Faithful flatness of left adjoint to almostification of algebras

I have been reading Bhatt’s notes on perfectoid spaces and I have stumbled upon a fact whose proof I am unable to understand. Specifically, in Remark 4.2.8 Bhatt describes the functor $$Amapsto A_{!!}$$ from $$R^a$$-algebras (almost $$R$$-algebras) to $$R$$-algebras which is left adjoint to the almostification functor (which is just the localization functor to the category of almost algebras). Bhatt gives as an exercise the statement that even though $$(-)_{!!}$$ doesn’t preserve flatness, it does preserve faithful flatness.

I have been unable to solve this myself, so I followed the reference to Gabber-Ramero’s Almost ring theory, Remark 3.1.3. Their argument is short so I replicate it here:

Let $$phi:Ato B$$ be a morphism of almost algebras. Then $$phi$$ is a monomorphism iff $$phi_{!!}$$ is injective; moreover, $$B_{!!}/operatorname{Im}(A_{!!})cong B_!/A_!$$ is flat over $$A_{!!}$$ if and only if $$B/A$$ is flat over $$A$$, by proposition 2.4.35.

I am able to follow the claims in this argument, but I don’t see how it implies faithful flatness. I suspect there is some faithful flatness criterion at play which I am not aware of, but I couldn’t find it in the literature I’ve checked. Could someone explain how exactly one recovers the result from this argument?

oa.operator algebras – Schaten p norm of block matrices

Let $$A=Doplus 0$$ be a diagonal Hermitian matrix and $$B$$ is an invertible Hermitian matrix with $$(1,1)$$ block being $$B_{11}$$ and $$B_{11}$$ and $$D$$ have the same dimensions. Then is it true that if $$(1+|t|^p)^{q/p}=|A+tB|_q^q$$ for all $$tinmathbb R,$$ then $$B_{12}=0$$ where $$B_{12}$$ is in the $$(1,2)$$ block of $$B$$?

$$|.|_q$$ denotes the Schatten-q norm and $$1

I somehow think that the answer is yes but cannot really prove it.

logic – Completeness theorem in the light of Boolean algebras. Is it correct what I am saying here?

Recently I have looked at logics on introductory level and was seduced to develop a perspective on base of Boolean algebras.

I experienced that as “enlightening” but I still feel a bit unsecure. This also because I did not really engaged Boolean algebras in the material used for learning.

So I decided to expose my “findings” here for some testing.

Thank you very much in advance.

Let $$mathcal L$$ denote a language and let $$S(mathcal L)$$ denote the set of $$mathcal L$$-sentences.

Usually based on a deduction system an expression like $$Sigmavdashphi$$ states that $$phiin S(mathcal L)$$, that $$Sigmasubseteq S(mathcal L)$$ and that a deduction of $$phi$$ from $$Sigma$$ exists.

Abbreviating $${psi}vdashphi$$ by $$psivdashphi$$ let us look at $$vdash$$ as a relation on $$S(mathcal L)$$.

It is evident that in that context $$(S(mathcal L),vdash)$$ is a preorder with specific properties. As any preorder it induces a poset $$mathcal B_{vdash}$$ which can be shown to be a Boolean algebra. The elements of $$mathcal B_{vdash}$$ are equivalence classes of the relation $$sim$$ prescribed by $$phisimpsiiff phivdashpsitext{ and }psivdashphi$$ and in the sequel I denote such classes as $$(phi)_{vdash}$$.

Similarly we can construct a Boolean algebra $$mathcal B_{vDash}$$ on base of the semantic relation $$vDash$$.

It seems to me that proving the completeness theorem is actually the same as proving that $$mathcal B_{vdash}$$ and $$mathcal B_{vDash}$$ coincide, or equivalently that the two relation $$vdash$$ and $$vDash$$ defined on $$S(mathcal L)$$ coincide. If I understood well then the essence of proving the completeness theorem is proving that every ultrafilter of $$mathcal B_{vdash}$$ can be written as $${(chi)_{vdash}mid chiinmathsf{Th}(mathfrak A)}$$ where $$mathfrak A$$ denotes some $$mathcal L$$-structure.

The statement $$Sigmanvdashphi$$ can be translated to: “$$(phi)_{vdash}$$ is not an element of the filter generated by $${(psi)_{vdash}mid psiinSigma}$$“.

Now if the statement $$phivdashpsi$$ is false then some ultrafilter $$U$$ exists with $$(phi)_{vdash}in U$$ and $$(psi)_{vdash}notin U$$. Then $$U={(chi)_{vdash}mid chiinmathsf{Th}(mathfrak A)}$$ for some $$mathcal L$$-structure $$mathfrak A$$ so that $$(phi)_{vdash}inmathsf{Th}(mathfrak A)$$ and $$(psi)_{vdash}notinmathsf{Th}(mathfrak A)$$, proving that the statement $$phivDashpsi$$ is false as well.

That means that $$phivDashpsiimpliesphivdashpsi$$ and the opposite direction is a consequence of soundness.

Everything okay above? Finally I wondered: “why are Boolean algebras not mentioned?” Is it so maybe that logicians – busy with meta-mathematics – automatically avoid mathematics as much as possible to keep things unmixed?

oa.operator algebras – Lower bounds in the space of compact operators

Let $$H$$ be a separable Hilbert space, and $$K(H)$$ the corresponding space of compact operators. Consider the “unit sphere” $$S:={Tin K(H)|Tgeq 0text{ and }||T||=1}$$. Is it true that, given any pair of operators $$T_1,T_2in S$$, there exists another operator $$Tin S$$ such that $$Tleq T_1,T_2$$?.

ra.rings and algebras – Main ideal of a non-associative magma

Definitions of an ideal of an algebraic structure $$A$$ (as a substructure $$I$$ so that the product of $$A$$ and $$I$$ is a subset of $$I$$) do not imply associativity.

However, definitions of a main ideal that I know of (for a semi-group or a ring) assume associativity.
For example, the main ideal left $$S ^ 1a$$ of a semi-group $$S$$ is an ideal because $$S ^ 1 (S ^ 1a) = (S ^ 1S ^ 1) a$$.
The main bilateral ideal of a semi-group is the whole $$S ^ 1aS ^ 1$$ which is defined due to associativity.
https://en.wikipedia.org/wiki/Green%27s_relations

I am trying to generalize the definition of a main ideal to non-associative structures.

Would it be a correct generalization to say that a main ideal is an ideal which can be obtained by taking a single element and all the finished products with the element as one of the orders?

The main left ideal (right) of a non-associative magma $$M$$ generated by an element $$a$$ is the set that includes $$a$$ and all finished items of $$M$$ or $$a$$ is the rightmost operand (resp. leftmost).

The main bilateral ideal of $$M$$ generated by $$a$$ is the set that includes $$a$$ and all finished items of $$M$$ which contain $$a$$ as an operand.

I wonder if there is a notation for sets of finished products for a non-associative structure.
Instead of $$S ^ 1a$$ or $$S ^ 1aS ^ 1$$ it must include all possible combinations of $$… S ^ 1S ^ 1a$$ or $$… S ^ 1S ^ 1aS ^ 1S ^ 1 …$$.

Are there better approaches or formulations of a main ideal of a non-associative magma?
Are there generalizations of a main ideal for non-associative rings, algebras, etc.?

differential geometry – Integrability of Lie algebras

In the article https://arxiv.org/pdf/math/0611259.pdf, the integrability of a Lie algebra is defined as follows: a Lie algebra $$A$$ is integrable if it is isomorphic to the Lie algebra of a Lie groupoid $$mathcal {G}$$. I have two questions regarding this definition:

1] The Lie groupoid $$mathcal {G}$$ must be on the same basis as $$A$$?

2] What is an exact isomorphism of Lie algebroids? It is strange, because they define this notion just before defining what is an algebraic morphism. Is it because isomorphism is supposed to be on the same basis, as I ask in 1], then the notion of compatibility with anchors and hooks is trivial and, therefore, such isomorphism doesn’t ; is that a vector isomorphism of beams on the same basis with these compatibilities?

Thank you so much!

Jordan isomorphisms of type I von Ieumann algebras

Let $$mathcal M$$ and $$mathcal N$$ to be two von Neumann algebras. A linear map $$J: mathcal M to mathcal N$$ is said to be a Jordanian isomorphism if $$J$$ is bijective, $$*$$-preservation and $$J (xy) = J (yx)$$ for everyone $$x, y in mathcal M.$$

Is there a good classification of type I von Ieumann algebras up to Jordan's isomorphism? Is there also a classification of type I factors up to Jordanian isomorphism?

reference request – Isomorphism for Ext spaces for finite dimension algebras

Let $$A$$ be an Artin algebra with an enveloping algebra $$A ^ e$$.
Then, we have $$Hom_ {A ^ e} (X, A ^ e) cong Hom_A (D (A) otimes_A X, A)$$ for a bimodule $$X$$. (see for example in the article "A Green's theorem on the dual of transposition" by Auslander and Reiten in Corollary 4.2.)

Question:
When do we have the Ext analog:
$$Ext_ {A ^ e} ^ i (X, A ^ e) cong Ext_A ^ i (D (A) otimes_A X, A)$$ for everyone $$i geq 1$$?

This applies for example to $$X = A$$ but does not generally hold. Maybe there is a nice condition and a reference for such isomorphisms.

I guess it should be true for $$X = A ^ {*}$$, the dual of the bimodule $$A$$ in case $$A$$ is reflexive.

Question 2: is this true?

This would prove the equivalence of conditions a) and b) for finite dimension algebras in question 2 of On properties of a algebra as a bimodule

operator algebras – \$ W = {x in A: sigma_A (x) subset U } \$ is opened in \$ A \$.

Let $$A$$ to be a unitary Banach algebra.

Let $$U subset Bbb C$$ open set.

I have to prove that $$W = {x in A: sigma_A (x) subset U }$$ is open in $$A$$.

$$sigma_A (x) = { lambda in Bbb C: x- lambda I in A$$ is not reversible$$}$$.

So i know that $$sigma_A (x)$$ is compact and not empty for everyone $$x in A$$, so by compactness and by the fact that $$U$$ is open we can take $$lambda_1, dots, lambda_k in Bbb C$$ s.t. $$sigma (x) subset cup_ {i = 1} ^ k B_ {r_i} ( lambda_i) subset U$$.

I thought it might help, but I don't see how to continue in this direction.

rt.representation theory – Classification of small Frobenius algebras

Despite Frobenius algebras (commutative) on a field $$K$$ being a very popular class of algebraic objects, there does not seem to be any attempt at classification (up to $$K$$-algebra isomorphism) for them has already been attempted.

We can assume that they are local and of the form $$K (x_1, …, x_r) / I$$, where being Frobenius means here (without loss of generality, assuming they are local and divided) that $$J ^ n subseteq I subseteq J ^ 2$$ for $$J =$$ and that there is a unique 1-dimensional ideal in $$K (x_1, …, x_r) / I$$, or equivalently, a "single" non-unique polynomial of the highest degree survives in $$K (x_1, …, x_r) / I$$.

In particular, it is unknown whether there is only a finite number of Frobenius algebras of a given vector space dimension and whether their classification is independent of the given field.

This motivates the following question:

Question: Can we give a classification of all Frobenius algebras (up to the isomorphism of $$K$$-algebras) of vector space dimension $$d$$ for the little ones $$d$$ (Let's say $$1 leq d leq 10$$)? In particular, is there a computer algebra system that can do this in case the field is finite, say with two elements?

Here's a start:

$$d = 1$$: We only have $$K$$.

$$d = 2$$: We only have $$K (x) / (x ^ 2)$$.

$$d = 3$$: We only have $$K (x) / (x ^ 3)$$.

$$d = 4$$: We have $$K (x) / (x ^ 4)$$ and $$K (x, y) / (x ^ 2, y ^ 2)$$. (there is also $$K (x, y) / (xy, x ^ 2-y ^ 2)$$ but should be isomorphic to $$K (x, y) / (x ^ 2, y ^ 2)$$?)

$$d = 5$$: We have $$K (x) / (x ^ 5)$$ and $$K (x, y) / (xy, x ^ 2-y ^ 3)$$ and $$K (x, y, z) / (yz, xz, xy, y ^ 2-z ^ 2, x ^ 2-z ^ 2)$$, I forgot one?

$$d = 6$$: We have $$K (x) / (x ^ 6)$$, $$K (x, y) / (x ^ 2, y ^ 3)$$, $$K (x, y) / (xy, x ^ 3-y ^ 3)$$ and $$K (x, y, z) / (yz, xz, xy, y ^ 2-z ^ 3, x ^ 2-z ^ 3)$$ but here I don't know if there are more (there probably are).