# Contractive linear preservers of absolutely compatible pairs between $$\hbox {C}^*$$-algebras

Original Paper

## Abstract

Let a and b be elements in the closed ball of a unital C$$^*$$-algebra A (if A is not unital we consider its natural unitization). We shall say that a and b are domain (respectively, range) absolutely compatible ($$a\triangle _d b$$, respectively, $$a\triangle _r b$$, in short) if $$\Big | |a| -|b| \Big | + \Big | 1-|a|-|b| \Big | =1$$ (respectively, $$\Big | |a^*| -|b^*| \Big | + \Big | 1-|a^*|-|b^*| \Big | =1$$), where $$|a|^2= a^* a$$. We shall say that a and b are absolutely compatible ($$a\triangle b$$ in short) if they are both range and domain absolutely compatible. In general, $$a\triangle _d b$$ (respectively, $$a\triangle _r b$$ and $$a\triangle b$$) is strictly weaker than $$ab^*=0$$ (respectively, $$a^* b =0$$ and $$a\perp b$$). Let $$T: A\rightarrow B$$ be a non-expansive bounded linear mapping between C$$^*$$-algebras. We prove that if T preserves domain absolutely compatible elements (i.e., $$a\triangle _d b\Rightarrow T(a)\triangle _d T(b)$$) then T is a triple homomorphism. A similar statement is proved when T preserves range absolutely compatible elements. It is finally shown that T is a triple homomorphism if, and only if, T preserves absolutely compatible elements.

## Keywords

Absolute compatibility Commutativity $$\hbox {C}^{*}$$-algebra von Neumann algebra Projection Partial isometry Linear absolutely compatible preservers

## Mathematics Subject Classification

Primary 46L10 Secondary 46B40 46L05

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## Authors and Affiliations

• Nabin K. Jana
• 1
• Anil K. Karn
• 1
• Antonio M. Peralta
• 2
1. 1.School of Mathematical ScienceNational Institute of Science Education and Research, HBNIBhubaneswarIndia 