ct.category theory: the noncommutative geometry derived from Kontsevich and the noncommutative «spaces» of Rosenberg

It seems to me (although I may be wrong) that the common opinion is that the main difference between two is that Rosenberg's version of non-commutative algebraic geometry refers mainly to non-commutative spaces represented by abelian categories, while non-commutative geometry derived from Kontsevich, Orlov, Efimov, Toen, etc., studies more general spaces represented by improved triangulated categories (DG categories, $ A _ {infty} $-categories, stable $ infty $-categories etc.)

However, it seems that this is not the case: Rosenberg introduced his notion of a spectrum (left) not only for abelian categories, but also for triangulated categories (this corresponding derived geometry at the level of generality) and to correct the exact categories with weakness equivalences (which, apparently, is a generalization of Barwick's notion of an exact $ infty $-category where "accuracy" is unilateral (see here and here).

Apparently, the key difference is the notion of a spectrum of an underlying noncommutative space: somehow, derived noncommutative geometrists do without it. It seems that the presence of spectrum makes the idea that the category is the "category of quasi-sheaves in a non-commutative space" more rigorous in a category, since a non-commutative space is represented by a categorical analog of a ringed space locally, that is, a stack of certain categories.

I would like to know more about the differences and similarities between these two approaches. In particular

What does the spectrum do in Rosenberg's noncommutative geometry and how does the geometry derived from Kontsevich manage without it? Would it be possible to introduce the notion of spectrum into derived noncommutative geometry?

linear algebra – non-commutative diagonalizable matrices on R and C

I have the question: find diagonalizable matrices that will not be switched (elements in matrixes of $ mathbb {R} $ Y $ mathbb {C} $). But I think the matrices about the $ mathbb {R} $ and the $ mathbb {C} $ Always commute, because the elements on the diagonal travel. What am I missing?

Non-commutative geometry – Equivalence of two approaches to transversal measures for a foliation

Suppose that $ (V, F) $ It is a foliated variety. There are three equivalent approaches to the notion of transversal measure as described in this book (see pages 65-69). I would like to understand the last line of section 5.$ alpha $ where it is stated that one easily controls, as in the case of flows, that $ Lambda $ meets definition 2 & # 39; & # 39 ;. So the context is this: one starts with a closed current $ C $ (of degree $ = dim F $) positive in the direction of the sheet and with the help of this current defines a measure $ mu_U $ (locally in $ U $, the domain of the foliation table) in the set of plates by the fromula
$$ langle C, omega rangle = int Big ( int _ { pi} omega Big) d mu_U ( pi). $$
Once we have this measure we can define it. $ Lambda (B): = int Card (B cap pi) d mu_ {U} ( pi) $ for any transverse borel $ B $ (that is, subset of Borel $ B subset V $ such that for each sheet $ L $ of the foliation $ B cap L $ it is at most countable.

Why $ Lambda $ satisfy $ Lambda (B) = Lambda ( psi (B)) $ for any injection of Borel $ psi $ that keeps the leaves.

My guess is that somehow you must follow the condition of $ C $ be closed but I'm not sure how to perform the calculations (for example, one problem I found is that $ Lambda $ It is defined locally and I do not see how to go from $ U $ to another foliation letter $ U & # 39; $ (What can happen by general $ psi $).

measurement theory – Strong convergence of the support operator in spaces $ 0 ^ p $ non-commutative

Leave $ M $ be a von Neumann algebra with a normal faithful semifinite trace $ tau. $ Leave $ L ^ p (M) $be the non-commutative associated $ L ^ p $-space. Suppose, $ x geq 0 $ be an element of $ L ^ p (M) $ Y $ 0 leq x_n leq x $ such that $ x_n $ converges to $ x $ in measure It is true that $ text {support} x_n $ converge to support $ x $ in strong operator topology or weak operator topology or $ sigma $Strong topology?

How to get rid of the brackets that appear in non-commutative multiplication?

I'm getting output as

(-T1T2T3)T4T5

How to get rid of the brackets that appear in the output?

I want my exit as -T1T2T3T4T5

Symbolic non-commutative algebra

Consider 3 noncommutative elements. $ A, B, C $ and we have the relationship $ CB = BC $.

I want to compute things like: $ (A otimes BC) (B otimes B) = AB otimes alpha B ^ 2C $

Is there a package doing this?