Kaplansky density theorem

Last updated July 19, 2024

In the theory of von Neumann algebras, the Kaplansky density theorem, due to Irving Kaplansky, is a fundamental approximation theorem. The importance and ubiquity of this technical tool led Gert Pedersen to comment in one of his books^[1] that,

Formal statement

Let K⁻ denote the strong-operator closure of a set K in B(H), the set of bounded operators on the Hilbert space H, and let (K)₁ denote the intersection of K with the unit ball of B(H).

Kaplansky density theorem.^[2] If

A

is a self-adjoint algebra of operators in

B(H)

, then each element

a

in the unit ball of the strong-operator closure of

A

is in the strong-operator closure of the unit ball of

A

. In other words,

(A)_{1}^{-}=(A^{-})_{1}

. If

h

is a self-adjoint operator in

(A^{-})_{1}

, then

h

is in the strong-operator closure of the set of self-adjoint operators in

(A)_{1}

.

The Kaplansky density theorem can be used to formulate some approximations with respect to the strong operator topology.

1) If h is a positive operator in (A⁻)₁, then h is in the strong-operator closure of the set of self-adjoint operators in (A⁺)₁, where A⁺ denotes the set of positive operators in A.

2) If A is a C*-algebra acting on the Hilbert space H and u is a unitary operator in A⁻, then u is in the strong-operator closure of the set of unitary operators in A.

In the density theorem and 1) above, the results also hold if one considers a ball of radius r > 0, instead of the unit ball.

Proof

The standard proof uses the fact that a bounded continuous real-valued function f is strong-operator continuous. In other words, for a net {a_α} of self-adjoint operators in A, the continuous functional calculus a→f(a) satisfies,

\lim f(a_{\alpha })=f(\lim a_{\alpha })

in the strong operator topology. This shows that self-adjoint part of the unit ball in A⁻ can be approximated strongly by self-adjoint elements in A. A matrix computation in M₂(A) considering the self-adjoint operator with entries 0 on the diagonal and a and a^* at the other positions, then removes the self-adjointness restriction and proves the theorem.

Notes

↑ Pg. 25; Pedersen, G. K., C*-algebras and their automorphism groups, London Mathematical Society Monographs, ISBN 978-0125494502.
↑ Theorem 5.3.5; Richard Kadison, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. ISBN 978-0821808191.

Related Research Articles

In mathematics, specifically in functional analysis, a C^∗-algebra is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:

In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.

In functional analysis, a discipline within mathematics, given a C*-algebra A, the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic *-representations of A and certain linear functionals on A. The correspondence is shown by an explicit construction of the *-representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal.

In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint.

In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both mixed states and pure states. Density matrices in turn generalize state vectors, which only represent pure states. For M an operator system in a C*-algebra A with identity, the set of all states ofM, sometimes denoted by S(M), is convex, weak-* closed in the Banach dual space M^*. Thus the set of all states of M with the weak-* topology forms a compact Hausdorff space, known as the state space of M.

In the mathematical field of functional analysis there are several standard topologies which are given to the algebra $B(X)$ of bounded linear operators on a Banach space $X$ .

In mathematics, particularly in operator theory and C*-algebra theory, the continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra.

In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus, which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s² to T yields the operator T². Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator $-Δ$ or the exponential

In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space $and one-parameter families$

The spectrum of a linear operator $that operates on a Banach space is a fundamental concept of functional analysis. The spectrum consists of all scalars such that the operator does not have a bounded inverse on . The spectrum has a standard decomposition into three parts:$

In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.

In the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach.

Hilbert C*-modules are mathematical objects that generalise the notion of Hilbert spaces, in that they endow a linear space with an "inner product" that takes values in a C*-algebra. Hilbert C*-modules were first introduced in the work of Irving Kaplansky in 1953, which developed the theory for commutative, unital algebras. In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke and Marc Rieffel, the latter in a paper that used Hilbert C*-modules to construct a theory of induced representations of C*-algebras. Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory, and provide the right framework to extend the notion of Morita equivalence to C*-algebras. They can be viewed as the generalization of vector bundles to noncommutative C*-algebras and as such play an important role in noncommutative geometry, notably in C*-algebraic quantum group theory, and groupoid C*-algebras.

In mathematics, Hilbert spaces allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space. A Hilbert space is a special case of a Banach space.

In noncommutative geometry and related branches of mathematics and mathematical physics, a spectral triple is a set of data which encodes a geometric phenomenon in an analytic way. The definition typically involves a Hilbert space, an algebra of operators on it and an unbounded self-adjoint operator, endowed with supplemental structures. It was conceived by Alain Connes who was motivated by the Atiyah-Singer index theorem and sought its extension to 'noncommutative' spaces. Some authors refer to this notion as unbounded K-cycles or as unbounded Fredholm modules.

In mathematics, Jordan operator algebras are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are real numbers, the algebras are called Jordan Banach algebras. The theory has been extensively developed only for the subclass of JB algebras. The axioms for these algebras were devised by Alfsen, Shultz & Størmer (1978). Those that can be realised concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product and the operator norm are called JC algebras. The axioms for complex Jordan operator algebras, first suggested by Irving Kaplansky in 1976, require an involution and are called JB* algebras or Jordan C* algebras. By analogy with the abstract characterisation of von Neumann algebras as C* algebras for which the underlying Banach space is the dual of another, there is a corresponding definition of JBW algebras. Those that can be realised using ultraweakly closed Jordan algebras of self-adjoint operators with the operator Jordan product are called JW algebras. The JBW algebras with trivial center, so-called JBW factors, are classified in terms of von Neumann factors: apart from the exceptional 27 dimensional Albert algebra and the spin factors, all other JBW factors are isomorphic either to the self-adjoint part of a von Neumann factor or to its fixed point algebra under a period two *-anti-automorphism. Jordan operator algebras have been applied in quantum mechanics and in complex geometry, where Koecher's description of bounded symmetric domains using Jordan algebras has been extended to infinite dimensions.

In mathematics, Kadison transitivity theorem is a result in the theory of C*-algebras that, in effect, asserts the equivalence of the notions of topological irreducibility and algebraic irreducibility of representations of C*-algebras. It implies that, for irreducible representations of C*-algebras, the only non-zero linear invariant subspace is the whole space.

In the theory of C*-algebras, the universal representation of a C*-algebra is a faithful representation which is the direct sum of the GNS representations corresponding to the states of the C*-algebra. The various properties of the universal representation are used to obtain information about the ideals and quotients of the C*-algebra. The close relationship between an arbitrary representation of a C*-algebra and its universal representation can be exploited to obtain several criteria for determining whether a linear functional on the algebra is ultraweakly continuous. The method of using the properties of the universal representation as a tool to prove results about the C*-algebra and its representations is commonly referred to as universal representation techniques in the literature.

This is a glossary for the terminology in a mathematical field of functional analysis.

References

Kadison, Richard, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. ISBN 978-0821808191.
V.F.R.Jones von Neumann algebras; incomplete notes from a course.
M. Takesaki Theory of Operator Algebras I ISBN 3-540-42248-X

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Pg. 25; Pedersen, G. K., C*-algebras and their automorphism groups, London Mathematical Society Monographs, ISBN 978-0125494502.

[2] Theorem 5.3.5; Richard Kadison, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. ISBN 978-0821808191.

[1]

[2]