Approximate identity

Similarly, a left approximate identity in a Banach algebra A is a netThe net of all positive elements in A of norm ≤ 1 with its natural order is an approximate identity for any C*-algebra.For example, for compact operators acting on a Hilbert space, the net consisting of finite rank projections would be another approximate identity.In general, a C*-algebra A is σ-unital if and only if A contains a strictly positive element, i.e. there exists h in A+ such that the hereditary C*-subalgebra generated by h is A.One sometimes considers approximate identities consisting of specific types of elements.For example, a C*-algebra has real rank zero if and only if every hereditary C*-subalgebra has an approximate identity consisting of projections.For example, the Fejér kernels of Fourier series theory give rise to an approximate identity.
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