## 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?