Operator algebra

An operator algebra is an algebra of linear operators, with multiplication given by composition. In quantum examples it can be generated by creation and annihilation operators subject to the canonical commutation relations or canonical anticommutation relations. Its analytic realization requires specifying a common domain for unbounded operators.

For a lattice system, finite-region observable algebras ${\mathcal{A}}_{\Lambda }$ embed into larger ones by tensoring with the identity. Their union has a norm closure called the quasi-local observable algebra, a C-star algebra. It describes finite-support observables and norm limits of such observables. An operator flipping every spin of an infinite lattice need not belong to it: successive finite-region flips are not norm-Cauchy. Restrictions to this algebra can therefore lose relative-phase information retained by increasingly nonlocal finite-system observables.

A vector is cyclic for an operator algebra when its orbit under that algebra has dense linear span. This is distinct from requiring the powers of one chosen operator to span the space. If the algebra is linear, its orbit itself is already a vector subspace.

A vector is separating for an operator algebra when evaluation at that vector is injective. A cyclic vector for the commutant of an operator algebra is separating for the original algebra: if $a\Omega =0$, then $a$ vanishes on its dense commutant orbit.

For a Von Neumann algebra with a cyclic separating vector, this is a densely defined antilinear operator. It is closable: the commutant orbit lies in the domain of its antilinear adjoint and is dense. Write $S$ for its closure and $S=J{\Delta }^{1/2}$ for its polar decomposition. In finite dimension it is everywhere defined and invertible. For a tracial vector, $S_0$ is isometric and extends to an antiunitary involution.

The positive operator in the polar decomposition $S=J{\Delta }^{1/2}$ of the Tomita operator. In a matrix-algebra standard representation with cyclic vector $D>0$, it is $\Delta (X)=D^{2}X{D}^{-2}$. For a tracial vector it is the identity.

The antiunitary factor in the polar decomposition of the Tomita operator. In the finite-dimensional and tracial settings it satisfies $J^2=I$ and $JMJ=M^{\prime }$. In a matrix-algebra standard representation by left multiplication, with positive invertible cyclic vector $D$, it acts by $J(X)=X^{\ast }$.

A unital star-subalgebra of the bounded operators on a Hilbert space that is closed in the weak operator topology, equivalently in the strong operator topology. The Von Neumann double commutant theorem makes these conditions equivalent to equality with the double commutant of an operator algebra. It is norm closed and hence a C-star algebra.

Here a Hilbert space module means a Hilbert space with a unital star-representation of a Von Neumann algebra, rather than a Hilbert C-star module with an algebra-valued inner product. A closed invariant subspace is reducing because the algebra is closed under adjoints. Module equivalence means an intertwining unitary. The polar decomposition of a bounded operator converts an intertwiner into a unitary between its initial and final support subspaces.

Two Hilbert space modules are unitarily equivalent if each embeds isometrically as a closed submodule of the other. For isometric intertwiners $u:H_1\to H_2$ and $v:H_2\to H_1$, let $D=H_1\ominus v{H}_2$ and $K={⨁}_{n\ge 0}(vu{)}^{n}D$. These summands are orthogonal. The operator acting as $u$ on $K$ and as $v^{\ast }$ on $K^{\perp }$ is the required unitary onto $H_2$.

A Von Neumann algebra whose center consists exactly of scalar multiples of the identity. A group von Neumann algebra is a factor exactly when every nonidentity conjugacy class is infinite. A central operator has a square-summable coefficient vector constant on those classes; an infinite class must therefore have coefficient zero. Conversely, the sum of left translations over a finite nonidentity class is a nonscalar central operator.

For a discrete group, its group von Neumann algebra is generated on ${\ell }^2(\Gamma )$ by left translations $\lambda (g){\delta }_h={\delta }_{gh}$. The vector ${\delta }_e$ is cyclic and separating, and its vector functional is tracial. Its modular conjugation acts by $J{\delta }_g={\delta }_{g^{-1}}$ with conjugated scalar coefficients. The commutant is the algebra generated by right translations $\rho (g){\delta }_h={\delta }_{h{g}^{-1}}$.