There is a new stub E-theory with redirect asymptotic morphism, new entry semiprojective morphism (of separable $C^\ast$-algebras) and stub Brown–Douglas–Fillmore theory, together with some recent bibliography&links changes at Marius Dadarlat, shape theory etc. There should be soon a separate entry shape theory for operator algebras but I still did not do it.

]]>Bounded self-adjoint operators make an algebra, and once can put that into the dagger category setup, however the unbounded operators do not form algebras under composition, hence one needs to be more careful. I removed the redirects and opened a new entry self-adjoint operator redirecting also adjoint operator. This is in part prompted by the discussion in parallel entry on the need and no need for the domain of the definition. The matter is not that simple, but sometimes one can avoid domain discussion for some questions. We need to be more precise one when it is important and when and how one can avoid it.

]]>I added a reference to a paper of Connes and Rovelli (1994) and a link (in modular theory) to

- MathOverflow question tomita-takesaki-versus-frobenuis-where-is-the-similarity

where André Henriques asks about some Connes philosophy. But André quotes in explaining the background to his question, that in full generality there is a homomorphism from imaginary line into the 2-group of invertible bimodules of the given von Neumann algebra $M$, which *in the presence of state* lifts to the homomorphism into $Aut(M)$. I learned just the case when there is a state, and am delighted to hear that this is just a strengthening of some categorical structure which exists even more generally. If somebody is familiar or can dig more on that general case, it would be nice to have such categorical picture in the $n$Lab entry modular theory.

New entry affiliated operator of a $C^\ast$-algebra aka affiliated element. This is important for the circle of entries on algebraic QFT, as the operator algebras are formed by bounded operators, while we typically need unbounded operators like derivative operator to do quantum mechanics.

I sent a version of that entry but the $n$Lab stuck in the middle of the operation so I am not sure if I succeeded. So here is the copy:

Most of the applications of operator algebras stuck in the problem that (hermitean or not) unbounded operators do not form an algebra under composition (or under Jordan multiplication); while the algebras of bounded operators are insufficient as most of applications involve also unbounded operators like the partial derivative operator on $L^2(\mathbb{R}^n)$ which is proportional to the momentum operator in quantum mechanics.

The motivational problem is typically resolved by considering an operator algebra which contains operators which properly approximate the unbounded operators as close as one wishes, and formalize this by defining the larger class of “approximable” operators by means of operator algebra itself. One way to do this is to define the **affiliated elements** of $C^\ast$-algebra, or the operators affiliated with the $C^\ast$-algebra. The idea is that if there is an unbounded self-adjoint operator then we can consider its spectral projections; they are bounded and if we include them into the algebra, the convergence of the spectral decomposition will supply the approximation.

- S. L. Woronowicz, K. Napiórkowski,
*Operator theory in $C^\ast$-framework*, Reports on Mathematical Physics**31**, Issue 3 (1992), 353-371, doi, pdf - S. L. Woronowicz,
*$C^\ast$-algebras generated by unbounded elements*, pdf - wikipedia affiliated operator

New stub model structure on operator algebras with redirects model structures on operator algebras, homotopy theory for operator algebras, homotopy theory for C*-algebras etc.

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