Additive preservers of mutual strong Birkhoff-James orthogonality on finite-dimensional $C^\ast$-algebras

Technical Summary: Additive Preservers of Mutual Strong Birkhoff-James Orthogonality on Finite-Dimensional C-Algebras*

Problem Statement
The paper investigates the structure of additive surjective maps $\Phi: A \to A$ on finite-dimensional complex $C^*$-algebras that preserve "mutual strong Birkhoff-James orthogonality" in one direction. In a normed space $X$, two vectors $x, y$ are Birkhoff-James (BJ) orthogonal ($x \perp_{BJ} y$) if $\|x + \lambda y\| \ge \|x\|$ for all scalars $\lambda$. When $X$ is a right Hilbert $C^*$-module over a $C^*$-algebra $A$, $x$ is strongly BJ orthogonal to $y$ ($x \perp^S_{BJ} y$) if $\|x + yc\| \ge \|x\|$ for all $c \in A$. The authors focus on the mutual version, denoted $x \perp\perp^S_{BJ} y$, where both $x \perp^S_{BJ} y$ and $y \perp^S_{BJ} x$ hold.

The specific setting is a finite-dimensional $C^*$-algebra $A$, which is isomorphic to a direct sum of matrix algebras:
$$A \cong M_{n_1}(\mathbb{C}) \oplus M_{n_2}(\mathbb{C}) \oplus \cdots \oplus M_{n_k}(\mathbb{C}).$$
The central question is to classify all additive surjections $\Phi$ that satisfy:
$$x \perp\perp^S_{BJ} y \implies \Phi(x) \perp\perp^S_{BJ} \Phi(y).$$

Methodology
The authors employ a reduction strategy, linking the preservation of orthogonality to the preservation of singularity (non-invertibility).

1. Reduction to Singularity: The core technical tool is Theorem 1.4, which characterizes additive surjections on direct sums of matrix algebras that map singular elements to singular elements. The proof of this theorem relies on:

   • Rank-one preservation: Showing that such maps send rank-one operators to rank-one operators or zero (Lemma 2.2).
   • Block permutation: Demonstrating that the map permutes the block components of the algebra (Lemma 2.3).
   • Fundamental Theorem of Geometry of Matrices: Applying known results on additive maps preserving rank or singularity to determine the specific form of the map on each block (involving field endomorphisms and invertible matrices).

2. Orthogonality to Singularity: Lemma 3.1 establishes that for finite-dimensional $C^*$-algebras (excluding specific low-dimensional cases like $\mathbb{C} \oplus \mathbb{C}$), any additive map preserving mutual strong BJ orthogonality must also preserve singularity. This is proven by constructing specific orthogonality relations involving singular value decompositions and showing that mapping a singular element to an invertible one leads to a contradiction.

3. Structural Analysis of Blocks: Once the map is reduced to a form involving block permutations and field endomorphisms, the authors analyze the constraints imposed by the orthogonality condition on the specific components:

   • They eliminate transposition-based forms (e.g., $A \mapsto A^T$) for blocks of dimension $n \ge 2$ (Lemma 3.3).
   • They determine that the field endomorphism $\sigma$ must be either the identity or complex conjugation (Lemma 3.4).
   • They prove that the invertible matrices involved must be scalar multiples of unitary operators and that the scaling factors across blocks must be uniform (Lemmas 3.6, 3.7, 3.8).

Key Results

• Theorem 1.2 (Main Classification): Let $A$ be a finite-dimensional complex $C^*$-algebra not isomorphic to $\mathbb{C}$, $\mathbb{C} \oplus \mathbb{C}$, or $M_2(\mathbb{C})$. An additive surjection $\Phi: A \to A$ preserves mutual strong BJ orthogonality if and only if there exist:

  1. A permutation $\pi$ of the minimal ideals of $A$ respecting their dimensions.
  2. A positive scalar $\gamma$.
  3. Unitary elements $u, v \in A$.
  4. A map $a \mapsto a^\dagger$ which acts as the identity on some minimal ideals and as entrywise conjugation (with respect to a fixed orthonormal basis) on others.

  Such that:
  $$\Phi(a) = \gamma u \pi(a)^\dagger v.$$

• Corollary 1.3: Under the same assumptions, the classification for maps preserving strong BJ orthogonality (not necessarily mutual) is identical to the mutual case.

• Theorem 1.4 (Singularity Preservers): For a direct sum of matrix algebras over a field of characteristic zero, an additive surjection preserves singularity if and only if it permutes the blocks and acts on each block via a map of the form $A_i \mapsto P_i \pi(A_i)^{\sigma_i} Q_i$ or $A_i \mapsto P_i \pi(A_i^T)^{\sigma_i} Q_i$, where $\sigma_i$ is a field endomorphism.

Exceptions and Counterexamples
The paper explicitly notes that the main theorem fails for algebras of the form $\mathbb{C}$, $\mathbb{C} \oplus \mathbb{C}$, and $M_2(\mathbb{C})$:

• For $A = \mathbb{C}$, every additive map preserves the relation.
• For $A = \mathbb{C} \oplus \mathbb{C}$, non-standard linear preservers exist that do not fit the unitary/scalar form (Example 3.2, Example 4.2).
• For $A = M_2(\mathbb{C})$, the structure is different; specifically, any map of the form $\Phi(A) = PAQ$ with invertible $P, Q$ preserves mutual strong BJ orthogonality, regardless of whether $P$ and $Q$ are scalar multiples of unitaries (Example 4.3).

Significance and Scope
The authors position this work as a classification of additive preservers, extending previous results on linear preservers and orthogonality in normed spaces. The significance lies in:

1. Extending Orthogonality Theory: Moving from linear preservers to additive surjections in the context of strong BJ orthogonality on $C^*$-algebras.
2. Connecting Concepts: Establishing a rigorous link between the geometric notion of mutual strong BJ orthogonality and the algebraic notion of singularity preservation.
3. Completing the Classification: Providing a complete structural description for the general finite-dimensional case, while carefully delineating the exceptional low-dimensional cases where the structure is more permissive.

The paper does not propose new applications or experimental frameworks but focuses strictly on the algebraic characterization of these preservers. The authors note that without the additivity assumption, the class of preservers becomes "wild" (Remark 4.4), citing existing literature on non-linear preservers.