Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

Absolute Continuity in the Dual of a Banach Algebra

Stephen Jay Berman
Transactions of the American Mathematical Society
Vol. 243 (Sep., 1978), pp. 169-194
DOI: 10.2307/1997761
Stable URL: http://www.jstor.org/stable/1997761
Page Count: 26
Preview not available

Abstract

If $A$ is a Banach algebra, $G$ is in the dual space $A^\ast$, and $I$ is a closed ideal in $A$, then let $\|G\|_{I^\ast}$ denote the norm of the restriction of $G$ to $I$. We define a relation $ll$ in $A^\ast$ as follows: $G \ll L$ is for every $\varepsilon > 0$ there exists a $\delta > 0$ such that if $I$ is a closed ideal in $A$ and $\|L\|_{I^\ast} < \delta$ then $\|G\|^{I^\ast} < \varepsilon$. We explore this relation (which coincides with absolute continuity of measures when $A$ is the algebra of continuous functions on a compact space) and related concepts in the context of several Banach algebras, particularly the algebra $C^1\lbrack 0, 1\rbrack$ of differentiable functions and the algebra of continuous functions on the disc with holomorphic extensions to the interior. We also consider generalizations to noncommutative algebras and Banach modules.

• 169
• 170
• 171
• 172
• 173
• 174
• 175
• 176
• 177
• 178
• 179
• 180
• 181
• 182
• 183
• 184
• 185
• 186
• 187
• 188
• 189
• 190
• 191
• 192
• 193
• 194