Topos polar decomposition

Background and motivation: a polar decomposition of an element $T$ of a $C^{\ast }$-algebra is a product $T=UA$ where $U$ is a partial isometry, $A$ is positive, and $\text{Ann}(U)=\text{Ann}(A)$, where $\text{Ann}(U)=\{X\mid UX=0\}$. Such a decomposition is necessarily unique and we have $A=|T|=\sqrt{T^{\ast }T}$. Let us say that a $C^{\ast }$-algebra admits polar decomposition in the traditional sense if every element of it has such a polar decomposition. If a $C^{\ast }$-algebra admits polar decomposition in this sense, then it has the following property: for every element $T$ there is a projection $P$ such that $\text{Ann}(T)=\text{Ann}(P)$ (the algebra is said to be Rickart). However, not every $C^{\ast }$-algebra is Rickart, so we would like to broaden the scope of polar decomposition by seeking a generalization of it that agrees with the traditional notion in the Rickart case. It turns out that the criteria we find, which we shall call topos polar decomposition, unfolds as a representation of the (multiplicative) cosets of the partial isometries of the algebra by the positive elements.

A topos approach: the objects of the division category $ℳ$ associated with a unital ring (with unit $1$) are the elements of the ring, denoted $r,s$, etc. A morphism $u:r\to s$ of $ℳ$ is an element $u$ of the ring such that $\text{Ann}(u)=\text{Ann}(r)$, and $\exists au=sa$, which we write $s\mid u$. For instance, if $s\mid r$, then $r:r\to s$ is a morphism of $ℳ$. Composition in $ℳ$ is as follows, which makes sense because $s\mid u$: if $u=sa$, then $\frac{vu}{s}=va$.

Let us say that a subcategory $𝒞\to ℳ$ is étale if: i) for any two objects $r,s$ of $𝒞$, if $s\mid r$, then the morphism $r:r\to s$ is in $𝒞$, and ii) in a triangle of composable morphisms of $ℳ$ (as above) if $v,\frac{vu}{s}\in 𝒞$, then $u\in 𝒞$. A subcategory is wide if it has the same set of objects as the including category. Our analysis of polar decomposition uses the following basic tool: the wide étale subcategories of $ℳ$ correspond to quotients of the representable presheaf $\widehat{1}$ in the topos of presheaves on $ℳ$, where $\widehat{1}(r)=ℳ(r,1)=\{u\mid \text{Ann}(u)=\text{Ann}(r)\}$. The correspondence associates with a wide étale subcategory $𝒞\to ℳ$ its presheaf quotient of multiplicative cosets $\widehat{1}\to \widehat{1}/𝒞$. On the other hand, it associates with an element $x:\widehat{1}\to ℱ$ of a presheaf $ℱ$ on $ℳ$ what we call its principal fiber, denoted $\mathrm{𝑃𝑓}(x)\to ℳ$. By definition, the objects of $\mathrm{𝑃𝑓}(x)$ are the elements of the ring, with morphisms $u:r\to s$ such that $x\cdot r=x\cdot u$. The subcategory $\mathrm{𝑃𝑓}(x)\to ℳ$ is wide étale.

$C^{\ast }$-algebras: the partial isometries of a unital $C^{\ast }$-algebra (with unit $I$) collectively form a subcategory $\partial \to ℳ$ of the division category of the underlying ring of the algebra on the projections. The subcategory $\partial$ is étale, but it is not wide. Nevertheless, it generates a wide étale subcategory $⟨\partial ⟩\to ℳ$ depicted below (right). On the other hand, the positive elements of the algebra organize themselves as a quotient of the representable presheaf $\widehat{I}$, labeled $d$ in the triangle below (left). It follows that the principal fiber $\mathrm{𝑃𝑓}(d)$ of $d$ consists of all morphisms $U:R\to S$ of $ℳ$ such that $R^{\ast }R=U^{\ast }U$. Furthermore, $\partial$ is a full subcategory of $\mathrm{𝑃𝑓}(d)$, so that $⟨\partial ⟩$ is a subcategory of $\mathrm{𝑃𝑓}(d)$, and $\partial$ is full in $⟨\partial ⟩$.

The cosets of $⟨\partial ⟩$ form a quotient $q$ depicted above, and moreover, there is a canonical natural transformation $\epsilon$ comparing $q$ and $d$. Let us say that a $C^{\ast }$-algebra admits topos polar decomposition if $\epsilon$ is an isomorphism. This condition holds if and only if $⟨\partial ⟩=\mathrm{𝑃𝑓}(d)$.

Proposition: A $C^{\ast }$-algebra admits polar decomposition in the traditional sense if and only the algebra is Rickart, and it admits topos polar decomposition.