Monlib4

4 The inner product for the multi-matrix algebra

4.1 On Mn

Definition 4.1.1
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Given a linear functional ϕ:MnC, we define ϕQ=i,jϕ(eij)eji.

Lemma 4.1.2
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Given a linear functional ϕ:MnC, we get ϕ is given by xTr(ϕQx).

Proof
Definition 4.1.3
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Given C-algebras A,C, we say a linear map f:AC is positive when 0f(a) for all 0a. In other words, f maps positive elements in A to positive elements in C.

As any matrix xMn is positive if and only if there exists some yMn such that x=yy, we get that a linear functional f on Mn is a positive map when 0f(xx) for any matrix xMn.

Given a linear functional ϕ on Mn, we have,
ϕ is positive 0ϕQ.
Here, ϕQ is the matrix of ϕ defined in Definition 4.1.1.

Proof
Proposition 4.1.5

Given a linear functional ϕ:MnC, we get ϕ is real (star-preserving) if and only if ϕQ is self-adjoint.
Here, ϕQ is the matrix given by ϕ defined in Definition 4.1.1.

Proof
Corollary 4.1.6
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If AMn. Then
0A and is invertible A is positive-definite.

Proof
Lemma 4.1.7

Given a positive definite matrix QMn, we have Tr(Qxx)=0 if and only if x=0 for any xMn.

Proof
Definition 4.1.8
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A linear functional f on A is said to be faithful if f(xx)=0 if and only if x=0 for all xA.

Given a linear functional ϕ:MnC, we have
ϕ is a positive and faithful map ϕQ is positive-definite.
Again, ϕQ is the matrix given by ϕ defined in Definition 4.1.1.

Proof

Given a linear functional ϕ:MnC, then the following are equivalent,

  1. ϕ is positive and faithful,

  2. ϕQ is pos-def and xMn:ϕ(x)=Tr(Qx),

  3. Mn×MnC:(x,y)ϕ(xy) defines an inner product on Mn.

Proof

4.2 On iMni

Definition 4.2.1

For each i, we fix a faithful and positive linear functional ψi on Mni, and we let QiMni denote the positive definite matrix such that ψi:xTr(Qix) (so each Qi=j,kψi(ejk)ekj) – see Proposition 4.1.1.

Let ψ be a faithful positive linear functional on iMni given by ψ=iψipi, where each pi is the projection map jMnjMni, and we let Q=iQi. So then, given xiMni, we get ψ(x)=Tr(Qx), where Tr here is defined by the sum of the diagonals in each matrix block.

By Theorem 4.1.10, we define the inner product on each Mni by

xyψi=ψi(xy)=Tr(Qixy),

for all x,yMni. We denote (Mni,ψi) to be the Hilbert space given by this inner product.
We define the inner product on iMni by

xyψ=ψ(xy)=Tr(Qxy),

for all x,yiMni, where Tr here is defined as the sum of the diagonals in each matrix block.

Proposition 4.2.2

The adjoint of psi on (iMni,ψ) and C is given by CB:xx1. In other words, ψ=|1.

Proof
Proposition 4.2.3

We get [ιs(eijQs1/2)i,j=1ns]s=1K is an orthonormal basis of (iMni,ψ).
Here, Q=ψQ from Proposition 4.1.1.

Proof

Recall that, given an orthonormal basis f=(fi) on a Hilbert space H, we let Rf be the linear isomorphism HCdimH given by Rf(x)i=fix.

Proposition 4.2.4
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Let f be the orthonormal basis from Proposition 4.2.3. Then for xiMni, we get Rf(x)s,ij=(xQ1/2)s,ij.
Here, Q=ψQ from Proposition 4.1.1.

Proof

4.3 The modular automorphism

Let B=iMni in this section.

Definition 4.3.1
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Given tR, we define the algebra automorphism σt:BB to be given by aQtaQt with inverse aQtaQt (so σt1=σt).

Proposition 4.3.2

For any xB, we get σt(x)=σt(x).
In other words, σtr=σt.

Proof
Proposition 4.3.3

For any tR, σt is self-adjoint.

Proof

For any tR, we get σt is also a co-algebra homomorphism.

Proof
Lemma 4.3.5

For any x,y,zB, we get xyz=yxz.

Proof
Lemma 4.3.6

For any x,y,zB, we get xyz=xzσ1(y).

Proof
Lemma 4.3.7

We clearly get σ0=id.

Proof
Lemma 4.3.8

For any s,tR, we get σsσt=σs+t.

Proof

From Proposition 3.1.4, we see that our Hilbert space on B is a Frobenius algebra.

Proposition 4.3.9
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For any tR, we get σt=id if and only if t=0 or ψ is tracial.

Proof

4.4 Multiplication composed with co-multiplication

Proposition 4.4.1

Given αC, we get mm=αid if and only if Tr(Qi1)=α for all i[K].

Proof