Zero product preserving maps on operator-valued Banach algebras of differentiable functions

Abstract For Banach algebras <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">A</m:mi> </m:math> $\mathcal{A}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">B</m:mi> </m:math> $\mathcal{B}$ , we study zero product preserving linear maps <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="normal">Θ</m:mi> <m:mspace width="0.17em"/> <m:mo>:</m:mo> <m:mspace width="0.17em"/> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mn>0,1</m:mn> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi mathvariant="script">A</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>→</m:mo> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mn>0,1</m:mn> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi mathvariant="script">B</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> ${\Theta} : {C}^{1}\left(\left[0,1\right],\mathcal{A}\right)\to {C}^{1}\left(\left[0,1\right],\mathcal{B}\right)$ . In the case that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">B</m:mi> </m:math> $\mathcal{B}$ is either an operator algebra or an irreducible Banach algebra, and Θ is continuous and has a dense range, it has a description according to a pair of families of (jointly) zero product linear maps from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">A</m:mi> </m:math> $\mathcal{A}$ to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">B</m:mi> </m:math> $\mathcal{B}$ and, in particular, it is a separating map. The cases where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">A</m:mi> <m:mo>=</m:mo> <m:mi>B</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> $\mathcal{A}=B\left(E\right)$ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">B</m:mi> <m:mo>=</m:mo> <m:mi>B</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>F</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> $\mathcal{B}=B\left(F\right)$ for some Banach spaces E and F and Θ is bijective (not necessarily continuous) as well as the case where Θ is a continuous zero product preserving linear map into a Banach function algebra will also be considered. Similar results are valid for two-sided zero product preserving maps. An application of the results is given for surjective homomorphisms between Banach algebras of operator-valued continuously differentiable functions.