2.3 Projections and partial isometries

Projections and partial isometries are fundamental building blocks in von Neumann algebras. They allow us to decompose complex operators into simpler components, providing insights into the structure and properties of these algebraic systems.

These concepts play crucial roles in quantum mechanics, functional analysis, and operator theory. Understanding projections and partial isometries is essential for grasping the deeper aspects of von Neumann algebras and their applications in mathematics and physics.

Definition of projections

• Projections form fundamental building blocks in von Neumann algebras, allowing decomposition of complex operators into simpler components
• Understanding projections provides insights into the structure and properties of von Neumann algebras, crucial for analyzing operator algebras

Self-adjoint idempotents

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• Projections defined as operators $P$ satisfying $P^2 = P$ and $P^* = P$
• Self-adjointness ensures $\langle Px, y \rangle = \langle x, Py \rangle$ for all vectors $x$ and $y$
• Idempotence property implies applying projection multiple times yields same result as applying once
• Geometric interpretation visualizes projections as "squashing" vectors onto subspaces

Geometric interpretation

• Projections map vectors onto specific subspaces of Hilbert space
• Orthogonal projections create right angles between projected vectors and kernel
• Visualized as "shadows" of vectors cast onto lower-dimensional subspaces
• Non-orthogonal projections allow oblique projections onto subspaces (not perpendicular to kernel)

Orthogonal projections

• Special case where projection occurs perpendicular to the kernel
• Minimize distance between original vector and its projection
• Characterized by additional property $PQ = QP = 0$ for complementary projection $Q = I - P$
• Widely used in quantum mechanics to represent observables and measurement outcomes

Properties of projections

Range and kernel

• Range of projection $P$ consists of all vectors fixed by $P$ (Range$(P) = \{x : Px = x\}$)
• Kernel of projection contains vectors mapped to zero (Ker$(P) = \{x : Px = 0\}$)
• Direct sum decomposition: Hilbert space $H = \text{Range}(P) \oplus \text{Ker}(P)$
• Complementary projection $I - P$ has range equal to kernel of $P$ and vice versa

Norm of projections

• Norm of non-zero projection always equals 1 ($\|P\| = 1$ if $P \neq 0$)
• Follows from spectral properties and idempotence of projections
• Implies projections bounded operators, crucial for continuity in operator topology
• Exception: zero projection has norm 0, trivial case mapping everything to zero

Spectral properties

• Spectrum of projection contains only 0 and 1 ($\sigma(P) = \{0, 1\}$ for $P \neq 0, I$)
• Eigenvalues correspond to vectors in range (1) and kernel (0) of projection
• Spectral theorem for self-adjoint operators simplifies for projections
• Projections decompose Hilbert space into eigenspaces corresponding to 0 and 1

Partial isometries

Definition and examples

• Partial isometries preserve norms on their initial space, map to zero on orthogonal complement
• Characterized by $V^*V = P$ and $VV^* = Q$, where $P$ and $Q$ projections
• Unitary operators special case of partial isometries (initial and final projections both identity)
• Shift operator on $\ell^2(\mathbb{N})$ classical example of partial isometry (not surjective)

Polar decomposition

• Every bounded operator $T$ uniquely expressed as $T = V|T|$
• $V$ partial isometry, $|T| = \sqrt{T^*T}$ positive operator
• Initial projection of $V$ projection onto closure of range of $|T|$
• Generalizes polar form of complex numbers to operator setting

Initial and final projections

• Initial projection $P = V^*V$ determines domain where $V$ acts isometrically
• Final projection $Q = VV^*$ represents range of $V$
• Kernel of $V$ orthogonal complement of range of initial projection
• Partial isometry uniquely determined by its initial and final projections

Projections vs partial isometries

Similarities and differences

• Both projections and partial isometries idempotent in some sense ($P^2 = P$, $V(V^*V) = V$)
• Projections always self-adjoint, partial isometries generally not
• Partial isometries generalize notion of projection to non-orthogonal settings
• Projections special case of partial isometries where initial and final projections identical

Interplay between concepts

• Partial isometries decompose into product of projection and unitary operator
• Every projection gives rise to partial isometry (inclusion map of range into whole space)
• Polar decomposition relates partial isometries to positive operators and projections
• Murray-von Neumann equivalence of projections defined using partial isometries

Projections in von Neumann algebras

Lattice structure

• Projections in von Neumann algebra form complete lattice under partial order
• Meet ($P \wedge Q$) and join ($P \vee Q$) operations defined for any pair of projections
• Lattice structure crucial for studying structure of von Neumann algebras (type classification)
• Orthocomplementation given by $P^\perp = I - P$, satisfying De Morgan's laws

Central projections

• Projections commuting with all elements of von Neumann algebra
• Correspond to direct sum decompositions of algebra into smaller von Neumann algebras
• Essential for studying structure of von Neumann algebras (type decomposition)
• Minimal central projections correspond to factors (von Neumann algebras with trivial center)

Comparison theory

• Murray-von Neumann equivalence allows comparing "sizes" of projections
• Projections $P \sim Q$ if partial isometry $V$ exists with $V^*V = P$ and $VV^* = Q$
• Leads to dimension theory for von Neumann algebras (coupling constant)
• Fundamental for classification of von Neumann algebras (finite vs infinite algebras)

Partial isometries in von Neumann algebras

Unitary extension

• Partial isometries in von Neumann algebras can be extended to unitaries
• Extension possible within larger von Neumann algebra (crossed product construction)
• Allows representation of partial isometries as "restricted" unitaries
• Crucial for studying dynamics and automorphisms of von Neumann algebras

Partial isometries as generators

• Partial isometries generate important classes of von Neumann algebras (Cuntz algebras)
• Play role in constructing examples of factors (type III factors)
• Used in defining Jones index for subfactors of von Neumann algebras
• Essential for studying non-commutative probability and free probability theory

Applications

Quantum mechanics

• Projections represent observables with yes/no outcomes (spin measurements)
• Partial isometries describe certain quantum operations and measurements
• Projection-valued measures model general observables in quantum theory
• Von Neumann algebras provide mathematical framework for quantum systems with infinitely many degrees of freedom

Functional analysis

• Projections and partial isometries fundamental tools in spectral theory
• Used in constructing functional calculus for normal operators
• Essential for understanding geometry of Hilbert spaces and Banach spaces
• Applications in approximation theory and best approximation problems

Operator theory

• Projections and partial isometries building blocks for more complex operators
• Used in studying invariant subspace problem and reflexivity of operator algebras
• Important in index theory and K-theory for operator algebras
• Applications in non-commutative geometry and spectral triples

Advanced topics

Murray-von Neumann equivalence

• Equivalence relation on projections using partial isometries
• Fundamental for dimension theory of von Neumann algebras
• Leads to classification of factors into types I, II, and III
• Generalizes notion of dimension to infinite-dimensional settings

Projection-valued measures

• Generalization of projections to measure-theoretic setting
• Used to formulate spectral theorem for self-adjoint operators
• Essential for rigorous formulation of quantum mechanics
• Applications in stochastic processes and non-commutative probability

Partial isometry decomposition

• Every operator factors as product of partial isometries and positive operator
• Generalizes polar decomposition to products of multiple partial isometries
• Used in studying structure of C*-algebras and von Neumann algebras
• Applications in non-commutative Lp spaces and operator space theory