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In Rickart, page 50 Theorem 2.2.1, the statement is made: A linear subspace $\mathfrak{M}$ of the algebra $\mathfrak{A}-\mathfrak{L}$ is invariant with respect to the representation $a{\rightarrow}A_a^{\mathfrak{A}-\mathfrak{L}}$. And, from which, it is concluded that $\mathfrak{L}_1$ (defined below) is a left ideal in $\mathfrak{A}$ that contains $\mathfrak{L}$.

I try to arrive at the same conclusion in the following manner. First the representation $a{\rightarrow}A_a^{\mathfrak{A}-\mathfrak{L}}$ means that $\mathfrak{A}-\mathfrak{L}$ is a left ideal in the algebra $\mathfrak{A}$ and that the left regular representation $a{\rightarrow}A_a$ of $\mathfrak{A}$ is restricted to the $\mathfrak{A}-\mathfrak{L}$ left ideal (definition according to Rickart). Second, all images of operators in the restricted left representation (image of $x$ denoted by $x^{\prime}$) are left ideal images. Third, since $\mathfrak{M}$ is an invariant subspace of $\mathfrak{A}-\mathfrak{L}$ then $\mathfrak{L}_1=\{x\,{:}\,x^{\prime}\,{\in}\,\mathfrak{M}\}$ must be a left ideal in $\mathfrak{A}-\mathfrak{L}$ and hence, $\mathfrak{L}_1$ is also a left ideal in $\mathfrak{A}$. This result is not quite right. According to Rickart I should have, in addition, concluded that $\mathfrak{L}_1$ contains $\mathfrak{L}$ but this does not seem possible since we're constrained to $\mathfrak{A}-\mathfrak{L}$. Can anyone point out what I missed?

Additional question. Rickart makes the statement that $\mathfrak{M}$ is a linear subspace of the algebra $\mathfrak{A}-\mathfrak{L}$. Is he saying that $\mathfrak{M}$ is $\textit{NOT}$ to be taken as a subalgebra of $\mathfrak{A}-\mathfrak{L}$ ? Or does he really mean to say that $\mathfrak{M}$ is a subalgebra of $\mathfrak{A}-\mathfrak{L}$ ?

(Sidebar: definition of an invariant subspace. Take $\mathfrak{M}$ as a subalgebra of $\mathfrak{A}-\mathfrak{L}$. If $Tx\:{\in}\:M$ for every $T\:{\in}\:\mathfrak{M}$ where $M$ is the set of vectors that generate the algebra $\mathfrak{M}$, then $M$ is said to be invariant with respect to the algebra $\mathfrak{M}$.)

Rickart: Theorem 2.2.1 page 50 of "General theory of Banach algebras," 1960.

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    $\begingroup$ It would help if you didn't use Fraktur, and if you used modern notation; $\frak U - \frak L$ is confusing. $\endgroup$
    – David Handelman
    Commented Jul 20, 2021 at 2:53
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    $\begingroup$ Here $L$ is a left ideal in $A$ and $A-L$ denotes the quotient vector space (Rickart seems to write $A/L$ only when $A$ is an algebra and $L$ a two-sided ideal, whence $A/L$ is an algebra). So when you write "... means that $A-L$ is a left ideal in the algebra $A$..." there is no meaning: $A-L$ cannot be a left ideal in $A$ as these are not even comparable structures. $\endgroup$
    – Matthew Daws
    Commented Jul 20, 2021 at 9:47
  • $\begingroup$ In this case Rickart takes $L$ as an unspecified ideal; it could be a left, right, or two-sided ideal; sorry for the confusion. So, I believe $L$ cannot be assumed to be left ideal and I believe $A-L$ is not a quotient vector space. On page 49, Rickart defines the representation $a{\rightarrow}A_a^X$ of $\mathscr{A}$, where $X$ is taken as a left ideal in the algebra $\mathscr{A}$, as the left regular representation $a{\rightarrow}A_a$ of $\mathscr{A}$ restricted to the left ideal $X$. This definition still stands if $X$ is replaced with $A-L$ so $A-L$ is a left ideal. Right? $\endgroup$
    – user54738
    Commented Jul 23, 2021 at 1:24

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