Schröder–Bernstein theorems for operator algebras

Schröder–Bernstein theorems for operator algebras The Schröder–Bernstein theorem from set theory has analogs in the context operator algebras. This article discusses such operator-algebraic results.

Contents 1 For von Neumann algebras 2 Representations of C*-algebras 3 See also 4 References For von Neumann algebras Suppose M is a von Neumann algebra and E, F are projections in M. Let ~ denote the Murray-von Neumann equivalence relation on M. Define a partial order « on the family of projections by E « F if E ~ F' ≤ F. In other words, E « F if there exists a partial isometry U ∈ M such that U*U = E and UU* ≤ F.

For closed subspaces M and N where projections PM and PN, onto M and N respectively, are elements of M, M « N if PM « PN.

The Schröder–Bernstein theorem states that if M « N and N « M, then M ~ N.

A proof, one that is similar to a set-theoretic argument, can be sketched as follows. Colloquially, N « M means that N can be isometrically embedded in M. So {displaystyle M=M_{0}supset N_{0}} where N0 is an isometric copy of N in M. By assumption, it is also true that, N, therefore N0, contains an isometric copy M1 of M. Therefore, one can write {displaystyle M=M_{0}supset N_{0}supset M_{1}.} By induction, {displaystyle M=M_{0}supset N_{0}supset M_{1}supset N_{1}supset M_{2}supset N_{2}supset cdots .} It is clear that {displaystyle R=cap _{igeq 0}M_{i}=cap _{igeq 0}N_{i}.} Let {displaystyle Mominus N{stackrel {mathrm {def} }{=}}Mcap (N)^{perp }.} So {displaystyle M=oplus _{igeq 0}(M_{i}ominus N_{i})quad oplus quad oplus _{jgeq 0}(N_{j}ominus M_{j+1})quad oplus R} and {displaystyle N_{0}=oplus _{igeq 1}(M_{i}ominus N_{i})quad oplus quad oplus _{jgeq 0}(N_{j}ominus M_{j+1})quad oplus R.} Notice {displaystyle M_{i}ominus N_{i}sim Mominus Nquad {mbox{for all}}quad i.} The theorem now follows from the countable additivity of ~.

Representations of C*-algebras There is also an analog of Schröder–Bernstein for representations of C*-algebras. If A is a C*-algebra, a representation of A is a *-homomorphism φ from A into L(H), the bounded operators on some Hilbert space H.

If there exists a projection P in L(H) where P φ(a) = φ(a) P for every a in A, then a subrepresentation σ of φ can be defined in a natural way: σ(a) is φ(a) restricted to the range of P. So φ then can be expressed as a direct sum of two subrepresentations φ = φ' ⊕ σ.

Two representations φ1 and φ2, on H1 and H2 respectively, are said to be unitarily equivalent if there exists a unitary operator U: H2 → H1 such that φ1(a)U = Uφ2(a), for every a.

In this setting, the Schröder–Bernstein theorem reads: If two representations ρ and σ, on Hilbert spaces H and G respectively, are each unitarily equivalent to a subrepresentation of the other, then they are unitarily equivalent.

A proof that resembles the previous argument can be outlined. The assumption implies that there exist surjective partial isometries from H to G and from G to H. Fix two such partial isometries for the argument. One has {displaystyle rho =rho _{1}simeq rho _{1}'oplus sigma _{1}quad {mbox{where}}quad sigma _{1}simeq sigma .} In turn, {displaystyle rho _{1}simeq rho _{1}'oplus (sigma _{1}'oplus rho _{2})quad {mbox{where}}quad rho _{2}simeq rho .} By induction, {displaystyle rho _{1}simeq rho _{1}'oplus sigma _{1}'oplus rho _{2}'oplus sigma _{2}'cdots simeq (oplus _{igeq 1}rho _{i}')oplus (oplus _{igeq 1}sigma _{i}'),} and {displaystyle sigma _{1}simeq sigma _{1}'oplus rho _{2}'oplus sigma _{2}'cdots simeq (oplus _{igeq 2}rho _{i}')oplus (oplus _{igeq 1}sigma _{i}').} Now each additional summand in the direct sum expression is obtained using one of the two fixed partial isometries, so {displaystyle rho _{i}'simeq rho _{j}'quad {mbox{and}}quad sigma _{i}'simeq sigma _{j}'quad {mbox{for all}}quad i,j;.} This proves the theorem.

See also Schröder–Bernstein theorem for measurable spaces Schröder–Bernstein property References B. Blackadar, Operator Algebras, Springer, 2006. Categories: Von Neumann algebrasC*-algebras