Timeline for Contractive projections on operator algebras

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Apr 13, 2022 at 8:41	comment	added	Narutaka OZAWA		The proof is as in the earlier question by interlacing. For every separable subspace $X_0$ of every operator space $Y$ and every $n$, there are a separable subspace $X_1\subset Y$ that contains $X_0$ and a linear map $P\colon Y\to X_1^{**}$ with $P\|_{X_0}$ canonical inclusion and $\\| P\\|_n=1$. This fact follows from the same for Banach spaces (or by an argument involving operator space projective tensor product).
Apr 13, 2022 at 8:16	comment	added	Narutaka OZAWA		One can prove: for every separable subset $X$ of every operator algebra $A$, there is a separable subalgebra $B$ of $A$ that contains $X$ and that is weakly completely $1$-complemented in $A$, i.e., there is a completely contractive projection $P$ from $A^{}$ onto $B^{}$ (which has to be a conditional expectation provided that $A$ is unital and $B$ contains the unit).
Apr 13, 2022 at 7:25	history	edited	Onur Oktay	CC BY-SA 4.0	added 881 characters in body
Apr 11, 2022 at 5:16	history	edited	Onur Oktay	CC BY-SA 4.0	added 75 characters in body
Apr 11, 2022 at 5:06	comment	added	Onur Oktay		If $P$ is completely contractive, then $P$ is a conditional expectation by Corollary 4.2.9 in the book by Blecher & le Merdy. books.google.ca/books?id=oBh0EE0HQLQC&pg=PA155
Apr 11, 2022 at 5:06	history	asked	Onur Oktay	CC BY-SA 4.0