# Central carrier
**Central carrier** (also called **central support** or **central cover**) is the smallest projection lying in the center of a von Neumann algebra that dominates a given projection or, more generally, a given operator. If \( M \) is a von Neumann algebra with center \( Z(M) = M \cap M' \), and \( p \in M \) is a projection, the central carrier of \( p \), written \( c_M(p) \), \( z(p) \), or \( C_p \) depending on the author, is the least projection \( z \in Z(M) \) with \( p \le z \).[1](https://knowlpedia.com/operator-algebras/central-support/) • [2](https://www.opuscula.agh.edu.pl/vol35/2/art/opuscula_math_3517.pdf) The notion measures how far a projection reaches across the central summands of the algebra, and it underlies the comparison of projections, the ideal theory of von Neumann algebras, and the Murray–von Neumann type classification.[3](https://www.collectionscanada.gc.ca/obj/thesescanada/vol2/OWTU/TC-OWTU-3920.pdf)
| Key fact | Detail |
|---|---|
| Definition | \( c_M(p) = \bigwedge\{z \in Z(M) : z \text{ is a projection and } p \le z\} \), the least central projection dominating \( p \)[1](https://knowlpedia.com/operator-algebras/central-support/) |
| Existence | It exists because \( Z(M) \) is itself a von Neumann algebra, so its projections form a complete lattice[4](https://ndl.ethernet.edu.et/bitstream/123456789/77231/1/23.pdf) • [3](https://www.collectionscanada.gc.ca/obj/thesescanada/vol2/OWTU/TC-OWTU-3920.pdf) |
| Unitary description | \( c_M(p) = \bigvee_{u \in \mathcal{U}(M)} u p u^* \), the join of all unitary conjugates of \( p \)[5](https://www.math.ucla.edu/~popa/Books/IIunV15.pdf) |
| Spatial description | \( c_M(p) \) is the orthogonal projection onto the closure of \( \overline{\mathrm{span}}(Mp\mathcal{H}) \)[5](https://www.math.ucla.edu/~popa/Books/IIunV15.pdf) |
| For an operator \( T \) | The central support \( C_T \) is the smallest central projection \( P \) with \( \overline{T(\mathcal{H})} \subseteq P(\mathcal{H}) \), and satisfies \( C_T T = T C_T = T \)[6](https://web.math.ku.dk/~musat/Injective_and_semidiscrete_von_Neumann_algebras_final.pdf) |
| Full support | \( c_M(p) = 1 \) means no nonzero central summand of \( M \) is orthogonal to \( p \); \( p \) then has full central support[1](https://knowlpedia.com/operator-algebras/central-support/) |
| In a factor | \( Z(M) = \mathbb{C} \cdot 1 \), so the only central projections are 0 and 1, and every nonzero projection has central carrier 1[3](https://www.collectionscanada.gc.ca/obj/thesescanada/vol2/OWTU/TC-OWTU-3920.pdf) |
## Definition and basic properties
A projection in a von Neumann algebra \( M \) is a self-adjoint idempotent, and a projection in the center \( Z(M) \), meaning it commutes with every element of \( M \), is called a central projection.[7](https://www.sciencedirect.com/topics/mathematics/von-neumann-algebra) Because \( Z(M) = M \cap M' \) is itself a von Neumann algebra by the Double Commutant Theorem, its projections form a complete lattice: arbitrary suprema and infima of central projections exist.[3](https://www.collectionscanada.gc.ca/obj/thesescanada/vol2/OWTU/TC-OWTU-3920.pdf) The central carrier of \( p \) is the infimum of all central projections above \( p \),[1](https://knowlpedia.com/operator-algebras/central-support/) and it is equivalently the smallest central projection \( z \) with \( zp = p \).[5](https://www.math.ucla.edu/~popa/Books/IIunV15.pdf)
The definition depends on the ambient algebra, which is why the subscript \( M \) is recorded: the same projection \( p \) can have different central carriers in different von Neumann algebras containing it.[1](https://knowlpedia.com/operator-algebras/central-support/)
**Equivalent descriptions.** Three characterizations are used interchangeably. First, \( c_M(p) \) is the join of the unitary conjugates of \( p \), \( c_M(p) = \bigvee_{u \in \mathcal{U}(M)} u p u^* \); a central projection \( z \) dominating \( p \) satisfies \( z = u^* z u \ge u^* p u \) for every unitary \( u \), so it dominates all conjugates, and the join of the conjugates is itself central.[5](https://www.math.ucla.edu/~popa/Books/IIunV15.pdf) • [3](https://www.collectionscanada.gc.ca/obj/thesescanada/vol2/OWTU/TC-OWTU-3920.pdf) It follows immediately that \( p \) and any unitary rotation \( u p u^* \) have the same central carrier.[8](https://www.researchgate.net/publication/330227463_Contextuality_and_Noncommutative_Geometry_in_Quantum_Mechanics) Second, when \( M \) acts on a [Hilbert space](https://www.edgepedia.org/hilbert-space) \( \mathcal{H} \), \( c_M(p) \) is the orthogonal projection onto the norm closure of \( (Mp)\mathcal{H} \), the closed linear span of \( \{T \xi : T \in M, \xi \in p\mathcal{H}\} \); this subspace is invariant under both \( M \) and \( M' \), so the projection onto it lies in \( Z(M) \).[5](https://www.math.ucla.edu/~popa/Books/IIunV15.pdf) • [3](https://www.collectionscanada.gc.ca/obj/thesescanada/vol2/OWTU/TC-OWTU-3920.pdf) Third, the ultraweakly closed two-sided ideal generated by \( p \) is \( M c_M(p) \).[1](https://knowlpedia.com/operator-algebras/central-support/)
## Central support of an arbitrary operator
The notion extends from projections to any operator \( T \in M \). The left and right supports of \( T \), the smallest projections \( S_l(T) \) and \( S_r(T) \) satisfying \( T = S_l(T) T = T S_r(T) \), belong to \( M \), and the central support \( C_T \) is the smallest central projection majorizing both.[6](https://web.math.ku.dk/~musat/Injective_and_semidiscrete_von_Neumann_algebras_final.pdf) Equivalently, \( C_T \) is the smallest central projection \( P \) of \( M \) such that \( \overline{T(\mathcal{H})} \subseteq P(\mathcal{H}) \).[6](https://web.math.ku.dk/~musat/Injective_and_semidiscrete_von_Neumann_algebras_final.pdf)
Two properties follow directly. The support intertwines the operator, \( C_T T = T C_T = T \), and \( C_T = C_{T^*} \).[6](https://web.math.ku.dk/~musat/Injective_and_semidiscrete_von_Neumann_algebras_final.pdf) Moreover, if \( Q \) is a central projection with \( Q T = 0 \), then \( Q \) and \( C_T \) are orthogonal: \( Q C_T = 0 \).[6](https://web.math.ku.dk/~musat/Injective_and_semidiscrete_von_Neumann_algebras_final.pdf) In this sense \( C_T \) records exactly which central summands of \( M \) the operator \( T \) actually lives on. For a self-adjoint operator \( S \), the dual notion \( \underline{S} = \sup\{W \in Z(M) : W = W^*, W \le S\} \) is called the core, and a projection \( P \) is core-free when \( \underline{P} = 0 \), which holds exactly when \( \overline{I - P} = I \).[9](https://www.mdpi.com/2075-1680/15/9/688)
## Role in comparing projections
The central carrier is the bookkeeping device for the Murray–von Neumann equivalence of projections, where \( p \sim q \) means some partial isometry \( V \in M \) satisfies \( V^* V = p \) and \( V V^* = q \).[2](https://www.opuscula.agh.edu.pl/vol35/2/art/opuscula_math_3517.pdf)
Two projections \( p \) and \( q \) have nonzero equivalent subprojections if and only if \( z(p) z(q) \ne 0 \); equivalently, \( p \) and \( q \) are centrally orthogonal, meaning \( z(p) z(q) = 0 \), exactly when \( p M q = \{0\} \) and no nonzero subprojections of \( p \) and \( q \) are equivalent.[4](https://ndl.ethernet.edu.et/bitstream/123456789/77231/1/23.pdf) • [10](https://www.ipam.ucla.edu/wp-content/uploads/2024/07/W_-Notes-GOALS.pdf) Equivalence preserves the central carrier, \( p \sim q \) implies \( z(p) = z(q) \), and for abelian projections the converse holds: two abelian projections with the same central support are equivalent.[5](https://www.math.ucla.edu/~popa/Books/IIunV15.pdf)
The comparison theorem reduces arbitrary pairs of projections to comparable ones using a central cut: for any projections \( P, Q \in M \) there exists a central projection \( Z \) such that \( Q Z \preccurlyeq P Z \) and \( P(I - Z) \preccurlyeq Q(I - Z) \), where \( \preccurlyeq \) denotes Murray–von Neumann subequivalence.[2](https://www.opuscula.agh.edu.pl/vol35/2/art/opuscula_math_3517.pdf) In a factor the center is trivial, so \( Z \) is 0 or 1 and any two projections are comparable.[5](https://www.math.ucla.edu/~popa/Books/IIunV15.pdf)
## Ideals and central decomposition
Central projections decompose a von Neumann algebra into direct summands: a central projection \( c \) splits \( M = cM \oplus (1-c)M \).[11](https://noncommutativeanalysis.wordpress.com/2017/04/04/introduction-to-von-neumann-algebras-lecture-3-some-more-generalities-projection-constructions-abelian-von-neumann-algebras/) Every ultraweakly closed two-sided ideal \( I \subseteq M \) arises this way: there is a central projection \( p \) with \( I = pM \), and consequently a surjective normal \( * \)-homomorphism \( \varphi : M \to N \) gives \( M \cong \ker \varphi \oplus N \).[12](https://web.math.ku.dk/~rordam/students/Speciale-Martin.pdf) In a factor, where \( Z(M) = \mathbb{C} \cdot 1 \), this forces \( p = 0 \) or \( p = 1 \), so a factor has no nontrivial \( \sigma \)-weakly closed two-sided ideals.[3](https://www.collectionscanada.gc.ca/obj/thesescanada/vol2/OWTU/TC-OWTU-3920.pdf)
The central carrier also appears in the type classification. A von Neumann algebra is of type I if it has an abelian projection (one with \( pMp \) abelian) with central carrier \( \mathrm{Id} \), of type II if it has no nonzero abelian projections but has a finite projection with central carrier \( \mathrm{Id} \), and of type III if it is purely infinite; every von Neumann algebra is uniquely a direct sum of algebras of types I, \( \mathrm{II}_1 \), \( \mathrm{II}_\infty \), and III.[13](https://ediss.uni-goettingen.de/bitstream/handle/11858/00-1735-0000-000D-F16D-2/knebusch.pdf?sequence=1) The standard proof of this decomposition builds the type II summand as the central cover of a maximal family of centrally orthogonal finite projections.[14](https://noncommutativeanalysis.wordpress.com/2017/05/08/introduction-to-von-neumann-algebras-lecture-5-comparison-of-projections-and-classification-into-types-of-von-neumann-algebras/)
## Uses
Beyond the structure theory, the central carrier serves as a test in several settings. A projection \( e \) with central support 1 is full, and such projections control whether a \( * \)-homomorphism \( b \mapsto bp \) from the commutant is injective: if \( C_p = 1 \), the map \( b \mapsto bp = pb \) is a \( * \)-isomorphism from \( M' \) onto \( M' p \).[14](https://noncommutativeanalysis.wordpress.com/2017/05/08/introduction-to-von-neumann-algebras-lecture-5-comparison-of-projections-and-classification-into-types-of-von-neumann-algebras/) In approximation questions, if the center of a finite von Neumann algebra is finite dimensional, the central support \( z(p) \) can be written as a finite sum of subprojections \( p_i \preccurlyeq p \).[15](https://mathoverflow.net/questions/223662/approximation-of-the-central-support) Within a von Neumann algebra, a point is full-derivable exactly when its central carrier equals the unit.[16](https://arxiv.org/abs/1611.01633)
## References
1. [Central support of a projection, Knowlpedia](https://knowlpedia.com/operator-algebras/central-support/)
2. [Steering projections in von Neumann algebras, Opuscula Math. 35 (2015)](https://www.opuscula.agh.edu.pl/vol35/2/art/opuscula_math_3517.pdf)
3. [Von Neumann Algebras for Abstract (thesis, Library and Archives Canada)](https://www.collectionscanada.gc.ca/obj/thesescanada/vol2/OWTU/TC-OWTU-3920.pdf)
4. [III.1 Projections and Type Classification (Takesaki, chapter scan)](https://ndl.ethernet.edu.et/bitstream/123456789/77231/1/23.pdf)
5. [An introduction to II1 factors, S. Popa, UCLA](https://www.math.ucla.edu/~popa/Books/IIunV15.pdf)
6. [Injective and semidiscrete von Neumann algebras, University of Copenhagen notes](https://web.math.ku.dk/~musat/Injective_and_semidiscrete_von_Neumann_algebras_final.pdf)
7. [Von Neumann Algebra, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/von-neumann-algebra)
8. [Contextuality and Noncommutative Geometry in Quantum Mechanics, ResearchGate](https://www.researchgate.net/publication/330227463_Contextuality_and_Noncommutative_Geometry_in_Quantum_Mechanics)
9. [Higher Lie n-Centralizers and Generalized Higher Lie n-Derivations of von Neumann Algebras, MDPI Axioms 15 (2025)](https://www.mdpi.com/2075-1680/15/9/688)
10. [Von Neumann Algebras, GOALS notes, IPAM UCLA](https://www.ipam.ucla.edu/wp-content/uploads/2024/07/W_-Notes-GOALS.pdf)
11. [Introduction to von Neumann algebras, Lectures 3 and 5, Noncommutative Analysis blog](https://noncommutativeanalysis.wordpress.com/2017/04/04/introduction-to-von-neumann-algebras-lecture-3-some-more-generalities-projection-constructions-abelian-von-neumann-algebras/)
12. [Von Neumann Algebras: General overview, University of Copenhagen thesis notes](https://web.math.ku.dk/~rordam/students/Speciale-Martin.pdf)
13. [Approximation of (thesis, University of Göttingen)](https://ediss.uni-goettingen.de/bitstream/handle/11858/00-1735-0000-000D-F16D-2/knebusch.pdf?sequence=1)
14. [introduction to von neumann algebras, lecture 5 (comparison of projections and classification into types of von neumann algebras) | noncommutative analysis](https://noncommutativeanalysis.wordpress.com/2017/05/08/introduction-to-von-neumann-algebras-lecture-5-comparison-of-projections-and-classification-into-types-of-von-neumann-algebras/)
15. [Approximation of the central support, MathOverflow](https://mathoverflow.net/questions/223662/approximation-of-the-central-support)
16. [Characterizations of centralizers and derivations on some algebras, arXiv:1611.01633](https://arxiv.org/abs/1611.01633)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Operator algebras › Von Neumann algebras › Projection lattice and comparison theory*
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