Contents

Idea

In mathematics, one often has an equivalence of categories between algebra-like objects and space-like objects. Such an idea has many incarnations: Stone duality, Gelfand duality, etc., and in this article we make some observations that are common to these dualities.

Given an algebra-like object $A$, we assign to it its poset of ideals (typically defined as kernels of homomorphisms $A\to B$), which is interpreted as the poset of opens of some space $S$.

The technical term for such posets is locale, which is a notion very closely related to topological spaces. In particular, from any locale one can canonically extract a topological space, and this is the topological space $S$ produced in many classical Stone-type dualities. The points of $S$ are ideals corresponding to morphisms $A\to k$, where $k$ is often a particularly simple algebra. These often turn out to be maximal ideals in $A$.

Conversely, given a space-like object $S$, we assign to it the algebra of morphisms $S\to k$, where $k$ is often the “same” algebra $k$ as above, only this time its underlying object is a space, not just a set.

Some examples from general topology, measure theory, differential geometry, algebraic geometry, and complex geometry (the list is very much incomplete):

algebra	homomorphism	$k$	ideal	space	maps	duality
Boolean algebra	homomorphism	$\mathbf{Z}/2$	ideal	compact totally disconnected Hausdorff space (Stone space)	continuous map	Stone duality
complete Boolean algebra	complete homomorphism	$\mathbf{Z}/2$	closed ideal	compact extremally disconnected Hausdorff space (Stonean space)	open continuous map	Stonean duality
localizable Boolean algebra	complete homomorphism	$\mathbf{Z}/2$	closed ideal	hyperstonean space	open continuous map
localizable Boolean algebra	complete homomorphism	$\mathbf{Z}/2$	closed ideal	compact strictly localizable enhanced measurable space	measurable map
commutative von Neumann algebra	normal *-homomorphism	$\mathbf{C}$	ultraweakly closed *-ideal	compact strictly localizable enhanced measurable space	measurable map
commutative unital C*-algebra	*-homomorphism	$\mathbf{C}$	norm-closed *-ideal	compact Hausdorff space	continuous map	Gelfand duality
commutative algebra over $k$	homomorphism	$k$	ideal	coherent space / affine scheme	continuous map / morphism of schemes	Zariski duality
finitely generated germ-determined C^∞-ring	C$^\infty$-homomorphism	$\mathbf{R}$	germ-determined ideal	smooth locus (e.g., smooth manifold)	smooth map	Milnor duality
finitely presented complex EFC-algebra	EFC-homomorphism	$\mathbf{C}$	ideal	globally finitely presented Stein space	holomorphic map	Stein duality

and:

The duality relevant to the spectral theory is the duality between commutative von Neumann algebras and compact strictly localizable enhanced measurable spaces.

Given a normal operator $T$ on a Hilbert space $H$, $T$ generates a commutative von Neumann algebra $A$ inside $B(H)$, i.e., bounded operators on $H$. (This is precisely the point where normality is crucial; without the relation $T^*T=TT^*$ the algebra generated by $T$ will be noncommutative.)

By the cited duality, the commutative von Neumann algebra $A$ is dual to a compact strictly localizable enhanced measurable space $Spec A$. This is indeed the spectrum of $T$ in the usual sense. Under this equivalence, the element $T\in A$ corresponds to the measurable map $Spec A\to \mathbf{C}$ given by inclusion of $Spec A$ into $\mathbf{C}$.

One may ask whether we can recover the full spectral theorem for a normal operator in this manner. This is possible once Stone duality is upgraded to Serre–Swan-type duality between modules and vector bundle-like objects (including, e.g., sheaves of modules etc.).

Given a vector bundle-like object $V\to S$, we assign to it its module of sections, which is a module over the algebra of maps $S\to k$. Conversely, given a module $M$ over $A$, the corresponding vector bundle-like object $V\to S$ over $S=Spec A$ has as its fiber over some point $s\in S$ the vector space $M/IM$, where $I$ is the ideal corresponding to $s$. (Many details are necessarily omitted in this brief sketch.)

Typically, genuine vector bundles correspond to dualizable modules (dualizable with respect to the tensor product over $A$). Non-dualizable module tend to correspond to sheaves of modules that are not vector bundles, e.g., skyscraper sheaves etc.

module	vector bundle-like object
module over a Boolean algebra	sheaf of $\mathbf{Z}/2$-vector spaces
Hilbert W*-module over a commutative von Neumann algebra	measurable field of Hilbert spaces
W*-representations of a commutative von Neumann algebra on a Hilbert space	measurable field of Hilbert spaces
Hilbert C*-module over a commutative unital C*-algebra	continuous field of Hilbert spaces
module over a commutative ring	quasicoherent sheaf of $\mathcal{O}$-modules over an affine scheme
dualizable module over a commutative algebra over $k$	algebraic vector bundle
dualizable module over a finitely generated germ-determined C$^\infty$-ring	smooth vector bundle
dualizable module over finitely presented complex EFC-algebra	holomorphic vector bundle

The duality relevant to the spectral theory is the duality between representations of a commutative von Neumann algebras on a Hilbert space and measurable fields of Hilbert spaces.

Given a normal operator $T$ on a Hilbert space $H$, $T$ generates a commutative von Neumann algebra $A$ inside $B(H)$, whose spectrum $Spec A$ is a compact strictly localizable enhanced measurable space.

Furthermore, the inclusion of $A$ into $B(H)$ is a representation of $A$ on $H$. As such, it corresponds under the Serre–Swan-type duality to a measurable field of Hilbert spaces over $A$. This is precisely the measurable field produced by the classical spectral theorem.

Under the duality, the operator $T$ corresponds to the operator that multiplies a given section of this measurable field of Hilbert spaces by the complex-valued function $Spec A\to \mathbf{C}$ produced above. Thus, we recovered the entire content of the classical spectral theorem.

In fact, the above considerations work equally well to establish the spectral theorem for an arbitrary family (not necessarily finite) of commuting normal operators.

Related entries

• Isbell duality

• Gelfand duality

• Stone duality

• Zariski spectrum?

• modules over a ring are equivalent to quasicoherent sheaves over its spectrum

• Serre-Swan theorem, smooth Serre-Swan theorem

• noncommutative geometry

• reconstruction theorem, Gabriel-Rosenberg theorem