In mathematics, the real rank of a C*-algebra is a noncommutative analogue of Lebesgue covering dimension. The notion was first introduced by Lawrence G. Brown and Gert K. Pedersen. [1]
The real rank of a unital C*-algebra A is the smallest non-negative integer n, denoted RR(A), such that for every (n + 1)-tuple (x0, x1, ... ,xn) of self-adjoint elements of A and every ε > 0, there exists an (n + 1)-tuple (y0, y1, ... ,yn) of self-adjoint elements of A such that is invertible and . If no such integer exists, then the real rank of A is infinite. The real rank of a non-unital C*-algebra is defined to be the real rank of its unitalization.
If X is a locally compact Hausdorff space, then RR(C0(X)) = dim(X), where dim is the Lebesgue covering dimension of X. As a result, real rank is considered a noncommutative generalization of dimension, but real rank can be rather different when compared to dimension. For example, most noncommutative tori have real rank zero, despite being a noncommutative version of the two-dimensional torus. For locally compact Hausdorff spaces, being zero-dimensional is equivalent to being totally disconnected. The analogous relationship fails for C*-algebras; while AF-algebras have real rank zero, the converse is false. Formulas that hold for dimension may not generalize for real rank. For example, Brown and Pedersen conjectured that RR(A⊗B) ≤ RR(A) + RR(B), since it is true that dim(X × Y) ≤ dim(X) + dim(Y). They proved a special case that if A is AF and B has real rank zero, then A ⊗ B has real rank zero. But in general their conjecture is false, there are C*-algebras A and B with real rank zero such that A ⊗ B has real rank greater than zero. [2]
C*-algebras with real rank zero are of particular interest. By definition, a unital C*-algebra has real rank zero if and only if the invertible self-adjoint elements of A are dense in the self-adjoint elements of A. This condition is equivalent to the previously studied conditions:
This equivalence can be used to give many examples of C*-algebras with real rank zero including AW*-algebras, Bunce–Deddens algebras, [3] and von Neumann algebras. More broadly, simple unital purely infinite C*-algebras have real rank zero including the Cuntz algebras and Cuntz–Krieger algebras. Since simple graph C*-algebras are either AF or purely infinite, every simple graph C*-algebra has real rank zero.
Having real rank zero is a property closed under taking direct limits, hereditary C*-subalgebras, and strong Morita equivalence. In particular, if A has real rank zero, then Mn(A), the algebra of n × n matrices over A, has real rank zero for any integer n ≥ 1.
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, a Lie algebra is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. Otherwise said, a Lie algebra is an algebra over a field where the multiplication operation is now called Lie bracket and has two additional properties: it is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors and is denoted . The Lie bracket does not need to be associative, meaning that the Lie algebra can be non associative. Given an associative algebra, a Lie bracket can be and is often defined through the commutator, namely defining correctly defines a Lie bracket in addition to the already existing multiplication operation.
In mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of S is variously denoted as P(S), 𝒫(S), P(S), , , or 2S. The notation 2S, meaning the set of all functions from S to a given set of two elements (e.g., {0, 1}), is used because the powerset of S can be identified with, equivalent to, or bijective to the set of all the functions from S to the given two elements set.
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebraA which is simple, and for which the center is exactly K.
In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7. The designation E7 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, and G2. The E7 algebra is thus one of the five exceptional cases.
In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms:
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra of a Lie algebra that is self-normalising. They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra over a field of characteristic .
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras..
In mathematics, the Cuntz algebra, named after Joachim Cuntz, is the universal C*-algebra generated by isometries of an infinite-dimensional Hilbert space satisfying certain relations. These algebras were introduced as the first concrete examples of a separable infinite simple C*-algebra, meaning as a Hilbert space, is isometric to the sequence space
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × A → A which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (ab)(cd), (a(bc))d and a(b(cd)) may all yield different answers.
In mathematics, a hereditary C*-subalgebra of a C*-algebra is a particular type of C*-subalgebra whose structure is closely related to that of the larger C*-algebra. A C*-subalgebra B of A is a hereditary C*-subalgebra if for all a ∈ A and b ∈ B such that 0 ≤ a ≤ b, we have a ∈ B.
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.
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, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras.
In mathematics, a regular element of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible. For example, in a complex semisimple Lie algebra, an element is regular if its centralizer in has dimension equal to the rank of , which in turn equals the dimension of some Cartan subalgebra . An element a Lie group is regular if its centralizer has dimension equal to the rank of .
In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares in finite-dimensional real Euclidean Jordan algebras, originally studied and classified by Jordan, von Neumann & Wigner (1934). The tube domain associated with a symmetric cone is a noncompact Hermitian symmetric space of tube type. All the algebraic and geometric structures associated with the symmetric space can be expressed naturally in terms of the Jordan algebra. The other irreducible Hermitian symmetric spaces of noncompact type correspond to Siegel domains of the second kind. These can be described in terms of more complicated structures called Jordan triple systems, which generalize Jordan algebras without identity.
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, Schultz & 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, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorphic to each other have Lie algebras that are isomorphic to each other, but the converse is not necessarily true. One obvious counterexample is and which are non-isomorphic to each other as Lie groups but their Lie algebras are isomorphic to each other. However, by restricting our attention to the simply connected Lie groups, the Lie group-Lie algebra correspondence will be one-to-one.
