On the 𝐶*-algebra of Toeplitz operators on the quarterplane

Abstract

Using the device of the tensor product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^ \ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras, we study the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^ \ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra generated by the Toeplitz operators on the quarter-plane. We obtain necessary and sufficient conditions for such an operator to be Fredholm, but show in this case that not all such operators are invertible.