[Paper Review] Additivity of maps preserving products $AP\pm PA^{*}$ on $C^{*}$-algebras

Ali Taghavi, Vahid Darvish+1 more|arXiv (Cornell University)|May 19, 2014

Advanced Topics in Algebra11 references3 citations

TL;DR

This paper establishes the additivity and $*$-preserving property of bijective unital maps $Φ$ between prime $C^*$-algebras that preserve the non-standard products $AP \pm PA^*$ for a fixed nontrivial projection $P$ and $\lambda = \pm 1$. Using a decomposition of the algebra into matrix-like components relative to $P$ and its complement, the authors prove that such maps are necessarily $*$-additive, meaning $\Phi(A+B) = \Phi(A) + \Phi(B)$ and $\Phi(A^*) = \Phi(A)^*$, under the given product-preserving condition.

ABSTRACT

Let $\mathcal{A}$ and $\mathcal{B}$ be two prime $C^{*}$-algebras. In this paper, we investigate the additivity of map $Φ$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective unital and satisfies $$Φ(AP+λPA^{*})=Φ(A)Φ(P)+λΦ(P)Φ(A)^{*},$$ for all $A\in\mathcal{A}$ and $P\in\{P_{1},I_{\mathcal{A}}-P_{1}\}$ where $P_{1}$ is a nontrivial projection in $\mathcal{A}$ and $λ\in\{-1,+1\}$. Then, $Φ$ is $*$-additive.

Motivation & Objective

• To investigate whether bijective unital maps preserving the non-standard products $AP + \lambda PA^*$ for $\lambda = \pm 1$ on prime $C^*$-algebras are additive.
• To determine whether such maps are necessarily $*$-additive, i.e., additive and star-preserving, without assuming linearity or star-preservation a priori.
• To extend known results on product-preserving maps in operator algebras by introducing and analyzing the $AP \pm PA^*$ product structure.
• To establish that the primeness of the $C^*$-algebra and the preservation of the product for a single nontrivial projection suffice to force additivity and $*$-preservation.

Proposed method

• Decompose the $C^*$-algebra $\mathcal{A}$ into matrix subspaces $\mathcal{A}_{ij} = P_i \mathcal{A} P_j$ using a fixed nontrivial projection $P_1$ and its complement $P_2 = I - P_1$.
• Use the given functional equation $\Phi(AP + \lambda PA^*) = \Phi(A)\Phi(P) + \lambda \Phi(P)\Phi(A)^*$ for $P \in \{P_1, P_2\}$ and $\lambda = \pm 1$ to derive identities involving real and imaginary parts of elements.
• Apply injectivity and surjectivity of $\Phi$ to equate components of $\Phi(T)$ with $\Phi(A) + \Phi(B)$, leading to component-wise additivity in the $\mathcal{A}_{ij}$ blocks.
• Leverage the primeness of $\mathcal{A}$ and $\mathcal{B}$ to deduce that certain operator identities imply equality of components, especially in the $\mathcal{A}_{ij}$ blocks with $i \neq j$.
• Prove that $\Phi$ preserves projections and maps $\mathcal{A}_{ij}$ bijectively onto $\mathcal{B}_{ij}$, enabling component-wise analysis.
• Combine component additivity and the identity $\Phi(A + A^*) = \Phi(A) + \Phi(A)^*$ with $\Phi(A + A^*) = \Phi(A) + \Phi(A^*)$ to conclude $\Phi(A^*) = \Phi(A)^*$, hence $*$-additivity.

Experimental results

Research questions

• RQ1 Does a bijective unital map $\Phi$ on a prime $C^*$-algebra that preserves the product $AP + \lambda PA^*$ for a fixed nontrivial projection $P$ and $\lambda = \pm 1$ necessarily satisfy $\Phi(A + B) = \Phi(A) + \Phi(B)$?
• RQ2 Can such a map be shown to be $*$-additive, i.e., satisfy $\Phi(A^*) = \Phi(A)^*$, without assuming linearity or star-preservation?
• RQ3 To what extent does the structure of the $AP \pm PA^*$ product determine the algebraic and $*$-structure of the $C^*$-algebra?
• RQ4 Is the primeness of the $C^*$-algebra sufficient to force additivity from product preservation, even without linearity assumptions?

Key findings

• The map $\Phi$ is additive on the entire $C^*$-algebra $\mathcal{A}$, as shown by proving additivity on each component $\mathcal{A}_{ij}$ via component-wise analysis using the projection decomposition.
• The map $\Phi$ is $*$-additive, meaning $\Phi(A^*) = \Phi(A)^*$, which follows from combining the identities $\Phi(A + A^*) = \Phi(A) + \Phi(A)^*$ and $\Phi(A + A^*) = \Phi(A) + \Phi(A^*)$.
• The map $\Phi$ preserves the projections $P_1$ and $P_2 = I - P_1$, mapping them to projections $Q_1, Q_2$ in $\mathcal{B}$ with $Q_1 + Q_2 = I$, and maps $\mathcal{A}_{ij}$ bijectively onto $\mathcal{B}_{ij}$.
• The component $\mathcal{A}_{ij}$ for $i \neq j$ is preserved under $\Phi$, and the additivity of $\Phi$ on $\mathcal{A}_{ii}$ is established via the identity $\Phi(AP_i + BP_i) = \Phi(AP_i) + \Phi(BP_i)$.
• The primeness of the $C^*$-algebra is essential in deducing component equality from operator identities, especially in the $\mathcal{A}_{ij}$ blocks with $i \neq j$, ensuring injectivity of the component decomposition.
• The final conclusion that $\Phi$ is $*$-additive is derived from the equality $\Phi(A + A^*) = \Phi(A) + \Phi(A)^*$ and $\Phi(A + A^*) = \Phi(A) + \Phi(A^*)$, which together imply $\Phi(A^*) = \Phi(A)^*$.