[Paper Review] Completeness of $n$--tuple of projections in $C^*$--algebras

Shanwen Hu, Yifeng Xue|arXiv (Cornell University)|Oct 17, 2012

Advanced Operator Algebra Research8 references3 citations

TL;DR

This paper establishes necessary and sufficient conditions for an $n$-tuple of projections in a unital $C^*$-algebra to be complete, meaning the algebra decomposes as a direct sum of the closed subspaces $P_i\mathcal{A}$. It characterizes completeness via invertibility of certain operator combinations, proves perturbation stability under small off-diagonal norms, and shows path-connectedness of the set of complete $n$-tuples in key algebras like the Calkin algebra.

ABSTRACT

Let $(P_1,...,P_n)$ be an $n$--tuple of projections in a unital $C^*$--algebra $å$. We say $\pn$ is complete in $å$ if $å$ is the linear direct sum of the closed subspaces $P_1å,...,P_nå$. In this paper, we give some necessary and sufficient conditions for the completeness of $\pn$ and discuss the perturbation problem and topology of the set of all complete $n$--tuple of projections in $å$. Some interesting and significant results are obtained in this paper.

Motivation & Objective

• To characterize when an $n$-tuple of projections in a unital $C^*$-algebra is complete, i.e., when the algebra is the linear direct sum of the closed subspaces $P_i\mathcal{A}$.
• To extend known results for $n=2$ to $n \geq 3$, where the structure of projections summing to the identity is less understood.
• To study the perturbation stability of complete $n$-tuples, particularly under small off-diagonal norms between projections.
• To analyze the topology of the set of complete $n$-tuples, including path-connectedness and homotopy equivalence classes in specific $C^*$-algebras.

Proposed method

• Established equivalent conditions for completeness using invertibility of operator combinations: $\lambda \sum_{j \neq i} P_j + P_i \in GL(\mathcal{A})$ for $\lambda \in [1-n, 0)$.
• Used the transformation $Q_i = A^{-1/2} P_i A^{-1/2}$ with $A = \sum P_i$ to reduce the problem to mutually orthogonal projections.
• Introduced a homotopy equivalence relation $\sim_h$ to classify complete $n$-tuples via unitary conjugation and path-connectedness in the algebra.
• Applied the Calkin algebra construction to show path-connectedness of $\mathbf{PC}_n(\mathcal{A})$ when $\mathcal{A} = B(H)/\mathcal{K}(H)$.
• Used continuous paths of unitaries $U_i(t)$ to construct a homotopy between any complete $n$-tuple and a tuple of orthogonal projections.
• Derived a sharp perturbation bound: if $\|P_i A^{-1} P_j\| < \left[(n-1)\|A^{-1}\|\|A\|^2\right]^{-1}$ for $i \neq j$, then $P_i A^{-1} P_j = 0$, implying orthogonality.

Experimental results

Research questions

• RQ1 What are necessary and sufficient conditions for an $n$-tuple of projections $(P_1, \dots, P_n)$ in a unital $C^*$-algebra $\mathcal{A}$ to be complete, i.e., $\mathcal{A} = \bigoplus_{i=1}^n P_i\mathcal{A}$?
• RQ2 How does the completeness of an $n$-tuple behave under small perturbations, especially when off-diagonal terms $\|P_i A^{-1} P_j\|$ are small?
• RQ3 What is the topological structure of the set $\mathbf{PC}_n(\mathcal{A})$ of complete $n$-tuples in $\mathcal{A}$, particularly in the Calkin algebra?
• RQ4 When are two complete $n$-tuples homotopy equivalent, and what invariants classify their connected components?

Key findings

• An $n$-tuple $(P_1, \dots, P_n)$ is complete in $\mathcal{A}$ if and only if $\lambda \sum_{j \neq i} P_j + P_i \in GL(\mathcal{A})$ for all $i$ and $\lambda \in [1-n, 0)$.
• If $A = \sum_{i=1}^n P_i$ is invertible and $\|P_i A^{-1} P_j\| < \left[(n-1)\|A^{-1}\|\|A\|^2\right]^{-1}$ for $i \neq j$, then $P_i A^{-1} P_j = 0$, implying the projections are orthogonal in the transformed algebra.
• For any $\epsilon \in (0,1)$, if $\|P_i P_j\| < \epsilon$ for $i \neq j$, then there exists a tuple of mutually orthogonal projections $(P'_1, \dots, P'_n)$ such that $\|P_i - P'_i\| < 2(n-1)\epsilon$, improving prior estimates.
• In the Calkin algebra $\mathcal{A} = B(H)/\mathcal{K}(H)$, the set $\mathbf{PC}_n(\mathcal{A})$ is path-connected, meaning all complete $n$-tuples are homotopy equivalent.
• For $\mathcal{A} = M_k(\mathbb{C})$, the set $\mathbf{PC}_n(\mathcal{A})$ is connected if and only if $n = k$, and disconnected for $2 \leq n \leq k-1$, with connected components classified by the trace tuple $(\mathrm{Tr}(P_1), \dots, \mathrm{Tr}(P_{n-1}))$.
• The homotopy equivalence class of a complete $n$-tuple is determined by its unitary conjugacy class, and $\sim_h$-equivalence preserves completeness and trace invariants.