Isomorphisms between the multiplier algebras of certain topological algebras (Q6552237)

A topological algebra \(E\) is an associative algebra which is a topological vector space and the ring multiplication is separately continuous. \(E\) is called a locally $m$-convex algebra if the topology of \(E\) is defined by a family of submultiplicative seminorms. A locally \(C^*\)-algebra is a complete involutive locally convex algebra \(E\) whose topology is given by a family of \(C^*\)-seminorms \((q_i)_{i\in I}\). That is, for each \(i\in I\), we have \(q_i(x)^2 = q_i(x^* x)\) for each \(x\in E\). \N\NSegal algebras were introduced as subalgebras in the context of group algebras. An abstract Segal algebra is defined as a Banach algebra \((B, \|.\|_B)\) which is continuously embedded as a dense ideal in another Banach algebra \((A,\|.\|_A)\) satisfying also a norm condition with respect to both norms. In particular, a \(C^*\)-Segal algebra is a Segal algebra in a \(C^*\)-algebra. The notion of Segal topological algebras was introduced in [\textit{M.~Abel}, Period. Math. Hung. 77, No.~1, 58--68 (2018; Zbl 1424.46062)].\N\NIn this paper, the concept of an abstract Segal algebra for Banach algebras is generalized to locally $m$-convex algebras as follows. \N\NA left ideal \(S\) of a locally $m$-convex algebra \((E, (p_\lambda)_{\lambda\in \Lambda})\) is called a left Segal locally \(m\)-convex algebra in \(E\) if the following hold. \N\begin{itemize}\N\item[(i)] \(S\) is dense in \(E\). \N\item[(ii)] \(S\) is a locally \(m\)-convex algebra with respect to some (saturated) family \((q_i)_{i\in I}\) of submultiplicative seminorms. \N\item[(iii)] For each \(\lambda\in \Lambda\), there exist \(i\in I\) and \(k > 0\) such that \(p\lambda(x) \leq k q_i(x)\) for each \(x\in S\). \N\end{itemize}\N\NLet \(L(E)\) be the algebra of all continuous linear operators on an algebra \(E\). An element \(T\in L(E)\) is called a left (right) multiplier on \(E\) if \(T(xy) = T(x)y\) (\(T(xy) = xT(y)\)) for all \(x, y\in E\). The set of all left (right) multipliers on \(E\) are subalgebras of \(L(E)\), which is called left (right) multiplier algebra of \(E\). \N\NIn this paper, the topological algebra identification of the multiplier algebra of a certain algebra \(E\) and that of a closed left ideal in \(E\) is studied. The case when one of the algebras is a Segal topological algebra in the other is considered. In particular, this problem is studied in the context of locally \(C^*\)-algebras.