Double adjoint of $*$-homomorphism between $C^*$-algebras

Suppose that we have an inclusion of $C^*$-algebras, $Asubset B$ and let’s denote by $iota:Ato B$ the inclusion map. Is it true then that the double-adjoint map $iota^{**}:A^{**}to B^{**}$ is an injective $*$-homomorphism? Is it ultraweakly continuous?

My idea was to show that it is ultraweakly continuous and somehow use the fact that the ball of $A$ is ultraweakly dense in the ball of $A^{**}$ and then verify that the operations are preserved. The problem is that I cannot show that it is ultraweakly continuous (and I am a little suspicious about it). Whether it remains injective or not after extended, I have no clue.

If I’m not mistaken, $*$-homomorphisms are not ultraweakly continuous in general. I know that a surjective $*$-homomorphism between von Neumann algebras with ultraweakly closed kernel is ultraweakly continuous. But what about injective ones? Does the annihilation of the kernel have something to offer?

equation solving – Using Solve[] to find Eigenstates of a 1D Double Dirac Potential

I’d like to Solve

$$ k^2 equiv – frac{2mE}{hbar^2} = (- frac{mA}{hbar^2} (1+ e^{-2ka}))^2 $$

for E, in terms of m, $hbar$, A, a.

I tried using the following command:

Solve(-((2 m ene)/h^2) == (m^2 A^2)/h^4 (1 + E^(-2 a*Sqrt(-((2 m ene)/h^2)))), ene)

Isn’t working well for this task. What do you recommend? At first glance it seems it could not be simple to solve “by hand”.

Background: This problem comes from Solving a 1D Quantum well with 2 Symmetric Dirac’ Deltas $delta_a$ and $delta_{-a}$, where $A$ is the amplitude.

Como puedo puedo establecer un valor 0.0 si se ingresa un valor de tipo double como salario en C#

la idea es que voy a ingresar un salario y si se ingresa un valor negativo el resultado que se establezca en 0.0 ya que la variable es de tipo double, lo he intentado hacer ingresando un valor negativo como lo es -200 pero el resultado me da un cero con entero.

Este es el método que tengo.

ublic void comprobarSalarioMensual(){
      if(salarioMensual > 0.0){

         salarioMensual = salarioMensual;
     } else if(salarioMensual <= 0.0){
        
         salarioMensual = 0.0;
        }
        
    }

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bash – How do you bind double tap key in Oh My Zsh?

I’m trying to bind double tab to “autosuggest-accept” from the auto suggest plugin:

https://github.com/zsh-users/zsh-autosuggestions

Having to press right arrow to accept suggestion is a bit cumbersome and just double tapping tab would make it so much faster.

Any ideas how to bind a double tap key in Oh My Zsh?

Thanks!

swift – Float and Double network byte order

The Swift library includes the function bigEndian that can be used on integer types (such as Int, UInt, UInt8, UInt64, Int64, etc) to convert them from host order (which might presumably be anything, but realistically will be big or little endian) to network byte order (which is big endian). There’re some good SO answers referring to this, and a particularly complete one is here.

However, I’ve not found a good resource that covers arranging a Float (32 bit) or Double (64 bit) type in to network byte order. Given that these types don’t have a bigEndian method, I’m wondering if there is some subtlety involved? (The linked question does discuss floating point types, but I’m not sure it is definitely covering all details that might be relevant).

Specifically, I want to handle the 64 bit Double floating point type. I’d like a solution that will work on any platform where Swift is available.

Thank you.

Why does the size/extents of a bounding box of a BoxCollider in Unity show the Z element being double what it is?

As you can see from the picture below, I fetch the BoxCollider inside CollisionHandler and Debug.Log() it to the screen.

In this case, I’m logging the size, but the same thing happens with extents. There are 4 logs because I have 4 instances of the object in the scene.

The Z value of the Vector3 is double what it should be, whereas the other values are, for the most part, normal. Also, it seems that each object prints a slightly different value for extents.z. Most stay around 1.3, but one gave me 1.07~.

Additionally, it inverts the order for 2 of the objects. I don’t know why.

None of the objects are scaled and all use the exact same values for their collider.

Screenshot of Unity

I tried drawing a cube with the same size as the BoxCollider’s bounding box, and here’s what I got:

Bounding box drawn with Gizmo.DrawBox from the top

Bounding box drawn with Gizmo.DrawBox from the side

The bounding box gizmo and the box collider, side-by-side

The bounding box gizmo and the box collider, side-by-side from the top

I’m not entirely sure if the misalignment with the Collider in the scene is a sign of a problem or just how Gizmos.DrawCube() works. Here’s the code that does this:

private void OnDrawGizmosSelected() {
    Vector3 origin = coll.bounds.center + transform.TransformDirection(new 
    Vector3(0f, 0f, 0.675323f + skinWidth));
    // The hardcoded number above is Z extent of the box.
    Vector3 direction = transform.TransformDirection(Vector3.forward) * 2;

    Gizmos.color = Color.red;
    Gizmos.DrawRay(origin, direction);
    Gizmos.DrawCube(origin + direction, coll.bounds.size);
    // Where coll is a reference to the BoxCollider, cached in Awake().
}

Unity version 2019.3.15f1.

For the X value of 2.286476, it rounded to 2.1 in the 3rd log for some reason. And in the first log, it rounded the doubled value of Z to 2.4 rather than 2.6…

fa.functional analysis – Inflating the double dual of a C*-algebra (matrix algebra of double dual)

in this post I would like to discuss the fact that If $A$ is a $C^*$-algebra, then $M_n(A^{**})cong M_n(A)^{**}$, as mentioned in Brown and Ozawa. I can’t really see it. Actually, it is enough for me to find a bijective linear map that respects positivity in both directions, so this is what I’m trying (I define the multiplication and involution on the double dual through Sherman-Takeda and not with Arens products, so it’s hard for me to check if these operations are being preserved).

I recently studied the Sherman-Takeda theorem that states that if $A$ is a $C^*$-algebra, then $A^{**}$ is isometrically isomorphic to $pi_u(A)”$, where $(H_u,pi_u)$ is the universal representation of $A$. It is also important to know how this is done: we define the map $sigma_*:A^*to(pi_u(A)”)_*$ first on $S(A)$ and then extending linearly to $pi_u(A)”$: if $omegain S(A)$ then there exists a unique normal state $bar{omega}$ on $mathcal{B}(H_u)$ satisfying $bar{omega}circpi_u=omega$. Actually $bar{omega}$ is a vector state, $bar{omega}(T)=langle Teta_{omega},eta_{omega}rangle$, where $eta_{omega}in H_u$. We set $sigma_*(omega)=bar{omega}$ and extend linearly and this is proven to be a linear isometry that is onto. The adjoint map $sigma:pi_u(A)”to A^{**}$ is what we are looking for.

After proving this, we are able to define multiplication and involution on $A^{**}$ by the preimages through $sigma$, making it a $C^*$-algebra (a von Neumann algebra actually, since it is $*$-isomorphic to $pi_u(A)”$. Moreover, the ultraweak topology of $A^{**}$ is simply the weak-* topology it inherits as the dual of $A^*$. Finally, I was able to prove something extra– that an element $chiin A^{**}$ is positive in the $C^*$-algebra sense if and only if $chi(tau)geq0$ for all the positive linear functionals $tau$ on $A$.

Now I was able to find a bijective linear map $rho:M_n(A^{**})to M_n(A)^{**}$. Let $epsilon_{i,j}:Ato M_n(A)$ denote the positive maps $amapsto aotimes e_{i,j}$ (where $e_{i,j}$ are matrix units). Then we define
$$rho(x_{i,j}):M_n(A)^*tomathbb{C}$$ by $$rho(x_{i,j})(Phi)=sum_{i,j=1}^nx_{i,j}(Phicircepsilon_{i,j}).$$

The problem is I cannot prove that $rho$ and $rho^{-1}$ respect positivity. I guess that the easy half is proving that $rho$ is positive, but I can’t even do that. Here’s what I’ve tried:

It suffices to show that elements of the form $(x_i^*x_j)_{i,j}in M_n(A^{**})$ are mapped to positive elements of $M_n(A)^{**}$, so it suffices to show that $rho(x_i^*x_j)(tau)geq0$ for all states $tauin S(M_n(A))$. Computing, we have to show that $sum_{i,j}x_i^*x_j(tau_{i,j})geq0$. Passing everything through the isomorphism $sigma^{-1}:A^{**}topi_u(A)”$ of Sherman-Takeda, this is equivalent to proving that
$$sum_{i,j}|taucircepsilon_{i,j}|cdotoverline{bigg(frac{taucircepsilon_{i,j}}{|taucircepsilon_{i,j}|}bigg)}(B_i^*B_j)geq0$$

for all operators $B_1,dots, B_ninmathcal{H}_u$, i.e.
$$sum_{i,j}|taucircepsilon_{i,j}|biglangle B_i^*B_jeta_{i,j},eta_{i,j}rangle_{H_u}geq0$$
where $eta_{i,j}$ is the vector that corresponds to the state $taucircepsilon_{i,j}/|taucircepsilon_{i,j}|$. But unfortunately I can’t prove this. As for checking that $rho^{-1}$ preserves positivity, I have absolutely no clue on how to manage that. Why is this so hard? It feels like this approach should be abandoned from scratch. Does anyone have any idea or a reference to a proof of this result?

asymptotics – Time complexity of pairs in array double loop

The time complexity of an algorithm depends on the model of computation. Algorithms are usually analyzed in the RAM machine, in which basic operations on machine words (such as assignment, arithmetic and comparison) cost $O(1)$. A machine word has length $O(log n)$, where $n$ is the size of the input.

In your case, the size of the input is at least $n$ (defined to be the length of array), and so count fits in a single machine word. Each of the basic operations in the algorithm cost $O(1)$, and so the overall time complexity is $Theta(n^2)$, since the algorithm executes this many basic operations.

magento2 – Magento 2 API invoice items showing as double

For some reason on a invoice more QTY is returned than what exists in the items array. It seems to only do this on configurable products however how should one handle this from the API?

Full example: https://devdocs.magento.com/guides/v2.4/rest/tutorials/orders/order-create-invoice.html

Short part in question:

    {
    "base_discount_tax_compensation_amount": 0,
    "base_price": 52,
    "base_price_incl_tax": 52,
    "base_row_total": 52,
    "base_row_total_incl_tax": 52,
    "base_tax_amount": 0,
    "entity_id": 10,
    "discount_tax_compensation_amount": 0,
    "name": "Chaz Kangeroo Hoodie",
    "parent_id": 3,
    "price": 52,
    "price_incl_tax": 52,
    "product_id": 67,
    "row_total": 52,
    "row_total_incl_tax": 52,
    "sku": "MH01-S-Gray",
    "tax_amount": 0,
    "order_item_id": 10,
    "qty": 1
    },
    {
    "base_price": 0,
    "entity_id": 11,
    "name": "Chaz Kangeroo Hoodie-S-Gray",
    "parent_id": 3,
    "price": 0,
    "product_id": 56,
    "sku": "MH01-S-Gray",
    "order_item_id": 11,
    "qty": 1
    }

Full API response:

{
"base_currency_code": "USD",
"base_discount_amount": 0,
"base_grand_total": 165,
"base_discount_tax_compensation_amount": 0,
"base_shipping_amount": 5,
"base_shipping_incl_tax": 5,
"base_shipping_tax_amount": 0,
"base_subtotal": 160,
"base_subtotal_incl_tax": 160,
"base_tax_amount": 0,
"base_to_global_rate": 1,
"base_to_order_rate": 1,
"billing_address_id": 6,
"can_void_flag": 0,
"created_at": "2017-08-21 22:36:02",
"discount_amount": 0,
"email_sent": 1,
"entity_id": 3,
"global_currency_code": "USD",
"grand_total": 165,
"discount_tax_compensation_amount": 0,
"increment_id": "000000003",
"order_currency_code": "USD",
"order_id": 3,
"shipping_address_id": 5,
"shipping_amount": 5,
"shipping_discount_tax_compensation_amount": 0,
"shipping_incl_tax": 5,
"shipping_tax_amount": 0,
"state": 2,
"store_currency_code": "USD",
"store_id": 1,
"store_to_base_rate": 0,
"store_to_order_rate": 0,
"subtotal": 160,
"subtotal_incl_tax": 160,
"tax_amount": 0,
"total_qty": 9,
"updated_at": "2017-08-21 22:36:03",
"items": (
    {
    "base_discount_tax_compensation_amount": 0,
    "base_price": 22,
    "base_price_incl_tax": 22,
    "base_row_total": 22,
    "base_row_total_incl_tax": 22,
    "base_tax_amount": 0,
    "entity_id": 3,
    "discount_tax_compensation_amount": 0,
    "name": "Radiant Tee-M-Orange",
    "parent_id": 3,
    "price": 22,
    "price_incl_tax": 22,
    "product_id": 1553,
    "row_total": 22,
    "row_total_incl_tax": 22,
    "sku": "WS12-M-Orange",
    "tax_amount": 0,
    "order_item_id": 3,
    "qty": 1
    },
    {
    "base_discount_tax_compensation_amount": 0,
    "base_price": 18,
    "base_price_incl_tax": 18,
    "base_row_total": 18,
    "base_row_total_incl_tax": 18,
    "base_tax_amount": 0,
    "entity_id": 4,
    "discount_tax_compensation_amount": 0,
    "name": "Advanced Pilates & Yoga (Strength)",
    "parent_id": 3,
    "price": 18,
    "price_incl_tax": 18,
    "product_id": 49,
    "row_total": 18,
    "row_total_incl_tax": 18,
    "sku": "240-LV08",
    "tax_amount": 0,
    "order_item_id": 4,
    "qty": 1
    },
    {
    "base_price": 68,
    "base_price_incl_tax": 68,
    "entity_id": 5,
    "name": "Sprite Yoga Companion Kit",
    "parent_id": 3,
    "price": 68,
    "price_incl_tax": 68,
    "product_id": 51,
    "sku": "24-WG080-24-WG084-24-WG088-24-WG082-blue-24-WG086",
    "order_item_id": 5,
    "qty": 1
    },
    {
    "base_discount_tax_compensation_amount": 0,
    "base_price": 27,
    "base_price_incl_tax": 27,
    "base_row_total": 27,
    "base_row_total_incl_tax": 27,
    "base_tax_amount": 0,
    "entity_id": 6,
    "discount_tax_compensation_amount": 0,
    "name": "Sprite Stasis Ball 65 cm",
    "parent_id": 3,
    "price": 27,
    "price_incl_tax": 27,
    "product_id": 29,
    "row_total": 27,
    "row_total_incl_tax": 27,
    "sku": "24-WG082-blue",
    "tax_amount": 0,
    "order_item_id": 6,
    "qty": 1
    },
    {
    "base_discount_tax_compensation_amount": 0,
    "base_price": 5,
    "base_price_incl_tax": 5,
    "base_row_total": 5,
    "base_row_total_incl_tax": 5,
    "base_tax_amount": 0,
    "entity_id": 7,
    "discount_tax_compensation_amount": 0,
    "name": "Sprite Foam Yoga Brick",
    "parent_id": 3,
    "price": 5,
    "price_incl_tax": 5,
    "product_id": 21,
    "row_total": 5,
    "row_total_incl_tax": 5,
    "sku": "24-WG084",
    "tax_amount": 0,
    "order_item_id": 7,
    "qty": 1
    },
    {
    "base_discount_tax_compensation_amount": 0,
    "base_price": 17,
    "base_price_incl_tax": 17,
    "base_row_total": 17,
    "base_row_total_incl_tax": 17,
    "base_tax_amount": 0,
    "entity_id": 8,
    "discount_tax_compensation_amount": 0,
    "name": "Sprite Yoga Strap 8 foot",
    "parent_id": 3,
    "price": 17,
    "price_incl_tax": 17,
    "product_id": 34,
    "row_total": 17,
    "row_total_incl_tax": 17,
    "sku": "24-WG086",
    "tax_amount": 0,
    "order_item_id": 8,
    "qty": 1
    },
    {
    "base_discount_tax_compensation_amount": 0,
    "base_price": 19,
    "base_price_incl_tax": 19,
    "base_row_total": 19,
    "base_row_total_incl_tax": 19,
    "base_tax_amount": 0,
    "entity_id": 9,
    "discount_tax_compensation_amount": 0,
    "name": "Sprite Foam Roller",
    "parent_id": 3,
    "price": 19,
    "price_incl_tax": 19,
    "product_id": 22,
    "row_total": 19,
    "row_total_incl_tax": 19,
    "sku": "24-WG088",
    "tax_amount": 0,
    "order_item_id": 9,
    "qty": 1
    },
    {
    "base_discount_tax_compensation_amount": 0,
    "base_price": 52,
    "base_price_incl_tax": 52,
    "base_row_total": 52,
    "base_row_total_incl_tax": 52,
    "base_tax_amount": 0,
    "entity_id": 10,
    "discount_tax_compensation_amount": 0,
    "name": "Chaz Kangeroo Hoodie",
    "parent_id": 3,
    "price": 52,
    "price_incl_tax": 52,
    "product_id": 67,
    "row_total": 52,
    "row_total_incl_tax": 52,
    "sku": "MH01-S-Gray",
    "tax_amount": 0,
    "order_item_id": 10,
    "qty": 1
    },
    {
    "base_price": 0,
    "entity_id": 11,
    "name": "Chaz Kangeroo Hoodie-S-Gray",
    "parent_id": 3,
    "price": 0,
    "product_id": 56,
    "sku": "MH01-S-Gray",
    "order_item_id": 11,
    "qty": 1
    }
),
"comments": ()
}