A characterization of abelian $$C^*$$-algebras in terms of strong Birkhoff-James orthogonality

Abstract We present a characterization of abelian $$C^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebras among all $$C^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebras in terms of strong Birkhoff-James orthogonality. As an application, it is shown that $$C^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebras $$\mathfrak {A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> and $$\mathfrak {B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> are $$*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:math> -isomorphic provided that either $$\mathfrak {A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> or $$\mathfrak {B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> is abelian and $$\mathfrak {A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> and $$\mathfrak {B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> are (nonlinearly) strong Birkhoff-James isomorphic to each other.