Graded C*-algebra - Imported Knowlpedia Content

Definition

A graded $C^*$ -algebra is a $C^*$ -algebra $A$ with closed linear subspaces $A^0,A^1$ such that

A=A^0\oplus A^1,\qquad A^iA^j\subseteq A^{i+j},\qquad (A^i)^*\subseteq A^i,

where indices are taken modulo $2$ . Elements of $A^0$ are even and elements of $A^1$ are odd; either kind is homogeneous, of degree $0$ or $1$ . A homomorphism of graded $C^*$ -algebras is graded when it preserves degrees. The trivial grading is $A^0=A$ , $A^1=0$ .

Grading automorphism

Equivalently, a grading is a $*$ -automorphism $\gamma:A\to A$ satisfying $\gamma^2=\operatorname{id}_A$ . Its $+1$ -eigenspace is $A^0$ , its $-1$ -eigenspace is $A^1$ , and

a^0=\frac{a+\gamma(a)}2,\qquad a^1=\frac{a-\gamma(a)}2

recover the homogeneous parts of $a$ . This formulation also shows that both summands are closed and that the decomposition is unique.

Graded signs and tensor products

For homogeneous elements, the graded commutator is $[a,b]_{\mathrm{gr}}=ab-(-1)^{|a||b|}ba$ . The algebraic graded tensor product uses

(a\widehat\otimes b)(a'\widehat\otimes b') =(-1)^{|b||a'|}aa'\widehat\otimes bb',

with a corresponding sign in the involution. These signs are structural: forgetting them produces the ordinary tensor product, not the graded one Blackadar, §14.4.

Examples and scope

Every $C^*$ -algebra has the trivial grading. A bounded-operator algebra on a graded Hilbert space is graded by conjugation with the grading operator; its even operators preserve the two summands and its odd operators interchange them. Gradings by larger groups require additional homogeneous components and are not meant by “graded” here unless explicitly stated.

References

Bruce Blackadar, K-Theory for Operator Algebras, 2nd ed., Cambridge University Press, 1998. Publisher DOI record. Relevant: §14.4 on graded $C^*$ -algebras and graded tensor products.
Nigel Higson and John Roe, Analytic K-Homology, Oxford University Press, 2000. Publisher DOI record. Relevant: Chapter 8 on graded algebras, modules, and operators.