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

Abstract

Abstract. C * -algebras form rather general and rich mathematical structures that can be studied with different morphisms (preserving multiplication, or not), and with different properties (commutative, or not). These various options can be used to incorporate various styles of computation (set-theoretic, probabilistic, quantum) inside categories of C * -algebras. At first, this paper concentrates on the commutative case and shows that there are functors from several Kleisli categories, of monads that are relevant to model probabilistic computations, to categories of C * -algebras. This yields a new probabilistic version of Gelfand duality, involving the "Radon" monad on the category of compact Hausdorff spaces. We then show that the state space functor from C * -algebras to Eilenberg-Moore algebras of the Radon monad is full and faithful. This allows us to obtain an appropriately commuting state-and-effect triangle for C * -algebras.

Quantum Programs as Kleisli Maps

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Furber and Jacobs have shown in their study of quantum computation that the category of commutative C * -algebras and PU-maps (positive linear maps which preserve the unit) is isomorphic to the Kleisli category of a comonad on the category of commutative C * -algebras with MIU-maps (linear maps which preserve multiplication, involution and unit). [3] In this paper, we prove a non-commutative variant of this result: the category of C * -algebras and PU-maps is isomorphic to the Kleisli category of a comonad on the subcategory of MIU-maps.A variation on this result has been used to construct a model of Selinger and Valiron's quantum lambda calculus using von Neumann algebras.[1]

Quantum Programs as Kleisli MapsWe will also see that the opposite (W * NCPsU ) op of the category of normal completely positive subunital maps between von Neumann algebras is Kleislian over the subcategory (W * NMIU ) op of normal unital *homomorphisms. This fact is used in [1] to construct an adequate model of Selinger and Valiron's quantum lambda calculus using von Neumann algebras.

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“…Let CC * MIU denote the category of MIU-maps between commutative C * -algebras and let CC * PU denote the category of PU-maps between commutative C * -algebras. From the work in [3] it follows that the embedding CC * MIU −→ CC * PU has a left adjoint F and moreover that F A = CStatA , where StatA is the topological space of PU-maps from A to C with pointwise convergence and CStatA is the C * -algebra of continuous functions from StatA to C.…”

“…The semantics of a non-deterministic program that takes two bits and returns three bits can be described as a multimap (= binary relation) from {0, 1} 2 to {0, 1} 3 . Similarly, a program that takes two qubits and returns three qubits can be modelled as a positive linear unit-preserving map from M 2 ⊗ M 2 ⊗ M 2 to M 2 ⊗ M 2 , where M 2 is the C * -algebra of 2 × 2-matrices over C.…”

Quotient-Comprehension Chains

“…For example, we get a quotient-comprehension chain for the opposite category of commutative unital C * -algebras with unital * -homomorphisms via Gelfand's duality (see e.g. [6]), and for the opposite category of unital Archimedean Riesz spaces with Riesz homomorphisms via Yosida's duality [23]. Interestingly, there are quotient-comprehension chains for 'algebraic' categories which do not seem to have a 'spacial' counterpart such as the category of commutative rings and homomorphisms, such as the category CRng op of commutative rings and homomorphisms, as we will see in Section 5.…”

“…The counit map π p : {X| p} → D(X +1) and the unit ξ p : X → D(X/p+1) are given by π p (x) = 1|x and ξ p (x) = p ⊥ (x)|x + p(x)| * . We can consider their combination, like in diagram (6), in K (D) +1 .…”