[Paper Review] Measures on projections in a $W^*$-algebra of type $I_2$

A. N. Sherstnev|arXiv (Cornell University)|Dec 23, 2011

Advanced Operator Algebra Research2 references4 citations

TL;DR

This paper establishes that every positive measure on projections in a $W^*$-algebra of type $I_2$ arises as the squared norm of a Hilbert-valued orthogonal vector measure. By constructing such a vector measure explicitly using the structure of $M_2$-valued projections and leveraging induction over spectral projections, the author extends Hamhalter's result to all $W^*$-algebras, including those with type $I_2$ summands, thereby completing the characterization of measures on projections via orthogonal vector measures.

ABSTRACT

It is shown that for every measure $m$ on projections in a $W^*$-algebra of type $I_2$, there exists a Hilbert-valued orthogonal vector measure $μ$ such that $\|μ(p)\|^2= m(p)$ for every projection $p$. With regard to J. Hamhalter's result (Proc. Amer. Math. Soc., 110 (1990), 803-806) it means that the assertion is valid for an arbitrary $W^*$-algebra.

Motivation & Objective

• To resolve the open problem of whether every positive measure on projections in a $W^*$-algebra of type $I_2$ can be represented as the squared norm of an orthogonal vector measure.
• To extend Hamhalter's result—previously valid only for $W^*$-algebras without type $I_2$ direct summands—to all $W^*$-algebras, including those of type $I_2$.
• To construct an explicit orthogonal vector measure $\mu$ such that $\|\mu(p)\|^2 = m(p)$ for every projection $p$ in a type $I_2$ $W^*$-algebra.
• To provide a unified framework for representing measures on projections via orthogonal vector measures across all $W^*$-algebras, completing the classification.

Proposed method

• Represent a $W^*$-algebra of type $I_2$ as $\mathcal{M} \otimes M_2$, where $\mathcal{M}$ is commutative and $M_2$ is the algebra of $2 \times 2$ complex matrices.
• Parameterize projections in $\mathcal{N} = \mathcal{M} \otimes M_2$ using triples $(x, v, \pi)$ with $x \in \mathcal{M}^+$, $v \in \mathcal{M}^{\text{un}}$, and $\pi \in \mathcal{M}^{\text{pr}}$, satisfying $0 \leq x \leq \pi$ and $\text{rp}(x(\pi - x)) = \pi$.
• Define a finitely additive orthogonal vector measure $\mu$ on projections by assigning values in a Hilbert space $H$ such that $\|\mu(p)\|^2 = m(p)$, using spectral decomposition and inductive extension over orthogonal projections.
• Use Zorn’s lemma to construct a maximal family of pairwise orthogonal projections $\{\pi_j\}$ in $\mathcal{M}^\text{pr}$ to decompose the problem into manageable components.
• Construct density functions $h_{\gamma 1}, h_{\gamma 2}, k_{\gamma 1}, k_{\gamma 2}$ in $L^2(\Omega, \nu)$ satisfying orthogonality and norm conditions to define $\mu(p)$ for each projection $p$, ensuring consistency across decompositions.
• Verify that the extended $\mu$ satisfies the axioms of an orthogonal vector measure: orthogonality of images of orthogonal projections and norm preservation of the measure $m$.

Experimental results

Research questions

• RQ1 Can every positive measure on projections in a $W^*$-algebra of type $I_2$ be represented as the squared norm of a Hilbert-valued orthogonal vector measure?
• RQ2 Does the representation result established by Hamhalter for $W^*$-algebras without type $I_2$ summands extend to those with such summands?
• RQ3 Is there a constructive method to build an orthogonal vector measure $\mu$ such that $\|\mu(p)\|^2 = m(p)$ for all projections $p$ in a type $I_2$ $W^*$-algebra?
• RQ4 How can the structure of projections in $\mathcal{M} \otimes M_2$ be exploited to define and extend such a vector measure consistently?

Key findings

• For every measure $m$ on projections in a $W^*$-algebra of type $I_2$, there exists a Hilbert-valued orthogonal vector measure $\mu$ such that $\|\mu(p)\|^2 = m(p)$ for all projections $p$.
• The construction is explicit and relies on the decomposition of projections in $\mathcal{M} \otimes M_2$ using triples $(x, v, \pi)$, enabling inductive extension over orthogonal components.
• The proof uses spectral decomposition and $L^2$-valued density functions to define $\mu(p)$, ensuring norm preservation and orthogonality of images under orthogonal projections.
• The result completes Hamhalter's earlier result by showing that the representation via orthogonal vector measures holds for all $W^*$-algebras, not just those without type $I_2$ summands.
• The extension of $\mu$ to the entire projection lattice is achieved via Zorn’s lemma and consistent gluing over orthogonal families of projections.
• The final result implies that every positive measure on projections in an arbitrary $W^*$-algebra arises as the squared norm of an orthogonal vector measure into some complex Hilbert space $H$.