible dynamics that dilates the irreversible one (meaning that compressing the reversible dynamics with a conditional expectation, we get the irreversible one). In the classical case, the double inductive limit construction is the dual of the classical Daniell-Kolmogorov construction (a projective limit!) of a classical Markov process from its semigroup of transition probabilities. In this talk we illustrate that $C^*$-correspondences fully merit to be considered generalized classical correspondences. They would not only merit to be called quantum correspondences when cosidering them over noncommutative $C^*$-algebras; in particular, in the classical case when the algebras are commutative, $C^*$-correspondences turn out to be the most noncommutative objects imaginable: In fact, the $C^*$-correspondence associated with the infinitesimal generator of the Markov semigroup of a classical Markov process on a finite state space $S$ with $\#S=n$ (in which case the $C^*$-algebra is ${\bf C}^n$, the diagonal subalgebra of $M_n$), does not contain a single nonzero element that commutes with the algebra elements: $$ bx ~=~ xb ~~~\forall b\in{\bf C}^n~~~ ~\Longrightarrow~ ~~~ x=0. $$ Some properties may be pushed forward from classical to $C^*$-correspondences (where some are easy, and some are surprisingly tricky leading to new notions for $C^*$correspondences). Some properties may be pushed forward only cum grano salis (under topological constraints). The notions of the transpose of a correspondence and the composition of correspondences, entirely resist to find a full analogue. At least, in the commutative case, the transposition of a correspondence is reflected by the commutant of von Neumann correspondences. (It is work in progress to do this also in the noncommutative case.) The composition of correspondences is, beyond doubt, closely related to the (internal) tensor product of $C^*$-correspondences. But the latter is ``bigger'' than the $C^*$-correspondence of the classical composition. (This is behind the inductive limit construction dual to the Daniell-Kolmogorov construction for a aclassical Markov semigroup. It leads to a new type of classical correspondences, the weighted correspondences.)

Area: CS12 - Quantum probability and related fields (Vitonofrio Crismale and Veronica Umanità)

Keywords: Classical and quantum probability; classical and quantum Markov chains; operator algebras; C*-correspondences; quantum correspondences

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