Timeline for Operator algebra on an invariant subset

6 events

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Jul 23, 2021 at 1:24	comment	added	user54738		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?
Jul 20, 2021 at 9:47	comment	added	Matthew Daws		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.
Jul 20, 2021 at 7:11	history	edited	YCor	CC BY-SA 4.0	formatting
Jul 20, 2021 at 2:53	comment	added	David Handelman		It would help if you didn't use Fraktur, and if you used modern notation; $\frak U - \frak L$ is confusing.
Jul 20, 2021 at 0:21	review	First posts
Jul 20, 2021 at 6:18
Jul 20, 2021 at 0:20	history	asked	user54738	CC BY-SA 4.0