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$\varrho$-dilations and hypo-Dirichlet algebras. (English) Zbl 0803.46061
If $A$ is a uniform algebra contained in $C(X)$, where $X$ is a compact Hausdorff space then $A$ is called hypo-Dirichlet if the uniform closure of $A+\overline A$ is of finite codimension in $C(X)$ (if the closure of $A+\overline A= C(X)$ then $A$ is Dirichlet). A representation $f\to T\sb f$ of $A$ on the Hilbert space $H$ is said to have a $\rho$-dilation $\phi\to U\sb \phi$ (where $\rho\in \bbfR\sb +$) to $C(X)$ (and a larger Hilbert space $K$) if $T\sb f= \rho PU\sb f\vert H$, where $P: K\to H$ is the orthogonal projection. While it is known that only two hypo-Dirichlet (non-Dirichlet) algebras have 1-dilations, it is shown here that if there is a non-zero complex homomorphism $\tau$ of $A$ for which the representing measures are given by $hdm$, where $m$ is the core measure and $h$ is continuous on $X$ (and not just essentially bounded with respect to $m$), then any representation of $A$ has a $\rho$-dilation for some $\rho$. The argument is based on Naimark’s dilation theorem and on the observation that with the assumption that the space $N\sb \tau$ of representing measures for $\tau$ is a subspace of $C(X)$, $C(X)$ is spanned by $A$, $\overline A$, and $N\sb \tau$. Several examples, e.g. certain subalgebras of the disc algebra, and corollaries are included.
46J30Subalgebras of commutative topological algebras