nLab measurable Gelfand duality

Idea

The adjunction known as measurable Gelfand duality is a Gelfand-type duality between measurable spaces and particular (σ-complete?) C*-algebras. It is an algebraic approach to point-free measure theory?.

It is closely related to the usual Gelfand duality, as well as to Loomis-Sikorski duality, which can be considered its Stone-type counterpart.

Main concepts

As in ordinary Gelfand duality (as well as in algebraic geometry and related fields), one wants to study the properties of a class of spaces by means of the algebras of functions defined on them.

In this case, the spaces we study are measurable spaces, that is, sets equipped with a σ-algebra. Bounded measurable functions on them form a particular algebraic structure (a commutative σ-C*-algebra?), and under some circumstances (in the center of the adjunction, see below) the spaces and these algebras determine one another.

From measurable spaces to σ-C*-algebras

Given a measurable space $X$, define by $\mathcal{L}^\infty(X)$ the set of bounded measurable functions $h:X\to\mathbb{C}$. With its usual algebraic structure and norm, the set $\mathcal{L}^\infty$ forms a commutative C*-algebra, and since it is closed under countable bounded monotone suprema, it is moreover a σ-C*-algebra?.

Given a measurable function $f:X\to Y$, the precomposition $h\mapsto h\circ f$ is a homomorphism of σ-C*-algebras $f^*:\mathcal{L}^\infty(Y)\to \mathcal{L}^\infty(X)$. This gives a functor $Meas^{op}\to \sigma C^*Alg$.

From σ-C*-algebras to measurable spaces

Given a commutative σ-C*-algebra $A$, define its spectrum $Sp(A)$ as the set of σ-C*-homomorphisms $p:A\to\mathbb{C}$ (a.k.a. points of $A$). For every $h\in A$, define also the map We equip $Sp(A)$ with the coarsest σ-algebra making the maps $\epsilon_h$ measurable for all $h\in A$ (cf. the σ-algebra of the Giry monad).

Given a σ-C*-homomorphism $\phi:A\to B$, precomposition $p\mapsto p\circ\phi$ gives a function $\phi^*:Sp(B)\to Sp(A)$, which is measurable for the σ-algebras defined above. We therefore have a functor $Sp:\sigma C^*Alg^\op\to Meas$.

The adjunction

The functors $\mathcal{L}^\infty:Meas^{op}\to \sigma C^*Alg$ and $Sp:\sigma C^*Alg^{op}\to Meas$ form a (contravariant) idempotent adjunction:

• The unit has as components measurable functions $\eta:X\to Sp(\mathcal{L}^\infty(X))$ which map a point $x\in X$ to the Dirac delta functional

• The counit has as components morphisms of $\sigma C^*Alg^{op}$ in the form $\epsilon^{op}:\mathcal{L}^\infty(Sp(A))\to A$, or equivalently, morphisms of $\sigma C^*Alg$ of the form $\epsilon:A\to \mathcal{L}^\infty(Sp(A))$, mapping $h\in A$ to the functional $\epsilon_h:Sp(A)\to\mathbb{C}$ defined above.

The adjunction induces an idempotent monad on $Meas$ (the monad of zero-one measures), and an idempotent monad on $\sigma C^*Alg$ as well (or equivalently, an idempotent comonad on $\sigma C*Alg^{op}$).

As usual with idempotent adjunctions, we have that the center is equivalently given by

• The full subcategory of $Meas$ whose objects are those spaces for which the unit $\eta:X\to Sp(\mathcal{L}^\infty(X))$ is an isomorphism. These are sometimes called sober measurable spaces. Analogously to sober topological spaces, these are the spaces whose points are uniquely determined by the σ-algebra.
• The full subcategory of $\sigma C^*Alg$ whose objects are those σ-C*-algebras for which the counit $\epsilon:A\to \mathcal{L}^\infty(Sp(A))$ is an isomorphism. These are sometimes called spatial or concrete σ-C*-algebras.

Because of this, sometimes one calls the monad on $Meas$ the sobrification monad. It is the same monad as the one arising from Loomis-Sikorski duality, see below.

Further results

Equivalence with Loomis-Sikorski duality

In ordinary measure theory, it is well known that the measurable subsets $A\subseteq X$ are “included” in the measurable functions via the indicator functions $1_A$. Those are exactly the measurable functions which take values only in $0$ and $1$. Conversely, from the σ-algebra one can recover exactly the measurable functions on $X$ (for example, by taking suprema of linear combinations of indicators).

We can abstract this idea of “functions with values $0$ and $1$” as the projection elements of a C*-algebra, that is, those elements $h\in A$ such that $h=h*$ (self-adjoint, generalizing real-valued) and $hh=h$ (idempotent). As one can show, the projection elements of a commutative σ-C*algebra indeed form a σ-complete Boolean algebra, the abstract analogue of a σ-algebra via the Loomis-Sikorski duality. Moreover, every σ-complete Boolean algebra can be shown as arising in this way.

We therefore have an equivalence of categories between σ-C*algebras and σ-complete Boolean algebras, commuting with the left and right-adjoints as in the following diagram.

In other words, we have an isomorphism of adjunctions, and so, the two dualities (measurable Gelfand and Loomis-Sikorski) can be seen as a Gelfand-type and a Stone-type description of one and the same duality.

More details in FL’25, Section 4.

Stochastic version

(…)

See also

• point-free measure theory?, Loomis-Sikorski duality
• Stochastic Gelfand duality?, Markov kernel, Stoch
• monotone-complete C*-algebra?, σ-complete Boolean algebra
• Gelfand duality, point-free topology
• zero-one measure, zero-one kernel, Markov kernel, Stoch
• Markov category, probability monad

References

• Kazuyuki Saitô and J. D. Maitland Wright, Monotone-Complete C*-algebras and Generic Dynamics, Springer, 2015.

• Gert K. Pedersen. C*-algebras and their automorphism groups, Academic Press, 2018.

• Tobias Fritz and Antonio Lorenzin, Categories of abstract and noncommutative measurable spaces, 2025. (arXiv)

category: probability