A generalization of Hoffman-Wermer theorem on algebras of operator fields (Q1896824)

Let \(T\) be a metrizable compact, \(A\) a \(C^*\)-algebra with identity, and \(C(T,A)\) the \(C^*\)-algebra of all continuous maps from \(T\) into \(A\) endowed with the natural operations and norm. A closed subalgebra \(\mathcal M\) of \(C(T,A)\) which contains the constants and separates the points of \(T\) (for every \(t_1,t_2\in T\), \(t_1\neq t_2\), and \(a_1,a_2\in A\) there exists \(x\in\mathcal M\) such that \(x(t_1)=a_1\), \(x(t_2)=a_2\)) is called a uniform algebra on \(T\). The fibre \(A\) is naturally embedded in \(C(T,A)\) by identifying the element \(a\in A\) with the function \(a(t)\equiv a\), \(t\in T\); a uniform algebra \(\mathcal M\) containing \(A\) is called \(A\)-algebra. Denote \(\text{Re} \mathcal M=\{\text{Re} x; x\in\mathcal M\}\). The classical Hoffman-Wermer theorem asserts that for \(A=\mathbb{C}\), the condition that \(\text{Re} \mathcal M\) be closed in \(C(T)\) for a uniform algebra \(\mathcal M\subset C(T)\) implies \(\mathcal M=C(T)\). This theorem has been extended by \textit{D.C. Taylor} [J. Funct. Anal. 10, 159-190 (1972; Zbl 0245.46083)]\ to general uniform algebras, but with a supplementary condition on the algebra \(\mathcal M\), which can fail even in rather simple cases. In the paper under review it is shown that the Hoffman-Wermer theorem remains in force for \(A\)-algebras \(\mathcal M\subset C(T,A)\) if the fiber \(A\) is a simple \(C^*\)-algebra or if \(A=B(H)\), the algebra of all bounded linear operators on a separable Hilbert space \(H\).