From Kleisli Categories to Commutative C*-algebras: Probabilistic Gelfand Duality

Furber, Robert; Jacobs, Bart

doi:10.2168/lmcs-11(2:5)2015

Cited by 29 publications

(40 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [22,Lemma 4.1] it is shown that E(X) can equivalently be described as the set of states Stat( ∞ (X)) on the commutative C * -algebra ∞ (X) of bounded functions X → C. This gives a clear similarity with the Radon monad described below, in Subsection 3.5.…”

Section: The Expectation Monad E On Setsmentioning

confidence: 97%

“…There are two equivalent ways to define the expectation monad E, using maps of effect modules (as in the original description from [38]), or using maps of C * -algebras, see [22]. Here we shall follow the first approach, mainly because the second approach is very similar to the one used below for the Radon monad.…”

Section: The Expectation Monad E On Setsmentioning

confidence: 99%

“…The Radon monad occurs in [57,21,22]. The main result of [21], presented as a probabilistic version of Gelfand duality, states that the Kleisli category K (R) of the Radon monad is the opposite (CCstar PU ) op of the category of commutative C * -algebras, with positive unital maps between them.…”

Section: The Radon Monad R On Chmentioning

confidence: 99%

See 2 more Smart Citations

From probability monads to commutative effectuses

Jacobs

2018

Journal of Logical and Algebraic Methods in Programming

Self Cite

View full text Add to dashboard Cite

Effectuses have recently been introduced as categorical models for quantum computation, with probabilistic and Boolean (classical) computation as special cases. These 'probabilistic' models are called commutative effectuses, and are the focus of attention here. The paper describes the main known 'probability' monads: the monad of discrete probability measures, the Giry monad, the expectation monad, the probabilistic power domain monad, the Radon monad, and the Kantorovich monad. It also introduces successive properties that a monad should satisfy so that its Kleisli category is a commutative effectus. The main properties are: partial additivity, strong affineness, and commutativity. It is shown that the resulting commutative effectus provides a categorical model of probability theory, including a logic using effect modules with parallel and sequential conjunction, predicate-and state-transformers, normalisation and conditioning of states.

show abstract

Section: The Expectation Monad E On Setsmentioning

confidence: 97%

Section: The Expectation Monad E On Setsmentioning

confidence: 99%

See 1 more Smart Citation

From probability monads to commutative effectuses

Jacobs

2018

Journal of Logical and Algebraic Methods in Programming

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the case of deterministic computing it can be axiomatized [26]. For related work in the case of probabilistic computing, see also [27], [28]. Finally, the special case of von Neumann algebras has been studied domain-theoretically [29], but forms a somewhat degenerate setting: the domain-theoretic notions listed above are not equivalent there, and come down to finite-dimensionality, which is relatively uninteresting from the perspective of information approximation; see Section IX for a detailed comparison.…”

Section: B Related Workmentioning

confidence: 99%

Domains of Commutative C*-Subalgebras

Heunen

Lindenhovius

2015

2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science

View full text Add to dashboard Cite

Abstract-Operator algebras provide uniform semantics for deterministic, reversible, probabilistic, and quantum computing, where intermediate results of partial computations are given by commutative subalgebras. We study this setting using domain theory, and show that a given operator algebra is scattered if and only if its associated partial order is, equivalently: continuous (a domain), algebraic, atomistic, quasi-continuous, or quasialgebraic. In that case, conversely, we prove that the Lawson topology, modelling information approximation, allows one to associate an operator algebra to the domain.

show abstract

“…This functor is full and faithful, see [8]. On the other side, the state functor sends a C * -algebra A to the (convex) set of its states, given by the homomorphisms A → C. This diagram is enriched over convex sets.…”

Section: Quantum Computation Brieflymentioning

confidence: 99%

Dijkstra and Hoare monads in monadic computation

Jacobs

2015

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

The Dijkstra and Hoare monads have been introduced recently for capturing weakest precondition computations and computations with pre-and post-conditions, within the context of program verification, supported by a theorem prover. Here we give a more general description of such monads in a categorical setting. We first elaborate the recently developed view on program semantics in terms of a triangle of computations, state transformers, and predicate transformers. Instantiating this triangle for different computational monads T shows how to define the Dijkstra monad associated with T , via the logic involved.Subsequently we give abstract definitions of the Dijkstra and Hoare monad, parametrised by a computational monad. These definitions presuppose a suitable (categorical) predicate logic, defined on the Kleisli category of the underlying monad. When all this structure exists, we show that there are maps of monads (Hoare) ⇒ (State) ⇒ (Dijkstra), all parametrised by a monad T .

show abstract

From Kleisli Categories to Commutative C*-algebras: Probabilistic Gelfand Duality

Cited by 29 publications

References 27 publications

From probability monads to commutative effectuses

From probability monads to commutative effectuses

Domains of Commutative C*-Subalgebras

Dijkstra and Hoare monads in monadic computation

Contact Info

Product

Resources

About