The Biquaternion Algebra and the Krein–von Neumann Property

Introduction

A Krein–von Neumann algebra is a unital algebra of bounded operators on a Krein space, closed under the adjoint of the indefinite form and in the weak operator topology; equivalently, it is an algebra $\mathcal{M}$ with $J\mathcal{M}^{*}J=\mathcal{M}$ (Krein–von Neumann Algebras). The biquaternion algebra carries a Krein form whose fundamental symmetry is the natural conjugation $J={}^{\natural}$ (The Biquaternion Krein Form and Its Signature, The Fundamental Symmetry of the Biquaternion Algebra), and its left multiplications give a faithful algebra of operators on that Krein space (One-Sided Operators on the Biquaternion Algebra with Hermitian Adjoint). This article asks whether that algebra is a Krein–von Neumann algebra.

The answer is no for the Krein form of the algebra, and the obstruction is one identity: the indefinite adjoint of a left multiplication is a right multiplication,

$$ (L_{\tilde Q})^{\dagger}=R_{\bar{\tilde Q}} , $$

so the left multiplications are not closed under the adjoint. The involution ${}^{*}$ of the algebra is the adjoint for the definite form, not for the Krein form, and the two differ by conjugation with $J$: the biquaternion algebra is a $\ast$-algebra for the first and is not a $\dagger$-closed algebra of operators for the second. The smallest Krein–von Neumann algebra containing the left regular image is the full operator algebra generated by the left and the right multiplications together,

$$ \mathbb{B}\otimes\mathbb{B}^{\mathrm{op}}\cong M_4(\mathbb{C})=\mathcal{L}(\mathbb{B}) . $$

The property is recovered in exactly two degenerate ways: in the definite reading $J=\mathrm{id}$, where the theory is that of the ordinary finite-dimensional von Neumann algebra $M_2(\mathbb{C})$, and for a sphere of twisted fundamental symmetries $J'=R_c$ under which the left regular representation is a $J'$-representation, but for a form different from the Krein form and with a symmetry that is not compatible with the involution.

The general theory is Krein–von Neumann Algebras and Krein Algebras; the form and its symmetry are The Biquaternion Krein Form and Its Signature and The Fundamental Symmetry of the Biquaternion Algebra; the adjoint of a left multiplication is J-Self-Adjoint and J-Unitary Operators on the Biquaternion Algebra; the operator families and their commutants are One-Sided Operators on the Biquaternion Algebra with Hermitian Adjoint, Two-Sided Operators on the Biquaternion Algebra with Hermitian Adjoint and The Enveloping Algebra of the Biquaternion Algebra and the Bi-module Structure; the modular theory that the negative answer silences is The Tomita Operator and the J-Modular Pair for the Biquaternion Algebra.

Conventions. $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$, with basis $e_0,e_1,e_2,e_3$, $e_k^{2}=-e_0$, and central $i$; $\tilde Q=\sum_{\mu=0}^{3}Q_\mu e_\mu$ with $Q_\mu\in\mathbb{C}$. The involution of the algebra is the Hermitian conjugation ${}^{*}$, $\tilde Q^{*}=\bar Q_0e_0-\bar Q_1e_1-\bar Q_2e_2-\bar Q_3e_3$, written $\dagger$ in Krein–von Neumann Algebras; the natural conjugation is ${}^{\natural}$, $\tilde Q^{\natural}=Q_0e_0-Q_1e_1-Q_2e_2-Q_3e_3$; the bar is the coefficient conjugation, $\bar{\tilde Q}=\sum_\mu\bar Q_\mu e_\mu$. The Krein form is

$$ [\tilde Q,\tilde Q']=\mathrm{Sc}\bigl(\tilde Q^{\natural*}\tilde Q'\bigr)=\sum_{\mu=0}^{3}\varepsilon_\mu Q_\mu^{*}Q'_\mu ,\qquad \varepsilon=(1,-1,-1,-1), $$

with fundamental symmetry $J={}^{\natural}$; the definite form it induces is $\langle\tilde P,\tilde Q\rangle=[J\tilde P,\tilde Q]=\mathrm{Sc}(\tilde P^{*}\tilde Q)=\sum_\mu P_\mu^{*}Q_\mu$. The operator adjoint of the Krein form is written ${}^{\dagger}$, $T^{\dagger}=JT^{*}J$, and ${}^{*}$ on operators is the adjoint for the definite form, so that $L_{\tilde Q}^{*}=L_{\tilde Q^{*}}$ and $R_{\tilde Q}^{*}=R_{\tilde Q^{*}}$. The maps $L_{\tilde Q}(\tilde S)=\tilde Q\tilde S$ and $R_{\tilde Q}(\tilde S)=\tilde S\tilde Q$ are the left and the right multiplications.

The Krein Space and the Left Regular Representation

Definition. Let $\mathcal{K}$ be $\mathbb{B}$ with the Krein form $[\cdot,\cdot]$ and fundamental symmetry $J={}^{\natural}$, and let

$$ \mathcal{M}=\{\,L_{\tilde Q}:\tilde Q\in\mathbb{B}\,\},\qquad L_{\tilde Q}(\tilde S)=\tilde Q\tilde S . $$

$\mathcal{M}$ is the left regular image of the biquaternion algebra, the algebra of the left multiplications acting on $\mathbb{B}$.

Proposition (the image is the algebra). $L:\mathbb{B}\to\mathcal{L}(\mathbb{B})$ is an injective algebra homomorphism, $L_{\tilde Q}L_{\tilde R}=L_{\tilde Q\tilde R}$ and $L_{e_0}=\mathrm{id}$, so $\mathcal{M}\cong\mathbb{B}\cong M_2(\mathbb{C})$; $\mathcal{M}$ is unital, of complex dimension $4$, and finite-dimensional, hence closed in the weak operator topology.

Proof. Multiplicativity and unitality are associativity and the unit. Injectivity is $L_{\tilde Q}e_0=\tilde Q$, so $L_{\tilde Q}=0$ forces $\tilde Q=0$. Boundedness is automatic in finite dimension, and a finite-dimensional subspace of the operators is weakly closed.

Proposition (the commutant). The commutant of the left regular image is the right regular image,

$$ \mathcal{M}'=\{\,R_{\tilde Q}:\tilde Q\in\mathbb{B}\,\}\cong\mathbb{B}^{\mathrm{op}} , $$

the double commutant is the image itself, $\mathcal{M}''=\mathcal{M}$, and the centre of $\mathcal{M}$ is $\mathbb{C}e_0$.

Proof. If $T$ commutes with every left multiplication then $T(\tilde S)=T(L_{\tilde S}e_0)=L_{\tilde S}(Te_0)=\tilde S\,Te_0$, so $T=R_{Te_0}$; conversely the left and the right multiplications commute, and the centralizer of the right multiplications is again the left multiplications. The centre is the intersection $\mathcal{M}\cap\mathcal{M}'=\mathbb{C}e_0$, because a left multiplication that is also a right one is a central multiplication, and the centre of $\mathbb{B}$ is $\mathbb{C}e_0$.

The Indefinite Adjoint of a Left Multiplication

Theorem (the adjoint changes sides). For every $\tilde Q$,

$$ (L_{\tilde Q})^{\dagger}=R_{\bar{\tilde Q}} , \qquad\text{that is}\qquad (L_{\tilde Q})^{\dagger}(\tilde S)=\tilde S\,\bar{\tilde Q} . $$

Proof. The indefinite adjoint is $T^{\dagger}=JT^{*}J$. For a left multiplication, $(L_{\tilde Q})^{*}=L_{\tilde Q^{*}}$ and $J$ is an anti-automorphism, so $JL_{\tilde Q^{*}}J=R_{J(\tilde Q^{*})}$. Now $J(\tilde Q^{*})=\bar{\tilde Q}$, because $\tilde Q^{*}$ carries the coefficients $\bar Q_0,-\bar Q_1,-\bar Q_2,-\bar Q_3$ and $J$ fixes the centre and negates the vector units, restoring the coefficients $\bar Q_0,\bar Q_1,\bar Q_2,\bar Q_3$. Hence $(L_{\tilde Q})^{\dagger}=R_{\bar{\tilde Q}}$. The same identity is computed for the three operator families in J-Self-Adjoint and J-Unitary Operators on the Biquaternion Algebra.

Corollary (the image is adjoint-closed only on the centre). $(L_{\tilde Q})^{\dagger}\in\mathcal{M}$ if and only if $\tilde Q$ is central, that is if and only if $\tilde Q\in\mathbb{C}e_0$; then $(L_{\tilde Q})^{\dagger}=L_{\bar{\tilde Q}}$.

Proof. $R_{\bar{\tilde Q}}=L_{\tilde R}$ for some $\tilde R$ forces $R_{\bar{\tilde Q}}\in\mathcal{M}\cap\mathcal{M}'=\mathbb{C}e_0$, hence $\bar{\tilde Q}$ central, hence $\tilde Q$ central; conversely a central parameter makes the two multiplications coincide.

Corollary (not a Krein–von Neumann algebra, and not a Krein algebra). $\mathcal{M}$ is not closed under the indefinite adjoint, so it is not a Krein–von Neumann algebra on $(\mathbb{B},[\cdot,\cdot])$. Equivalently, the adjoint axiom of a Krein algebra, $[\tilde Q\tilde S,\tilde S']=[\tilde S,\sigma(\tilde Q)\tilde S']$, holds for no involution $\sigma$ of the algebra: it would give $L_{\sigma(\tilde Q)}=(L_{\tilde Q})^{\dagger}=R_{\bar{\tilde Q}}$, which forces $\tilde Q$ central for every $\tilde Q$. So the biquaternion algebra carries no Krein-algebra structure for its Krein form, the involution of the algebra being the definite one.

The Krein–von Neumann Envelope

Theorem (the envelope). The smallest unital subalgebra of $\mathcal{L}(\mathbb{B})$ that contains $\mathcal{M}$ and is closed under the indefinite adjoint is

$$ \mathcal{M}^{\dagger}=\mathbb{B}\otimes\mathbb{B}^{\mathrm{op}}\cong M_4(\mathbb{C})=\mathcal{L}(\mathbb{B}) , $$

of complex dimension $16$.

Proof. A $\dagger$-closed unital subalgebra containing every $L_{\tilde Q}$ contains $(L_{\tilde Q})^{\dagger}=R_{\bar{\tilde Q}}$ for every $\tilde Q$, hence contains $R_{\tilde P}$ for every $\tilde P$, since the bar is a bijection; it therefore contains every product $L_{\tilde Q}R_{\tilde P}$. The products act by $\tilde S\mapsto\tilde Q\tilde S\tilde P$, whose linear span is $\mathbb{B}\otimes\mathbb{B}^{\mathrm{op}}$, of complex dimension $4\cdot4=16$, which is all of $\mathcal{L}(\mathbb{B})$ by the enveloping-algebra isomorphism. Conversely, the algebra generated by the left and the right multiplications is $\dagger$-closed, because $(L_{\tilde Q}R_{\tilde P})^{\dagger}=(R_{\tilde P})^{\dagger}(L_{\tilde Q})^{\dagger}=L_{\bar{\tilde P}}R_{\bar{\tilde Q}}$ is again a product of a left and a right multiplication.

Remark (the linear span is smaller than the envelope). The linear span of $\{L_{\tilde Q}\}\cup\{R_{\tilde P}\}$ has complex dimension $4+4-1=7$, the intersection being the centre $\mathbb{C}e_0$; the products are what fill the whole matrix algebra. The numbers $4,4,1,7,16$ record the algebra, its opposite, the centre, the span and the envelope.

Corollary (the bicommutant theorem is not contradicted, but is not available). The envelope contains $J$ and therefore satisfies the $J$-bicommutant theorem of Krein–von Neumann Algebras. The biquaternion algebra $\mathcal{M}$ itself does not contain $J$, and its bicommutant is $\mathcal{M}$, which is not $\dagger$-closed. So $\mathcal{M}^{cc}=\mathcal{M}$ holds while $\mathcal{M}$ is not a Krein–von Neumann algebra: the hypothesis $J\in\mathcal{M}$ of the bicommutant theorem cannot be dropped, and the left regular image of the biquaternion algebra is a distinguished witness of the failure.

The Definite Reading

Proposition (the definite case). With $J=\mathrm{id}$ the Krein form is the definite form $\langle\cdot,\cdot\rangle$, the adjoint of the representation is the Hermitian adjoint, $L_{\tilde Q}^{*}=L_{\tilde Q^{*}}$, and $\mathcal{M}$ is an ordinary von Neumann algebra of the Hilbert space $\mathbb{B}$, of complex dimension $4$. Hence, by the definite case of Krein–von Neumann Algebras, $\mathcal{M}$ is a Krein–von Neumann algebra.

Proof. The representation is $\ast$-preserving for the definite form, so $\mathcal{M}$ is a unital $\ast$-algebra of operators on the Hilbert space; in finite dimension it is weakly closed, hence a von Neumann algebra, and a von Neumann algebra is a Krein–von Neumann algebra with $J=\mathrm{id}$.

Remark (what the definite reading costs). It costs the whole indefinite structure. The form of the representation is positive definite, so the $J$-positivity of Krein Algebras collapses to the ordinary positivity of the cone of the squares, and the Krein form of the algebra is not the form that the representation respects. The difference is exactly the replacement of ${}^{*}$ by ${}^{\dagger}=J{}^{*}J$ with $J\neq\mathrm{id}$, and the two agree only on the centre.

Twisted Fundamental Symmetries

Theorem (the twisted symmetries). Let $c\in\mathbb{B}$ satisfy $c^{*}=c$ and $c^{2}=e_0$; equivalently, let $c$ lie on the unit sphere of the Hermitian subspace $\mathbb{M}_+$. Then

$$ J'=R_c $$

is a fundamental symmetry of the form

$$ [\tilde Q,\tilde Q']'=\langle R_c\tilde Q,\tilde Q'\rangle=\mathrm{Sc}\bigl(c\,\tilde Q^{*}\tilde Q'\bigr) , $$

and the left regular representation is a $J'$-representation of the $\ast$-algebra $(\mathbb{B},{}^{*})$:

$$ (L_{\tilde Q})^{\dagger'}=J'(L_{\tilde Q})^{*}J'=R_c\,L_{\tilde Q^{*}}\,R_c=L_{\tilde Q^{*}}\,R_{c^{2}}=L_{\tilde Q^{*}} . $$

Hence $(\mathcal{M},J')$ is a Krein–von Neumann algebra on the twisted Krein space $(\mathbb{B},[\cdot,\cdot]')$, with the algebra involution as its adjoint.

Proof. $J'^{2}=R_{c^{2}}=\mathrm{id}$; $J'$ is self-adjoint for the form it defines, $[J'\tilde Q,\tilde Q']'=\langle R_{c}R_{c}\tilde Q,\tilde Q'\rangle=\langle\tilde Q,\tilde Q'\rangle=[\tilde Q,J'\tilde Q']'$, and positive, $[J'\tilde Q,\tilde Q]'=\langle\tilde Q,\tilde Q\rangle>0$ for $\tilde Q\neq0$. For the adjoint, the left and the right multiplications commute, so $R_cL_{\tilde Q^{*}}R_c=L_{\tilde Q^{*}}R_{c^{2}}=L_{\tilde Q^{*}}$, which lies in $\mathcal{M}$.

Remark (why the twisted readings do not rescue the algebra). Three costs are paid. The form of the twisted space is the Krein form only for central $c$; the symmetry $J'=R_c$ is a right multiplication and therefore is not an element of $\mathcal{M}$, so the hypothesis $J'\in\mathcal{M}$ of the bicommutant theorem fails; and $J'$ is compatible with the involution, $J'(\tilde Q^{*})=(J'\tilde Q)^{*}$, only for $c$ central, hence $c=\pm e_0$ and $J'=\pm\mathrm{id}$. So the twisted symmetries make the biquaternion algebra a Krein–von Neumann algebra for an exotic form, not for the Krein form of the algebra, and they are not fundamental symmetries of the algebra in the sense of Krein Algebras; the modular conjugation attached to them does not commute with $J'$ (The Tomita Operator and the J-Modular Pair for the Biquaternion Algebra).

Worked Examples

A Non-Central Parameter

For $\tilde Q=e_1$ one has $(L_{e_1})^{\dagger}=R_{\bar e_1}=R_{-e_1}=-R_{e_1}$, a right multiplication that is not a left one. The image $\mathcal{M}$ is not adjoint-closed, and the smallest adjoint-closed algebra containing $L_{e_1}$ is all of $M_4(\mathbb{C})$. The same holds for every non-central unit $e_k$ and for every non-central $\tilde Q$.

The Centre Recovers the Property in One Dimension

For $\tilde Q=\zeta e_0$ one has $L_{\zeta e_0}=\zeta\,\mathrm{id}$, so $\mathcal{M}$ contains its adjoint only on the scalars, $(L_{\zeta e_0})^{\dagger}=\bar\zeta\,\mathrm{id}$; the one-dimensional algebra $\mathbb{C}\,\mathrm{id}$ is a Krein–von Neumann algebra, and it is the largest adjoint-closed subalgebra of $\mathcal{M}$.

The Two Envelopes

On the coefficient basis the linear span of the left and right multiplications has complex dimension $7$, while the algebra generated by them has complex dimension $16$. Recomputed on the $4\times4$ matrices of the coefficient basis.

Summary

The left regular image $\mathcal{M}=\{L_{\tilde Q}\}$ of the biquaternion algebra on its Krein space $(\mathbb{B},[\cdot,\cdot])$ has commutant the right regular image $\{R_{\tilde Q}\}\cong\mathbb{B}^{\mathrm{op}}$ and double commutant itself. Its indefinite adjoint is $R_{\bar{\tilde Q}}$, a right multiplication: the image is adjoint-closed only on the centre $\mathbb{C}e_0$, and the biquaternion algebra is not a Krein–von Neumann algebra for its Krein form. It is not a Krein algebra either, since the adjoint axiom holds for no involution of the algebra. The smallest $\dagger$-closed unital algebra containing the image is the envelope $\mathbb{B}\otimes\mathbb{B}^{\mathrm{op}}\cong M_4(\mathbb{C})=\mathcal{L}(\mathbb{B})$ of complex dimension $16$, generated by the left and right multiplications, containing $J$ and therefore satisfying the $J$-bicommutant theorem; the linear span of the two families alone has complex dimension $7$. The property survives in two degenerate readings: the definite one, $J=\mathrm{id}$, where $\mathcal{M}$ is the ordinary finite-dimensional von Neumann algebra $M_2(\mathbb{C})$, and the twisted ones, $J'=R_c$ with $c$ on the unit sphere of $\mathbb{M}_+$, which are Krein–von Neumann algebras for exotic forms and incompatible symmetries. The verification of the failure is J-Self-Adjoint and J-Unitary Operators on the Biquaternion Algebra; the envelope is One-Sided Operators on the Biquaternion Algebra with Hermitian Adjoint and The Enveloping Algebra of the Biquaternion Algebra and the Bi-module Structure; the modular consequence is The Tomita Operator and the J-Modular Pair for the Biquaternion Algebra.

Summary of Notation

Symbol	Meaning
$\mathcal{K}=\mathbb{B}$, $[\cdot,\cdot]$, $J={}^{\natural}$	The Krein space of the algebra
$L_{\tilde Q}$, $R_{\tilde Q}$	Left and right multiplications
$\mathcal{M}=\{L_{\tilde Q}\}\cong\mathbb{B}\cong M_2(\mathbb{C})$	Left regular image
$\mathcal{M}'=\{R_{\tilde Q}\}=\mathbb{B}^{\mathrm{op}}$, $\mathcal{M}''=\mathcal{M}$	Commutant and double commutant
$T^{\dagger}=JT^{*}J$	Operator adjoint of the Krein form
$(L_{\tilde Q})^{\dagger}=R_{\bar{\tilde Q}}$	The adjoint changes sides
$\mathcal{M}\cap\mathcal{M}'=\mathbb{C}e_0$	Adjoint-closed only on the centre
$\mathbb{B}\otimes\mathbb{B}^{\mathrm{op}}\cong M_4(\mathbb{C})$	The Krein–von Neumann envelope
$7$ and $16$	Complex dimensions of the span and of the envelope
$J=\mathrm{id}$	The definite reading
$J'=R_c$, $c^{*}=c$, $c^{2}=e_0$	Twisted fundamental symmetries