A new algebraic approach to the Szegő-Widom limit theorem.

We give another proof of the Szeg\H{o}–Widom Limit Theorem. This proof relies on a new Banach algebra method that can be directly applied to the asymptotic computation of the Toeplitz determinants. As a by-product, we establish an interesting identity for operator determinants of Toeplitz operators, namely if <MATH>a_1,\dots,a_R</MATH> are certain matrix valued functions defined on the unit circle, then <MATH> \det \B(e^{T(a_1)}\cdots e^{T(a_R)} T(e^{-a_R}\cdots e^{-a_1})\B) =\det\B( T(e^{\ta_1}\cdots e^{\ta_R}) e^{-T(\ta_R)}\cdots e^{-T(\ta_1)}\B) </MATH> where <MATH>\ta_r(e^{i\theta})\eg a_r(e^{-i\theta})</MATH> [ABSTRACT FROM AUTHOR]