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YCor
YCor
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If A$A$ is a Banach algebra and L$L$ a left ideal of A$A$, consider the representation T_{L}$T_{L}$ of A$A$ into the algebra B(A/L)$B(A/L)$ of bounded linear operators on the quotient space A/L$A/L$ defined by T_{L}(a)(b+L)=ab+L$T_{L}(a)(b+L)=ab+L$. When is T_{L}(A)$T_{L}(A)$ closed in the norm of B(A/L)$B(A/L)$? Certainly this happens if A/L$A/L$ is finite dimensional-dimensional. Are there other cases? Thanks, Paul.

If A is a Banach algebra and L a left ideal of A, consider the representation T_{L} of A into the algebra B(A/L) of bounded linear operators on the quotient space A/L defined by T_{L}(a)(b+L)=ab+L. When is T_{L}(A) closed in the norm of B(A/L)? Certainly this happens if A/L is finite dimensional. Are there other cases? Thanks, Paul.

If $A$ is a Banach algebra and $L$ a left ideal of $A$, consider the representation $T_{L}$ of $A$ into the algebra $B(A/L)$ of bounded linear operators on the quotient space $A/L$ defined by $T_{L}(a)(b+L)=ab+L$. When is $T_{L}(A)$ closed in the norm of $B(A/L)$? Certainly this happens if $A/L$ is finite-dimensional. Are there other cases? Thanks, Paul.

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Paul Visoianu
Paul Visoianu
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Representations of Banach algebras

If A is a Banach algebra and L a left ideal of A, consider the representation T_{L} of A into the algebra B(A/L) of bounded linear operators on the quotient space A/L defined by T_{L}(a)(b+L)=ab+L. When is T_{L}(A) closed in the norm of B(A/L)? Certainly this happens if A/L is finite dimensional. Are there other cases? Thanks, Paul.