1. CJM Online first
|The Bochner-Schoenberg-Eberlein property and spectral synthesis for certain Banach algebra products|
Associated with two commutative Banach algebras $A$ and $B$ and a character $\theta$ of $B$ is a certain Banach algebra product $A\times_\theta B$, which is a splitting extension of $B$ by $A$. We investigate two topics for the algebra $A\times_\theta B$ in relation to the corresponding ones of $A$ and $B$. The first one is the Bochner-Schoenberg-Eberlein property and the algebra of Bochner-Schoenberg-Eberlein functions on the spectrum, whereas the second one concerns the wide range of spectral synthesis problems for $A\times_\theta B$.
Keywords:commutative Banach algebra, splitting extension, Gelfand spectrum, set of synthesis, weak spectral set, multiplier algebra, BSE-algebra, BSE-function
Categories:46J10, 46J25, 43A30, 43A45
2. CJM 2004 (vol 56 pp. 344)
|Predual of the Multiplier Algebra of $A_p(G)$ and Amenability |
For a locally compact group $G$ and $1
3. CJM 2001 (vol 53 pp. 592)
|Ideal Structure of Multiplier Algebras of Simple $C^*$-algebras With Real Rank Zero |
We give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of $\sigma$-unital simple $C^\ast$-algebras $A$ with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of the multiplier algebra $\mul$, is therefore analyzed. In important cases it is shown that, if $A$ has finite scale then the quotient of $\mul$ modulo any closed ideal $I$ that properly contains $A$ has stable rank one. The intricacy of the ideal structure of $\mul$ is reflected in the fact that $\mul$ can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.
Keywords:$C^\ast$-algebra, multiplier algebra, real rank zero, stable rank, refinement monoid
Categories:46L05, 46L80, 06F05