Stable isomorphism and strong Morita equivalence ofC^∗-algebras

A CHARACTERIZATION OF SATURATED C*-ALGEBRAIC BUNDLES OVER FINITE GROUPS

“…We define an equivalence relation in N as follows: for all (G, β), (H, γ ) ∈ N we say that (G, β) is equivalent to (H, γ ), we write (G, β) ∼ (H, γ ), if there exists an isomorphism λ of G onto H satisfying the condition that (G, β) is exterior equivalent to (G, γ λ(·) ). We [3] A characterization of saturated C * -algebraic bundles 365 denote by [G, β] the equivalence class of (G, β) in N and denote by N /∼ the set of all equivalence classes of (G, β) in N . For each C * -algebra C, let M(C) be its multiplier algebra.…”

An Introduction to Noncommutative Spaces and their Geometries

“…Indeed, there is a complete classification of all separable 16 C * -algebras with a finite dual [6]. Given any finite T 0 -space P , it is possible to construct a C * -algebra A(P, d) of operators on a separable 17 Hilbert space H(P, d) which satisfies A(P, d) = P . Here d is a function on P with values in IN ∪ ∞ which is called defector.…”

“…16 Recall that a C * -algebra A is called separable if it admits a countable subset which is dense in the norm topology of A. 17 . .…”

“…Then, by using the projection (6.35) we get a projection 16) and an inner product on (II E ⊗ π)(E ⊗ A Ω p A) by 17) which is the analogue of the inner product (6.75). The corresponding orthogonal projector P has a range which can be identified with E ⊗ A Ω p D A.…”

“…We finish by mentioning that if A and B are two separable C * -algebras and K is the C * -algebra of compact operators on an infinite dimensional separable Hilbert space, then one proves [17] that the algebras A and B are strongly Morita equivalent if and only if A ⊗ K is isomorphic to B ⊗ K.…”