Wavelet and Linear Algebra

Abstract در این مقاله، انقباض ‎$\mathcal{MT}$-‎دوری هاردی-روگرز را معرفی خواهیم کرد و با استفاده از آن وجود بهترین نقطه تقریبی برای چنین نگاشتهایی در فضاهای متری بررسی خواهد شد. یکتایی این نقطه با افزودن یک شرط که آن را خاصیت ‎$UC$‎ خواهیم نامید حاصل خواهد شد.
در انتها یک کاربرد ارائه خواهیم داد تا نتایجمان را توصیف کند.

Abstract In this paper,  the notion of $n$-weak biamenability of Banach algebras is introduced and for every $n\geq 3$, it is shown that $n$-weak biamenability of the second dual $A^{**}$ of a Banach algebra $A$ implies $n$-weak biamenability of $A$ and this  is true for $n=1, 2$ under some mild conditions.   As a concrete example,  it is  shown that for every abelian locally compact group $G$, $L^1(G)$ is $1$-weakly biamenable and  $\ell^1(G)$ is $n$-weakly biamenable for every odd integer $n$.

Abstract A Banach algebra $\mathfrak{A}$ has a generalized Matrix representation if there exist the algebras $A, B$, $(A,B)$-module $M$ and $(B,A)$-module $N$ such that $\mathfrak{A}$ is isomorphic to the generalized matrix Banach algebra $\Big[\begin{array}{cc}
A & \ M \\
N & \ B%
\end{array}%
\Big]$.
In this paper, the algebras with generalized matrix representation will be characterized. Then we show that there is a unital permanently weakly amenable  Banach algebra $A$ without generalized matrix representation such that $H^1(A,A)=\{0\}$.
This implies that there is a unital Banach algebra $A$ without any triangular matrix representation such that $H^1(A,A)=\{0\}$  and gives a negative answer to the open question of \cite{D}.