arXiv subjects
Publications and source records attributed to J. G. Patel.
Let $\mathcal{A}$ be an algebra, and let $\mathcal{A}^2 =$ span$\{ab : a, b \in \mathcal{A}\}$ be a subalgebra of $\mathcal{A}$. In this paper, we prove that if $\mathcal{A}^2$ has infinite codimension in $\mathcal{A}$ iff $\mathcal{A}$ has discontinuous square annihilation property (DSAP). In fact, in this case, the algebra $\mathcal{A}$ admits infinitely many non-equivalent algebra norms.
We prove that the faithful and uniqueness of norm properties are stable in different product algebras such as direct-sum product algebra, convolution product algebra, and module product algebra. Further, we exhibit that these properties are not stable in null product algebra, and also give a common sufficient condition in terms of algebra norm for the co-dimension of $\mathcal{A}^2 = \text{span} \{ ab : a,b \in \mathcal{A}\}$ to be finite in $\mathcal{A}$ and $\mathcal{A}^{2} = \mathcal{A} \ ( \text{when } \overline{\mathcal{A}^2} = \mathcal{A})$.
Let $ω$ be a weight on a right cancellative semigroup $S$. Let $\|\cdot\|_ω$ be the weighted norm on the weighted discrete semigroup algebra $\ell^1(S, ω)$. In this paper, we prove that the weight $ω$ satisfies F-property if and only if the operator norm $\| \cdot \|_{ωop}$ of $\| \cdot \|_ω$ is exactly equal to another weighted norm $\| \cdot \|_{\widetildeω_1}$ [Theorem 2.5 ($iii$)]. Though its proof is elementary, the result is unexpectedly surprising. In particular, $\| \cdot \|_{1 op}$ is same as $\| \cdot \|_1$ on $\ell^1(S)$. Moreover, various examples are discussed to understand the relating among $\| \cdot \|_{ωop}$, $\| \cdot \|_ω$, and $\ell^1(S, ω)$.